(b) Write \(\left(64m^{12}\right)^{\frac{2}{3}}\) in the form \(am^b\) where \(a\) and \(b\) are integers. (2)
Mark scheme (a)
Answer
Mark
Mark scheme
2
M1
for a correct start, using one rule of indices, eg \(3^n \times 3^{12} = 3^{14}\) or \(3^n \times 3^{20} = 3^{22}\) or \(3^{20} \div 3^8 = 3^{12}\) or \(3^{14} \times 3^8 = 3^{22}\) or \(\dfrac{3^{n + 20}}{3^8} = 3^{14}\) or \(3^{n - 8} \times 3^{20} = 3^{14}\) or for forming an equation in \(n\), eg \(n + 20 - 8 = 14\) oe or (\(n =\)) \(14 + 8 - 20\)
A1
cao
SCB1 for an answer of \(3^2\) if M0 scored
Additional guidance
Allow an answer of \(3^n = 3^2\)
An answer of 9 or \(3^n = 9\), \(n \neq 2\) on its own is to be awarded 0 marks
Mark scheme (b)
Answer
Mark
Mark scheme
\(16m^8\)
M1
for an intention to find the cube root and square, eg \(\sqrt[3]{64m^{12}}^{\,2}\) or \(\sqrt[3]{\left(64m^{12}\right)^2}\) or \(\left(4m^4\right)^2\) or \(\sqrt[3]{4096m^{24}}\) or for \(am^8\) with \(a \neq 16\) or \(16m^b\) with \(b \neq 8\)
A1
cao
Additional guidance
Do not condone missing brackets
16 or \(m^8\) imply M1 Allow multiplication sign for M1
for \(2^x = 2^{n - \frac{1}{3}}\) or \(2^y = 2^{\frac{5}{2}}\) or \(x = n - \dfrac{1}{3}\) oe or \(y = \dfrac{5}{2}\) oe or for \((2^{x + y} =)\ \dfrac{2^n}{\sqrt[3]{2}} \times \left(\sqrt{2}\right)^5\)
P1
for \(2^{x + y} = 2^{n - \frac{1}{3} + \frac{5}{2}}\) or \(x + y = n - \dfrac{1}{3} + \dfrac{5}{2}\) oe or \(\dfrac{11}{2} = n - \dfrac{1}{3}\) oe
14 Work out the value of \(27^{\frac{2}{3}} + \left(\dfrac{1}{2}\right)^{-3}\) (3)
Mark scheme
Answer
Mark
Mark scheme
17
M1
for \(27^{\frac{2}{3}} = \left(\sqrt[3]{27}\right)^2\) or \(3^2\) or \(\sqrt[3]{27^2}\) or \(\sqrt[3]{729}\) or \(\left(\dfrac{1}{2}\right)^{-3} = \dfrac{1}{\left(\frac{1}{2}\right)^3}\) or \(2^3\)
M1
for a full method eg \(\text{``}3\text{''}^2 + \text{``}2\text{''}^3\)
or for correctly evaluating \(27^{\frac{2}{3}}\) or \(\left(\dfrac{1}{2}\right)^{-3}\) eg 9 or 8 seen
(a) Write \(\dfrac{1}{16}\) in the form \(4^n\) where \(n\) is an integer. (1)
(b) Work out the value of \(8^{\frac{5}{3}} - 9^{\frac{3}{2}}\) (3)
Mark scheme (a)
Answer
Mark
Mark scheme
\(4^{-2}\)
B1
for \(4^{-2}\)
Additional guidance
Accept \(n = -2\)
Mark scheme (b)
Answer
Mark
Mark scheme
5
M1
for \(8^{\frac{5}{3}} = \left(\sqrt[3]{8}\right)^5\) or \(2^5\) or \(\sqrt[3]{8^5}\) or \(\sqrt[3]{32768}\) or \(9^{\frac{3}{2}} = \left(\sqrt{9}\right)^3\) or \(3^3\) or \(\sqrt{9^3}\) or \(\sqrt{729}\)
M1
for correctly evaluating \(8^{\frac{5}{3}}\) or \(9^{\frac{3}{2}}\), eg 32 or 27 seen
11 Write \(\dfrac{(6x^5y^3)^2}{3x^2y^7 \times 4xy^{-3}}\) in the form \(ax^by^c\) where \(a\), \(b\) and \(c\) are integers. (3)
Mark scheme
Answer
Mark
Mark scheme
\(3x^7y^2\)
M1
for full evaluation of numerator or denominator with at least 2 of 3 terms correct in a product, eg \(36x^{10}y^6\) or \(12x^3y^4\)
or full evaluation of \(\dfrac{6x^5y^3}{3x^2y^7}\) or \(\dfrac{6x^5y^3}{4xy^{-3}}\) with at least 2 of 3 terms correct in a product, eg \(2x^3y^{-4}\) or \(1.5x^4y^6\)
M1
for correct evaluation of numerator and denominator,
eg \(36x^{10}y^6\) and \(12x^3y^4\)
or for full evaluation of numerator and denominator with no more than one error and a final answer of the form \(ax^by^c\) with two of \(a\), \(b\) and \(c\) correct or for correct evaluation of \(\dfrac{6x^5y^3}{3x^2y^7}\) and \(\dfrac{6x^5y^3}{4xy^{-3}}\) eg \(2x^3y^{-4}\) and \(1.5x^4y^6\)
or for full evaluation of \(\dfrac{6x^5y^3}{3x^2y^7}\) and \(\dfrac{6x^5y^3}{4xy^{-3}}\) with no more than one error and a final answer of the form \(ax^by^c\) with two of \(a\), \(b\) and \(c\) correct
17 Work out the value of \(\left(\dfrac{8}{27}\right)^{\frac{4}{3}}\) (2)
Mark scheme
Answer
Mark
Mark scheme
\(\dfrac{16}{81}\)
M1
for showing the cube root of 8 = 2 and the cube root of 27 = 3
or an intention to find the cube root and raise to power 4 eg \(\sqrt[3]{\left(\dfrac{8}{27}\right)^4}\) or \(\left(\sqrt[3]{\dfrac{8}{27}}\right)^4\) or \(\left(\dfrac{2}{3}\right)^4\)
18 Work out the value of \(\dfrac{\left(5\dfrac{4}{9}\right)^{-\frac{1}{2}} \times \left(4\dfrac{2}{3}\right)}{2^{-3}}\)
You must show all your working. (4)
Mark scheme
Answer
Mark
Mark scheme
16
M1
for working with square root or with reciprocal in \(\left(5\dfrac{4}{9}\right)^{-\frac{1}{2}}\) eg \(\left(\dfrac{9}{49}\right)^{\frac{1}{2}}\) or \(\dfrac{1}{\sqrt{\dfrac{49}{9}}}\) or \(\dfrac{1}{\left(\dfrac{49}{9}\right)^{\frac{1}{2}}}\) or \(\left(\dfrac{7}{3}\right)^{-1}\) or \(\dfrac{3}{7}\)
M1
for a full method to simplify the numerator eg \(\dfrac{3}{7} \times \dfrac{14}{3}\ (= 2)\)
M1
for showing \(\div\, 2^{-3}\) as \(\times\, 8\), eg \(\dfrac{3}{7} \times \dfrac{14}{3} \times 8\) or for \(2^1 \div 2^{-3}\ (= 2^4)\) or for correctly reducing the expression to a single calculation, eg \(\dfrac{336}{21}\) or \(\dfrac{112}{7}\) or \(2 \div \dfrac{1}{8}\)
A1
cao
Additional guidance
May be seen at any time during the calculation
Award 0 marks for a correct answer with no supportive working
(a) Express \(\sqrt{\dfrac{10^{360}}{10^{150} \times 10^{90}}}\) as a power of 10 (3)
Liam was asked to express \((12^{50})^2\) as a power of 12
Liam wrote \((12^{50})^2 = 12^{50^2} = 12^{2500}\)
Liam’s method is wrong.
