November 2021 Paper 1 Q22
22 Solve \(\quad 8^x = \dfrac{2^{56} - 4^{26}}{30}\) [4 marks]
| Answer | Mark | Comments |
|---|---|---|
| Alternative method 1 Working with powers of 2 | ||
| \(2^{3x}\) or \(2^{52}\) | B1 | from \(8^x\) and \(4^{26}\) |
| \(2^{52}(2^4 - 1)\) or \(2^{52} \times 15\) or \(2 \times 8^x = 2^{52}\) or \(2 \times 2^{3x} = 2^{52}\) | M1 | oe |
| \(2^{3x + 1} = 2^{52}\) or \(3x + 1 = 52\) | M1dep | oe eg \(2^{3x} = 2^{51}\) |
| 17 | A1 | |
| Alternative method 2 Working with powers of 4 | ||
| \(4^{1.5x}\) or \(4^{28}\) | B1 | from \(8^x\) and \(2^{56}\) |
| \(4^{26}(4^2 - 1)\) or \(4^{26} \times 15\) or \(2 \times 4^{1.5x} = 4^{26}\) | M1 | oe |
| \(4^{1.5x} = \dfrac{4^{26}}{4^{0.5}}\) or \(1.5x = 26 - 0.5\) | M1dep | oe eg \(4^{1.5x} = 4^{25.5}\) |
| 17 | A1 | |
| Alternative method 3 Working with powers of 8 | ||
| \(8^{\frac{56}{3}}\) or \(8^{\frac{52}{3}}\) | B1 | from \(2^{56}\) and \(4^{26}\) |
| \(8^{\frac{52}{3}}\left(8^{\frac{4}{3}} - 1\right)\) or \(8^{\frac{52}{3}} \times 15\) or \(2 \times 8^x = 8^{\frac{52}{3}}\) | M1 | oe |
| \(8^x = 8^{\frac{52}{3}} \div 8^{\frac{1}{3}}\) or \(x = \dfrac{52}{3} - \dfrac{1}{3}\) | M1dep | oe eg \(8^x = 8^{17}\) |
| 17 | A1 | |