(b) Explain why. (1)
Mark scheme (a)
Answer
Mark
Mark scheme
\(10^{60}\)
M1
for a correct first step using one of the rules of indices, eg \(10^{150} \times 10^{90} = 10^{240}\) or \(10^{360} \div 10^{150} = 10^{210}\) or \(10^{360} \div 10^{90} = 10^{270}\) or \(\sqrt{10^{360}} = 10^{180}\) or \(\sqrt{10^{150}} = 10^{75}\) or \(\sqrt{10^{90}} = 10^{45}\)
M1
for correct use of rules of indices leading as far as \(\sqrt{10^{120}}\) or \(\dfrac{10^{180}}{10^{120}}\)
A1
cao
Mark scheme (b)
Answer
Mark
Mark scheme
reason
C1
for correct reasoning
Acceptable examples eg should do \(50 \times 2\) (not \(50^2\)) because \((12^{50})^2 = 12^{100}\) because when you have a power inside and outside the bracket you times them because \((a^b)^c = a^{bc}\) (not \(a^{b^c}\))
Not acceptable examples because you need to multiply everything in the brackets by 2 because he should have squared 12 as well you add the powers instead of timesing
for any two of \(12 \div 6\ (= 2)\), \(x^{7-3}\ (= x^4)\), \(y^{3-1}\ (= y^2)\) in a product or written as a fraction with complete and correct cancelling of at least two terms
19 Given that \(9^{-\frac{1}{2}} = 27^{\frac{1}{4}} \div 3^{x+1}\) find the exact value of \(x\). (3)
Mark scheme
Answer
Mark
Mark scheme
\(\dfrac{3}{4}\) oe
P1
for a first step to converting to a common base with one correct conversion, eg. \(9^{-\frac{1}{2}} = 3^{-1}\) or \(\dfrac{1}{3}\) or \(27^{\frac{1}{4}} = 3^{\frac{3}{4}}\) oe
P1
(dep) for \(3^{-1} = 3^{\frac{3}{4}} \div 3^{x+1}\) oe
A1
cao
Additional guidance
\(9^{-\frac{1}{2}} = 3^{-1}\) (or \(\frac{1}{3}\)) oe or \(27^{\frac{1}{4}} = 3^{\frac{3}{4}}\) oe seen alone gets the P1
for multiplying both sides by 7 as a first step eg \(9 + x = 7(11 - x)\) or dividing each term on the left hand side by 7 eg \(\dfrac{9}{7} + \dfrac{x}{7} = 11 - x\)
M1
(dep M1) for method to isolate the \(x\) terms on one side
A1
oe
Additional guidance
\(\times 7\) written near the equation is not enough for this mark
for any two of \(12 \div 6\ (= 2)\), \(x^{7-3}\ (= x^4)\), \(y^{3-1}\ (= y^2)\) in a single product or written as a fraction with complete and correct cancelling of at least two terms
12 Patrick has to work out the exact value of \(64^{\frac{1}{4}}\)
Patrick says,
“\(\dfrac{1}{4}\) of 64 is 16 so \(64^{\frac{1}{4}} = 16\)”
Explain what is wrong with what Patrick says. (1)
Mark scheme
Answer
Mark
Mark scheme
Explanation
C1
for explanation eg needs to find 4th root or gives the correct answer of 2.828… Acceptable examples: He needs to find \(\sqrt[4]{64}\) It should be 2.8..(or \(2\sqrt{2}\)) It is not asking for \(64 \div 4\), it is asking what number to the power of 4 = 64 \(64^{\frac{1}{4}}\) means the fourth root not a quarter of 64 \(64^{\frac{1}{4}}\) means square root and square root again, not divide by 4 Not acceptable examples: It should be 2 The expression is 64 to the power of \(\frac{1}{4}\) \(64^{\frac{1}{4}}\) is not a \(\frac{1}{4}\) of 64
20 Work out the value of \(\dfrac{3^7 \times 3^{-2}}{3^3}\) (2)
Mark scheme
Answer
Mark
Mark scheme
9
M1
for a correct first step, using the laws of indices to simplify eg \(3^2\) or. \(3^{7 + -2}\) or \(3^{7-3}\) or \(3^{-2-3}\) OR for using exact values, eg. \(2187 \times \dfrac{1}{9}\ (= 243)\) or \(2187 \div 27\ (= 81)\) or \(\dfrac{1}{27 \times 9}\ \left(= \dfrac{1}{243}\right)\)
for showing the 4th root of 16 as 2 and the 4th root of 81 as 3 or \(\dfrac{8}{n}\ (n \neq 27)\) or \(\dfrac{n}{27}\ (n \neq 8)\) or an intention to find the 4th root and cube, eg. \(\sqrt[4]{\left(\dfrac{16}{81}\right)^3}\) or \(\left(\sqrt[4]{\dfrac{16}{81}}\right)^3\) oe
A1
cao
Mark scheme (b)
Answer
Mark
Mark scheme
0
M1
for writing \(\dfrac{1}{9} = 3^{-2}\), \(9\sqrt{3} = 3^{2.5}\), \(\dfrac{1}{\sqrt{3}} = 3^{-0.5}\) as powers of 3, with at least 2 correct or for working out \(\dfrac{1}{9} \times 9\sqrt{3} \times \dfrac{1}{\sqrt{3}} = 1\)
1 Work out the value of \(\dfrac{3^7 \times 3^{-2}}{3^3}\) (2)
Mark scheme
Answer
Mark
Mark scheme
9
M1
for a correct first step, using the laws of indices to simplify eg. \(3^2\) or \(3^{7 + -2}\) or \(3^{7-3}\) or \(3^{-2-3}\) OR for using exact values, eg. \(2187 \times \dfrac{1}{9}\ (= 243)\) or \(2187 \div 27\ (= 81)\) or \(\dfrac{1}{27 \times 9}\ \left(= \dfrac{1}{243}\right)\)
(b) Find the value of \(\left(\dfrac{64}{125}\right)^{\frac{2}{3}}\) (2)
Mark scheme (a)
Answer
Mark
Notes
\(\dfrac{1}{9}\)
M1
for showing a method using either reciprocal or square root e.g. \(\dfrac{1}{n}\) or 9 seen
A1
cao Accept \(\pm\dfrac{1}{9}\) or 0.1 recurring
Mark scheme (b)
Answer
Mark
Notes
\(\dfrac{16}{25}\)
M1
for showing cube root of 64 as 4 and the cube root of 125 as 5 or \(\dfrac{16}{n}\ (n \ne 25)\) or \(\dfrac{n}{25}\ (n \ne 16)\) or an intention to find the cube root and square.
(b) Make \(y\) the subject of the formula \[p = \sqrt{\frac{x + y}{5}}\] (3)
Mark scheme (a)
Answer
Mark
Notes
\(10a^5b^4\)
2
B2 cao (B1 for any two of 10, \(a^5\), \(b^4\) in a product)
Mark scheme (b)
Answer
Mark
Notes
\(y = 5p^2 - x\)
3
M1 for a correct first step to either remove the square root, eg. \(p^2 = \dfrac{x + y}{5}\) or clear the fraction, eg. \(\sqrt{5}\,p = \sqrt{x + y}\) M1 (dep) for a method to remove the square root and a method to clear the fraction, eg. \(5p^2 = x + y\) A1 for \(y = 5p^2 - x\)
(c) Work out the value of \(\left(\sqrt{12} - \sqrt{3}\right)^2\) (2)
Mark scheme (a)
Answer
Mark
Notes
\(\dfrac{1}{8}\)
1
B1 for \(\dfrac{1}{8}\) oe
Mark scheme (b)
Answer
Mark
Notes
1.5
1
B1 for 1.5 oe
Mark scheme (c)
Answer
Mark
Notes
3
2
M1 for \(\sqrt{12} \times \sqrt{12} - \sqrt{12} \times \sqrt{3} - \sqrt{3} \times \sqrt{12} + \sqrt{3} \times \sqrt{3}\) or \(\sqrt{144} - \sqrt{36} - \sqrt{36} + \sqrt{9}\) oe. with no more than one sign error A1 cao
OR M1 for writing \((\sqrt{12} - \sqrt{3})\) as \((2\sqrt{3} - \sqrt{3})\) \((= \sqrt{3})\) A1 cao
(c) Write \(\dfrac{5}{x - 3} - \dfrac{4}{x + 3}\) as a single fraction in its simplest form. (3)
Mark scheme (a)
Answer
Mark
Notes
1
1
B1 cao
Mark scheme (b)
Answer
Mark
Notes
\(\dfrac{5y}{8x^3}\)
2
M1 for correct square root or correct use of reciprocal eg \(\dfrac{8x^3}{5y}\) or \(\dfrac{25y^2}{64x^6}\) A1 for \(\dfrac{5y}{8x^3}\) or \(\dfrac{5}{8}yx^{-3}\) oe
Mark scheme (c)
Answer
Mark
Notes
\(\dfrac{x + 27}{(x - 3)(x + 3)}\)
3
M1 for denominator \((x - 3)(x + 3)\) or \(x^2 - 9\) M1 for \(\dfrac{5(x + 3)}{(x - 3)(x + 3)}\) oe or \(\dfrac{4(x - 3)}{(x - 3)(x + 3)}\) oe (NB The denominator must be \((x - 3)(x + 3)\) or \(x^2 - 9\) or another suitable common denominator) A1 for \(\dfrac{x + 27}{(x - 3)(x + 3)}\) or \(\dfrac{x + 27}{x^2 - 9}\)
19 Write these numbers in order of size. Start with the smallest number. (2)
\(5^{-1}\)
0.5
\(-5\)
\(5^0\)
Mark scheme
Working
Answer
Mark
Notes
\(-5\), 0.2, 0.5, 1
\(-5\), \(5^{-1}\), 0.5, \(5^0\)
2
M1 for either \(5^{-1}\) or \(5^0\) evaluated correctly A1 for a fully correct list from correct working, accept original numbers or evaluated (SC B1 for one error in position or correct list in reverse order)
(a) Work out the value of \(\quad \left(\dfrac{9}{16}\right)^{-\frac{3}{2}}\) [3 marks]
(b) \(\sqrt{125} = 5^n\)
Work out the value of \(n\). [2 marks]
Mark scheme (a)
Answer
Mark
Comments
\(\dfrac{64}{27}\) or \(2\dfrac{10}{27}\)
B3
B2 shows understanding of at least two of • a negative index means reciprocal • an index of \(\dfrac{1}{2}\) means square root • an index of 3 means cube B1 shows understanding of one of the above
Additional guidance
Condone missing brackets
Allow equivalent decimals, fractions or mixed numbers throughout
Ignore any attempt to simplify, change to a mixed number or convert to a decimal after a correct answer is seen eg \(\dfrac{64}{27} = 2\dfrac{20}{27}\)
B3
Understanding can be shown in expressions which are not equivalent to the original eg \(\dfrac{27}{64}\) implies understanding of square root and cubing, but not reciprocal
B2
Example B2 expressions: \(\left(\dfrac{1/\sqrt{9}}{1/\sqrt{16}}\right)^3\) or \(\left(\dfrac{4096}{729}\right)^{\frac{1}{2}}\) or \(\left(\dfrac{27}{64}\right)^{-1}\)
B2
Example B1 expressions: \(\left(\dfrac{16}{9}\right)\) or \(\left(\dfrac{3}{4}\right)^{-3}\) or \(\left(\dfrac{729}{4096}\right)^{-\frac{1}{2}}\)
B1
To show understanding, correct method or value for calculation of a cube is required, but calculation of a square root is not required eg1 \(\left(\dfrac{\sqrt{9}}{\sqrt{16}}\right)^3\) only shows understanding of the square root
B1
eg2 \(\left(\dfrac{\sqrt{9}}{\sqrt{16}}\right) \times \left(\dfrac{\sqrt{9}}{\sqrt{16}}\right) \times \left(\dfrac{\sqrt{9}}{\sqrt{16}}\right)\) shows understanding of the square root and cubing
B2
Mark scheme (b)
Answer
Mark
Comments
1.5 or \(\dfrac{3}{2}\) or \(1\dfrac{1}{2}\)
B2
B1 \(125^{\frac{1}{2}}\) or \(\left(5^3\right)^{\frac{1}{2}}\) or \(\left(5^{\frac{1}{2}}\right)^3\) or \(5^{\frac{3}{2}}\) or \(125 = 5^{2n}\) or \(5^3 = 5^{2n}\) or \(2n = 3\)
(a) Work out the value of \(\quad \left(5\dfrac{1}{16}\right)^{\frac{1}{4}}\) [2 marks]
(b) Write \(\quad \left(49^m\right)^{2.5} \quad\) as a power of 7 in terms of \(m\). [2 marks]
Mark scheme (a)
Answer
Mark
Comments
\(\sqrt[4]{5\dfrac{1}{16}}\) or \(\left(5\dfrac{1}{16} =\right) \dfrac{81}{16}\) or \(\dfrac{81^{\frac{1}{4}}}{16^{\frac{1}{4}}}\) or \(\left(81^{\frac{1}{4}} =\right) 3\) or \(\left(16^{\frac{1}{4}} =\right) 2\)
25 Write \(\quad (3^6 \times 3^5) : 3^7 \quad\) in the form \(\quad n : 1 \quad\) where \(n\) is an integer. [3 marks]
Mark scheme
Answer
Mark
Comments
\(3^{11}\ (: 3^7)\) or \(3^6 : 3^2\) or \(3^5 : 3^{(1)}\) or \(\dfrac{a}{3^7}\) or \(177\,147 : 2187\)
M1
oe eg 729 : 9 or 243 : 3 \(3^n\) may be implied by a multiplication string of \(n\) 3s \(a\) can be any value other than \(3^7\)
\(\dfrac{3^{11}}{3^7}\ (: 1)\) or \(\dfrac{3^6}{3^2}\ (: 1)\) or \(3^6 \times 3^{-2}\ (: 1)\) or \(\dfrac{3^5}{3^{(1)}}\ (: 1)\) or \(3^{-1} \times 3^5\ (: 1)\) or \(3^4\ (: 1)\) or \(\dfrac{177\,147}{2187}\ (: 1)\)
M1dep
oe left-hand side with one or two components eg \(\dfrac{729}{9} : 1\) or \(243 \times \dfrac{1}{3} : 1\) allow (: 1) to be (: \(3^0\)) \(3^n\) may be implied by a multiplication string of \(n\) 3s
81 : 1
A1
Additional guidance
\(\dfrac{3^6 \times 3^5}{3^7}\ (: 1)\) with no further work
M1M0A0
81 : 1 or \(3^4\ (: 1)\) could be seen from incorrect working eg \(\dfrac{9^{11}}{3^7} = 3^4\) Answer 81 : 1
(a) Work out the value of \(\quad \left(\dfrac{5}{4}\right)^{-2}\) [2 marks]
(b) Work out the value of \(\quad \left(\dfrac{9}{100}\right)^{\frac{3}{2}}\) [2 marks]
Mark scheme (a)
Answer
Mark
Comments
\(\left(\dfrac{4}{5}\right)^2\) or \(\dfrac{4^2}{5^2}\) or \(\left(\dfrac{25}{16}\right)^{-1}\) or \(\dfrac{1}{\left(5/4\right)^2}\) or \(\dfrac{1}{5^2/4^2}\) or \(\left(\dfrac{1}{5/4}\right)^2\) or \(\dfrac{1}{25/16}\) or \(\dfrac{1/25}{1/16}\)
M1
missing brackets must be recovered accept a correct decimal or mixed number for any fraction eg \(\dfrac{1}{1.25^2}\)
\(\dfrac{16}{25}\)
A1
oe fraction or decimal
Additional guidance
Ignore any attempt to convert a correct fraction into a decimal
M1A1
Mark scheme (b)
Answer
Mark
Comments
\(\left(\sqrt{\dfrac{9}{100}}\right)^3\) or \(\dfrac{3^3}{10^3}\) or \(\left(\dfrac{3}{10}\right)^3\) or \(\sqrt{\dfrac{9^3}{100^3}}\) or \(\sqrt{\left(\dfrac{9}{100}\right)^3}\) or \(\dfrac{\left(\sqrt{9}\right)^3}{\left(\sqrt{100}\right)^3}\) or \(\sqrt{\dfrac{729}{1\,000\,000}}\) or \(\dfrac{\sqrt{729}}{\sqrt{1\,000\,000}}\)
M1
oe with 0.09 for \(\dfrac{9}{100}\) or 0.3 for \(\dfrac{3}{10}\) or \(3^2\) for 9 or \(10^2\) for 100 missing brackets must be recovered
\(\dfrac{27}{1000}\) or 0.027
A1
Additional guidance
Ignore any attempt to convert a correct fraction into a decimal
M1A1
For M1 do not allow power \(\dfrac{1}{2}\) with no square root
11 Write \(\quad (3^6 \times 3^5) : 3^7 \quad\) in the form \(\quad n : 1 \quad\) where \(n\) is an integer. [3 marks]
Mark scheme
Answer
Mark
Comments
\(3^{11}\ (: 3^7)\) or \(3^6 : 3^2\) or \(3^5 : 3^{(1)}\) or \(\dfrac{a}{3^7}\) or 177 147 : 2187
M1
oe eg 729 : 9 or 243 : 3 \(3^n\) may be implied by a multiplication string of \(n\) 3s \(a\) can be any value other than \(3^7\)
\(\dfrac{3^{11}}{3^7}\ (: 1)\) or \(\dfrac{3^6}{3^2}\ (: 1)\) or \(3^6 \times 3^{-2}\ (: 1)\) or \(\dfrac{3^5}{3^{(1)}}\ (: 1)\) or \(3^{-1} \times 3^5\ (: 1)\) or \(3^4\ (: 1)\) or \(\dfrac{177\,147}{2187}\ (: 1)\)
M1dep
oe left-hand side with one or two components eg \(\dfrac{729}{9} : 1\) or \(243 \times \dfrac{1}{3} : 1\) allow (: 1) to be (: \(3^0\)) \(3^n\) may be implied by a multiplication string of \(n\) 3s
81 : 1
A1
Additional guidance
\(\dfrac{3^6 \times 3^5}{3^7}\ (: 1)\) with no further work
M1M0A0
81 : 1 or \(3^4\ (: 1)\) could be seen from incorrect working eg \(\dfrac{9^{11}}{3^7} = 3^4 \qquad\) Answer 81 : 1
B1 \(3^{12-7}\) or \(3^5\) oe single index or \(3 \times 3 \times 3 \times 3 \times 3\) oe multiplication string or 531 441 seen as \(3^{12}\) or as a numerator or 2187 seen as \(3^7\) or as a denominator or \(3^n\) correctly evaluated, where \(n\) is an integer \(\geqslant 4\)
Additional guidance
Condone \(3^5\) and 243 on the answer line, in either order
B2
\(3^5\) only on the answer line
B1
Do not allow a misread
\(12 - 7\) is insufficient for B1 unless \(3^{12-7}\) or \(3^5\) is also seen
Do not award B1 for a correct evaluation of \(3^n\) not ascribed to a particular value of \(n\) eg a list 3, 9, 27, 81 … does not score the mark unless 81 is identified as \(3^4\)
Mark scheme (b)
Answer
Mark
Comments
\(2^{13}\)
B2
B1 \(2^{3+6+4}\) or (\(8 =\)) \(2 \times 2 \times 2\) or \(2^3\) or (\(2^6 \times 2^4 =\)) \(2^{6+4}\) or (\(2^6 \times 2^4 =\)) \(2^{10}\) or \(2^9\) (\(\times\, 2^4\)) or \(2^7\) (\(\times\, 2^6\)) or 8192
Additional guidance
8192 and \(2^{13}\) on answer line, in either order
B2
8192 only on the answer line
B1
Correctly combined powers can be implied eg \(8 = 2^4\) with answer \(2^{14}\) implies \(2^6 \times 2^4 = 2^{10}\)
B1
Evaluations other than 8192 do not score eg \(8 \times 1024\) without seeing \(8 \times 2^{10}\) eg \(8 \times 64 \times 16\)
B0 B0
Do not award B1 for 8192 if it is in a list of powers of 2 unless it is indicated or it is the highest power evaluated
Changing terms to numbers with a base of 8 scores zero unless converted to a number with a base of 2
B1 \(3^{12-7}\) or \(3^5\) oe single index or \(3 \times 3 \times 3 \times 3 \times 3\) oe multiplication string or 531 441 seen as \(3^{12}\) or as a numerator or 2187 seen as \(3^7\) or as a denominator or \(3^n\) correctly evaluated, where \(n\) is an integer \(\geqslant 4\)
Additional guidance
Condone \(3^5\) and 243 on the answer line, in either order
B2
\(3^5\) only on the answer line
B1
Do not allow a misread
\(12 - 7\) is insufficient for B1 unless \(3^{12-7}\) or \(3^5\) is also seen
Do not award B1 for a correct evaluation of \(3^n\) not ascribed to a particular value of \(n\) eg a list 3, 9, 27, 81 … does not score the mark unless 81 is identified as \(3^4\)
Mark scheme (b)
Answer
Mark
Comments
\(2^{13}\)
B2
B1 \(2^{3+6+4}\) or \((8 =)\ 2 \times 2 \times 2\) or \(2^3\) or \((2^6 \times 2^4 =)\ 2^{6+4}\) or \((2^6 \times 2^4 =)\ 2^{10}\) or \(2^9\ (\times 2^4)\) or \(2^7\ (\times 2^6)\) or 8192
Additional guidance
8192 and \(2^{13}\) on answer line, in either order
B2
8192 only on the answer line
B1
Correctly combined powers can be implied eg \(8 = 2^4\) with answer \(2^{14}\) implies \(2^6 \times 2^4 = 2^{10}\)
B1
Evaluations other than 8192 do not score eg \(8 \times 1024\) without seeing \(8 \times 2^{10}\) eg \(8 \times 64 \times 16\)
B0 B0
Do not award B1 for 8192 if it is in a list of powers of 2 unless it is indicated or it is the highest power evaluated
Changing terms to numbers with a base of 8 scores zero unless converted to a number with a base of 2
22 Work out the value of \(\quad \left(\dfrac{5}{7}\right)^{-2}\)
Give your answer as a mixed number. [3 marks]
Mark scheme
Answer
Mark
Comments
\(1\dfrac{24}{25}\)
B3
oe mixed number B2 \(\dfrac{49}{25}\) B1 \(\left(\dfrac{7}{5}\right)^2\) or \(\dfrac{1}{\left(\frac{5}{7}\right)^2}\) or \(\left(\dfrac{1}{5/7}\right)^2\) or \(\dfrac{1}{25/49}\) or \(\left(\dfrac{25}{49}\right)^{-1}\) or \(\dfrac{1}{5^2} \div \dfrac{1}{7^2}\)
Additional guidance
For B2 or B1 allow equivalent fractions or decimals eg 1.96 for \(\dfrac{49}{25}\)
(a) Write down the value of \(\quad 7^0\) [1 mark]
(b) Work out the value of \(\quad 32^{-\frac{3}{5}}\) [2 marks]
Mark scheme (a)
Answer
Mark
Comments
1
B1
Mark scheme (b)
Answer
Mark
Comments
\(\dfrac{1}{8}\) or 0.125
B2
B1 correct expression including at least one of changes 32 to \(2^5\) shows that the negative index means the reciprocal shows that index \(\dfrac{1}{5}\) means 5th root splits the index into the multiplication of two indices
28 Work out the value of \(\quad 100^{-\frac{1}{2}}\) [2 marks]
Mark scheme
Answer
Mark
Comments
\(\dfrac{1}{10}\) or 0.1
B2
B1 \(\sqrt{100}^{\,-1}\) or \(10^{-1}\) or \(\left(\dfrac{1}{100}\right)^{\frac{1}{2}}\) or \(\dfrac{1}{100^{\frac{1}{2}}}\) or \(\dfrac{1}{\sqrt{100}}\) or \(\sqrt{\dfrac{1}{100}}\)
Additional guidance
Accept \(\pm\) or \(-\) for B1 only, eg \(\pm 10^{-1}\) or \(-\dfrac{1}{10}\)
Give your answer in the form \(\quad 2^m \quad\) where \(m\) is an integer. [3 marks]
Mark scheme
Answer
Mark
Comments
Alternative method 1
\((8^4 =)\ (2^3)^4\) or \(2^{12}\) or \(\left(32^{\frac{2}{5}} =\right) (2^5)^{\frac{2}{5}}\) or \(2^2\)
M1
\(2^{12}\) and \(2^2\)
M1dep
or calculation in the form \(2^a \div 2^b\) where \(a - b = 10\) \(2^c \times 2^d\) where \(c + d = 10\)
\(2^{10}\)
A1
Accept \(m = 10\)
Alternative method 2
\((8^4 =)\ 4096\) or \(\left(32^{\frac{2}{5}} =\right) 4\)
M1
1024
M1dep
\(2^{10}\)
A1
Accept \(m = 10\)
Additional guidance
Note that 1024 from \(32 \times 32\) scores 2 marks if 1024 is their final numerical answer However, if they then try to find \(\sqrt[5]{1024}\) they are clearly processing \(\left(32^{\frac{2}{5}} =\right)\), so this would only score 0 marks without further work
If a numerical method and an index method are both attempted and an incorrect answer is given, award up to M1M1 from the better method
\((6^2 =)\ 36\) or \((8^2 =)\ 64\) or 100 or \(\sqrt{100}\)
M1
10
A1
their \(10 = 5a\) or (their \(10)^3 = 125a^3\) or \(1000 = 125a^3\) or \(8 = a^3\)
M1
2
A1ft
ft their 10 with both method marks scored
Alternative method 2
5 or \(a\)
M1
\(5a\)
A1
their \(5a = \sqrt{100}\) or their \(5a = 10\)
M1
\((a =)\ \dfrac{\sqrt{100}}{5}\) or \((a =)\ \dfrac{10}{5}\) implies M1A1M1
2
A1ft
ft their \(5a\) with both method marks scored
Additional guidance
Use the scheme that gives the better mark eg1 \(\sqrt{14^2} = 5a\), \(14 = 5a\), \(a = 2.8\) scores M0A0M1A0 on alt 1 and M1A1M0A0 on alt 2 eg2 \(\sqrt{100} = 5a^3\), \(10 = 5a^3\), \(a = \sqrt[3]{2}\) scores M1A1M0A0 on alt 1 and M1A0M1A1ft on alt 2
17 Work out the exact value of \(\quad \left(\dfrac{3}{4}\right)^{-3}\) [2 marks]
Mark scheme
Answer
Mark
Comments
\(\left(\dfrac{4}{3}\right)^3\) or \(\dfrac{4^3}{3^3}\) or \(\left(\dfrac{27}{64}\right)^{-1}\) or \(\dfrac{1}{\frac{27}{64}}\) or \(\dfrac{1}{\left(\frac{3}{4}\right)^3}\) or \(\dfrac{1}{0.75^3}\) or \(\left(\dfrac{1}{0.75}\right)^3\)
M1
\(\dfrac{64}{27}\) or \(2\dfrac{10}{27}\)
A1
oe fraction, mixed number or decimal
Additional guidance
\(\dfrac{64}{27}\) followed by an incorrect attempt to convert to a mixed number
\((6^2 =)\ 36\) or \((8^2 =)\ 64\) or 100 or \(\sqrt{100}\)
M1
10
A1
their \(10 = 5a\) or (their 10)\(^3 = 125a^3\) or \(1000 = 125a^3\) or \(8 = a^3\)
M1
2
A1ft
ft their 10 with both method marks scored
Alternative method 2
5 or \(a\)
M1
\(5a\)
A1
their \(5a = \sqrt{100}\) or their \(5a = 10\)
M1
\((a =)\ \dfrac{\sqrt{100}}{5}\) or \((a =)\ \dfrac{10}{5}\) implies M1A1M1
2
A1ft
ft their \(5a\) with both method marks scored
Additional guidance
Use the scheme that gives the better mark eg1 \(\ \sqrt{14^2} = 5a,\ 14 = 5a,\ a = 2.8\) scores M0A0M1A0 on alt 1 and M1A1M0A0 on alt 2 eg2 \(\ \sqrt{100} = 5a^3,\ 10 = 5a^3,\ a = \sqrt[3]{2}\) scores M1A1M0A0 on alt 1 and M1A0M1A1ft on alt 2
(a) Work out the value of \(\quad 2^{14} \div \left(2^9\right)^2\)
Give your answer as a fraction in its simplest form. [3 marks]
(b) Work out the value of \(\quad 25^{\frac{3}{2}}\) [2 marks]
Mark scheme (a)
Answer
Mark
Comments
\(\dfrac{1}{16}\)
B3
B2 \(2^{-4}\) or \(\dfrac{1}{2^4}\) or \(4^{-2}\) or \(\dfrac{1}{4^2}\) or \(16^{-1}\) or \(0.5^4\) or \(\dfrac{16\,384}{262\,144}\) oe fraction B1 \(2^{18}\) or \(2^5 \div 2^9\) or \(\left(2^2\right)^{-2}\) or \(4^7 \div 4^9\)
Mark scheme (b)
Answer
Mark
Comments
\(25 \times 25^{\frac{1}{2}}\) or \(\left(25^{\frac{1}{2}}\right)^3\) or \(\left(25^3\right)^{\frac{1}{2}}\) or \(25\,(\times)\sqrt{25}\) or \(25 \times 5\) or \(5^3\) or \(\sqrt{25^3}\) or \(\left(\sqrt{25}\right)^3\) or \(\sqrt{15\,625}\) or \(15\,625^{\frac{1}{2}}\) or \(\sqrt{25 \times 25^2}\) or \(\sqrt{25 \times 625}\)
18 The solution of \(\quad 3^x = 300 \quad\) lies between two consecutive integers.
Work out the two integers. [1 mark]
Mark scheme
Answer
Mark
Comments
5 and 6 with no incorrect evaluation seen for \(3^5\) or \(3^6\) or 5 and 6 with no incorrect evaluation seen for \(\sqrt[5]{300}\) or \(\sqrt[6]{300}\)
B1
5 and 6 in either order allow any evaluations truncated or rounded to 2 sf or 1 sf
Additional guidance
5 and 6 with either \(3^5\) or \(3^6\) evaluated incorrectly
B0
\(3^5\) or \(3^6\)
B0
243 and 729
B0
\(3^5 = 243\) Allow 240 or 200 (with no incorrect value seen) \(3^6 = 729\) Allow 720 or 730 or 700 (with no incorrect value seen)
\(\sqrt[5]{300} = 3.1(2\ldots)\) or 3.13 \(\sqrt[6]{300} = 2.5(8\ldots)\) or 2.59 or 2.6
(a) Work out the value of \(\quad 81^{-\frac{1}{4}}\) [2 marks]
(b) Write \(\quad 16 \times 8^{2x} \quad\) as a power of 2 in terms of \(x\). [3 marks]
Mark scheme (a)
Answer
Mark
Comments
\(\dfrac{1}{81^{\frac{1}{4}}}\) or \(\dfrac{1}{\sqrt[4]{81}}\) or \(\sqrt[4]{\dfrac{1}{81}}\) or \(3^{-1}\) or \(9^{-\frac{1}{2}}\) or \(81^{\frac{1}{4}} = 3\) or \(\sqrt[4]{81} = 3\) or \(3^4 = 81\)
M1
\(\dfrac{1}{3}\)
A1
Additional guidance
3 without \(81^{\frac{1}{4}}\) or \(\sqrt[4]{81}\)
M0A0
Mark scheme (b)
Answer
Mark
Comments
Alternative method 1
(16 =) \(2^4\) or \(\left(2^3\right)^{2x}\) or \(2^{6x}\)
M1
oe with consistent base 2
(16 =) \(2^4\) and \(\left(2^3\right)^{2x}\) or \(2^{6x}\)
28 \(y = p \times q^{\,x - 1} \qquad\) where \(p\) and \(q\) are numbers.
\(y = 10 \;\) when \(\; x = 1\) \(y = 0.3125 \;\) when \(\; x = 6\)
Work out the value of \(y\) when \(x = 3\) [5 marks]
Mark scheme
Answer
Mark
Comments
\(p \times q^{1 - 1} = 10\) or \(p \times q^0 = 10\) or \(p \times q^{6 - 1} = 0.3125\) or \(p \times q^5 = 0.3125\)
M1
oe
\(p = 10\) or \(10 \times q^{6 - 1} = 0.3125\) or \(q^5 = 0.3125 \div\) their 10 or \(q^5 = 0.03125\)
M1dep
\(\sqrt[5]{\text{their } 0.03125}\) or 0.5
M1dep
oe
their \(10 \times\) their \(0.5^2\) or their \(10 \times\) their \(\left(\sqrt[5]{\text{their } 0.03125}\right)^2\) or their \(10 \times\) their \(0.03125^{\frac{2}{5}}\)
B1 for \(2^5\) oe or for \(1024^{\left[\frac{1}{2}\right]}\) or M1 for \(\sqrt[2]{4}^{\,5}\) or better
M1 accept e.g. 4 × 4 × 2 \(\left(4^{\frac{1}{2}}\right)^5\) alone is not enough for M1
Mark scheme (b)
Answer
Marks
Part marks and guidance
\(\frac{1}{3}x^2\) or \(\frac{x^2}{3}\) final answer
4
B3 for answer \(\frac{1}{3}x^k\) \((k \ne 0)\) or \(\frac{9}{27}x^2\) or B2 for answer \(kx^2\) \((k \ne 0)\) or \(\frac{9}{27}x^k\) or B1 for \(\frac{1}{27}\) seen or \(\frac{9}{27}\) seen or \(x^{-6}\) oe seen
B1 oe e.g. \(x^6\) seen in the denominator of a fraction
For M2 accept equivalent work with all terms in other bases e.g. 4 Accept \((2^2)^y\) for \(2^{2y}\) Allow B1 for writing one other term correctly in base 4 or base 16 e.g. [\(2^x =\)] \(4^{\frac{x}{2}}\) or [\(4^y =\)] \(16^{\frac{y}{2}}\)
\(x + 2y = 4\)
M1
M1 dep on M2
For M1 accept correct equivalent equation
one further step leading to \(y = 2 - \frac{x}{2}\)
Alternative methods: eg M2 for finding \(\frac{1}{5}\), \(\frac{1}{4}\), \(\frac{1}{50}\) or 20[%], 25[%], 2[%] or other comparable forms or M1 for two of these
\(2 \times 10^{-2}\), 0.2, \(2^{-2}\)
B1
accept answer in alternate form e.g fractions or decimals
Alternative methods: eg M2 for finding \(\frac{1}{5}\), \(\frac{1}{4}\), \(\frac{1}{50}\) or 20[%], 25[%], 2[%] or other comparable forms or M1 for two of these
\(2 \times 10^{-2}\), 0.2, \(2^{-2}\)
B1
accept answer in alternate form e.g fractions or decimals
\[x^{-\frac{1}{6}} = \frac{5x^{\frac{1}{3}}}{x^{\frac{3}{4}}}, \text{ where } x \neq 0\]
[3]
Mark scheme
Answer
Marks
Part marks and guidance
625 with no extras
3
M1 for \(\dfrac{x^{-\frac{1}{6}} \times x^{\frac{3}{4}}}{x^{\frac{1}{3}}} = 5\) or better
M1 for \(-\frac{1}{6} + \frac{3}{4} - \frac{1}{3} = \frac{1}{4}\) or better e.g. \(x^{\frac{1}{4}} = 5\)
Alternative method : M1 for \(x^{\frac{1}{3} - \frac{3}{4}}\) or \(x^{\frac{-5}{12}}\) or \(x^{\frac{-1}{6} + \frac{3}{4}}\) or \(x^{\frac{7}{12}}\) and M1 for \(x^{\frac{-1}{6} - \textit{their} \frac{-5}{12}}\) or \(x^{\frac{7}{12} - \frac{1}{3}}\) or \(x^{\frac{3}{12}}\) or \(x^{\frac{1}{4}}\)
could be \(x^{\frac{5}{12}}\) or \(x^{\frac{-1}{4}}\) depends on which side of the equation
18 \(\sqrt[5]{p^2} = \left(\sqrt[3]{m}\right)^2\) and \(p = m^x\), where \(p \gt 0\), \(m \gt 0\) and \(p \ne m\).
Show that the value of \(x\) is \(\frac{5}{3}\). [3]
Mark scheme
Answer
Marks
Part marks and guidance
Correct method to establish \(x = \frac{5}{3}\) e.g. \(p^{\frac{2}{5}} = m^{\frac{2}{3}}\) or better
or \(m^{\frac{2x}{5}} = m^{\frac{2}{3}}\) or \(\frac{2x}{5} = \frac{2}{3}\)
or \(p = \left(\sqrt{\left(\sqrt[3]{m}\right)^2}\right)^5\) or better
M2
M1 for \(p^{\frac{2}{5}}\) or \(m^{\frac{2}{3}}\) or [ \(\sqrt[5]{(m^x)^2}\) =] \(m^{\frac{2x}{5}}\)
or for first step in making \(p\) the subject \(p^2 = \left(\sqrt[3]{m^2}\right)^5\) or \(\sqrt[5]{p} = \sqrt{\left(\sqrt[3]{m^2}\right)}\) or better
If 0 scored, SC1 for [ \(p\) =] \(\sqrt[3]{m^5}\) written
or better e.g. \(\sqrt[5]{p} = \sqrt[6]{m^2}\)
e.g. or better for M2 \(p = \sqrt[6]{m^{10}}\)
Maximum mark is SC1 for those working backwards from \(x = \frac{5}{3}\) and this mark is for interpreting the index \(m^{\frac{5}{3}}\) as \(\sqrt[3]{m^5}\)
\(p^{\frac{1}{5}} = m^{\frac{1}{3}}\) leading to \(p = m^{\frac{5}{3}}\)
or \(\frac{2}{3} \times \frac{5}{2} = \frac{5}{3}\)
or \(6x = 10\) leading to \(\frac{5}{3}\)
or \(p = \left(\sqrt[3 \times 2]{m}\right)^{(2 \times 5)}\) or better leading to \(p = m^{\frac{5}{3}}\)
17 The \(k\)th term of a sequence is \(r^k\), where \(r \ne 0\).
The sixth term is equal to three times the second term.
Find the value of \(r\), giving your answer correct to 3 decimal places. [4]
Mark scheme
Answer
Marks
Part marks and guidance
1.316
4
B3 for answer 1.3160 to 1.3161
OR
M3 for [\(r\) =] \(\sqrt[4]{3}\) oe or M2 for \(r^4 = 3\) or M1 for \(r^6 = 3r^2\)
Trials or insufficient method: B4 for answer 1.316 or B3 for answer 1.3160 to 1.3161 or M2 for at least three correct trials of \(r^4\) oe or of \(r^6\) and \(3r^2\) oe or M1 for at least two correct trials of \(r^4\) oe or of \(r^6\) and \(3r^2\) oe
eg M3 for \(\sqrt{3}\) = 1.73[2…] and \(\sqrt{1.73[2\ldots]}\)
14 The number of bees, \(P\), in a colony is given by the formula
\[P = ab^x\]
where \(x\) is the number of months after the start of July.
At the start of July, there were 25 000 bees in the colony. After one month, there were 23 500 bees in the colony.
Find the value of \(a\) and the value of \(b\). Give the value of \(b\) as a decimal. [4]
Mark scheme
Answer
Marks
Part marks and guidance
[\(a\) = ] 25 000 [\(b\) = ] [0].94
4
B1 for 25 000
AND
B3 for 0.94 oe or B2 for \(\frac{23500}{25000}\) or 94% oe or \(\frac{25000 - 23500}{25000}\) or 0.06 oe or M1 for 23500 = \(ab^{[1]}\) or 23500 = \(25000b^{[1]}\)
17 Show that \(\sqrt[3]{a^4} \times \dfrac{1}{a}\) can be expressed as \(a^{\frac{1}{3}}\). [3]
Mark scheme
Answer
Marks
Part marks and guidance
\(a^{\frac{4}{3}} \times a^{-1} = a^{\frac{4}{3} - 1} = a^{\frac{1}{3}}\) or \(a^{\frac{4}{3}} \div a = a^{\frac{4}{3} - 1} = a^{\frac{1}{3}}\)
3
B1 for \(a^{\frac{4}{3}}\)
B1 for \(\left[\frac{1}{a} =\right] a^{-1}\)
e.g. \(a^{\frac{4}{3}} \times a^{-1} = a^{1/3}\) scores B1B1 only but \(a^{\frac{4}{3}} \times a^{-1} = a^{1/3}\) with \(\frac{4}{3} - 1 = \frac{1}{3}\) or arrow connecting \(\frac{4}{3}\) and –1 scores full marks.
14 Write \(\left(\sqrt[4]{8}\right)^5\) as a power of 2. [3]
Mark scheme
Answer
Marks
Part marks and guidance
\(2^{\frac{15}{4}}\) or \(2^{3.75}\)
3
B2 for \(2^{15}\) or \(2^{\frac{3}{4}}\) or \((2^3)^{\frac{5}{4}}\) or \(3 \times \frac{1}{4} \times 5\) seen or B1 for [8 =] \(2^3\) or \(8^{\frac{5}{4}}\) or \(8^{1.25}\) seen
B2 for answer \(ka^{11}\) or \(\dfrac{6a^{13}}{a^2}\) or \(6 \times a^6 \times a^5\) or \(6 \times a^8 \times a^3\) shown in working OR B1 for answer \(6a^k\) or \(ka^{13}\) or for \(3a^8 \times 2a^3\) or \(3a^6 \times 2a^5\) shown in working
Mark scheme (b)
Answer
Marks
Part marks and guidance
2.5 or \(2\frac{1}{2}\) or \(\frac{5}{2}\)
3
M1 for correct first step e.g. \(6x - 10 = 5\)
Do not accept \(\frac{15}{6}\) as final answer but allow to imply M1M1 Embedded answer scores M2 max
M1 for \(6x = 5 + 10\) FT their first step or for FT their \(ax = b\) to \(x = \dfrac{b}{a}\)
where \(\frac{b}{a}\) is a simplified fraction (improper or mixed number) or an integer e.g. M1 for \(6x = 1\) leading to \(x = \frac{1}{6}\)
B1 for [\((a^3)^2\) =] \(a^6\) or \(a^3 \times a^3\)
Alternative: B2 for [\(a^5 \times (a^3)^2\) =] \(a \times a \times \ldots \times a\) [\(= a^{11}\)] or B1 for [\((a^3)^2\) =] \(a \times a \times a \times a \times a \times a\)
\(a^{5+6}\) or \(a^{5+3+3}\) or intent to add indices stated or unambiguously indicated (eg 5 + 6, add indices etc)
written in full with eleven \(a\)’s.
written in full with six \(a\)’s May be implied by \((a \times a \times a \times a \times a \times a)\) seen within an incorrect lengthier product.
Mark scheme (b)
Answer
Marks
Part marks and guidance
\(5^{15}\)
3
B1 for \(\left[\dfrac{1}{125} =\right]\) \(5^{-3}\) or [125 =] \(5^3\)
12 Carol says that \(64^{-\frac{1}{2}} = \dfrac{1}{32}\).
Explain her error and give the correct value of \(64^{-\frac{1}{2}}\) in the form \(\dfrac{p}{q}\). [3]
Mark scheme
Answer
Marks
Part marks and guidance
She divided by 2 and did not square root oe
B1
Must mention divide by 2 oe accept in symbols e.g. ÷ 2 and square rooting. Accept \(\sqrt{\phantom{x}}\) or [square] root for square root Each step must be correctly evaluated but FT previous step Allow method marks in either order
\(\dfrac{1}{8}\)
2
M1 for square root [8 seen in answer] M1 for reciprocal [implied by answer of form \(\dfrac{1}{n}\), \(n \ne 0\) or 32]
B1 for [\((a^3)^2\) =] \(a^6\) or \(a^3 \times a^3\)
\(a^{5+6}\) or \(a^{5+3+3}\) or intent to add indices stated or unambiguously indicated (eg 5 + 6, add indices etc)
Alternative: B2 for [\(a^5 \times (a^3)^2 =\)] \(a \times a \times \ldots \times a\) [= \(a^{11}\)] or B1 for [\((a^3)^2\) =] \(a \times a \times a \times a \times a \times a\)
written in full with eleven \(a\)’s.
written in full with six \(a\)’s May be implied by \((a \times a \times a \times a \times a \times a)\) seen within an incorrect lengthier product.
Mark scheme (b)
Answer
Marks
Part marks and guidance
\(5^{15}\)
3
B1 for \(\left[\dfrac{1}{125} =\right]\ 5^{-3}\) or [125 =] \(5^3\)
In left-hand methods, M1M1 can be awarded if the denominator 3 is consistently omitted
\(= \dfrac{3^{\frac{4}{3}}}{3}\) or \(\dfrac{\sqrt[3]{3^3 \times 3}}{3} = \dfrac{3\sqrt[3]{3}}{3}\)
M1dep
\(= \sqrt[3]{\dfrac{81}{27}}\)
There may be other surd methods. M1 first productive step \(\sqrt[3]{81} = 81^{\frac{1}{3}}\) is not sufficiently productive as a first step M1dep second productive step from a correct first step
\(\left[= 3^{\frac{4}{3} - 1}\right] = 3^{\frac{1}{3}}\) or \(\sqrt[3]{3} = 3^{\frac{1}{3}}\)
