(a) Ans: 1.62 or 1.62…
Using a calculator, \(2^{0.7} \approx 1.6245\). Rounded to two decimal places, the answer is \(1.62\).
(b)
(i) Ans: 7
Since \(128 = 2^{7}\), we have \(x = 7\).
(ii) Ans: 4
Using exponent rules: \(2^{x+9} = 2^{13} \Rightarrow x + 9 = 13 \Rightarrow x = 4\).
(iii) Ans: 7
Rewriting: \(2^{9-x} = 2^{2} \Rightarrow 9 – x = 2 \Rightarrow x = 7\).
(iv) Ans: \(\frac{1}{3}\) oe
\(\sqrt[3]{2} = 2^{1/3}\), so \(x = \frac{1}{3}\).
(c)
(i) Ans:
| \(x\) | -3 | -2 | -1 | 0 | 1 | 2 | 3 |
| \(y\) | 0.125 | 0.25 | 0.5 | 1 | 2 | 4 | 8 |
For \(x = -2\), \(y = 2^{-2} = 0.25\). For \(x = 0\), \(y = 2^{0} = 1\).
(ii) Ans: Correct curve
Plot the points from the table and draw a smooth exponential curve through them.
(iii) Ans: 2.3
From the graph, the solution to \(2^{x} = 5\) is approximately \(x \approx 2.3\).
(iv) Ans: \(y = 3x – 1\)
Slope \(m = \frac{8-2}{3-1} = 3\). Using point \((1, 2)\): \(y – 2 = 3(x – 1) \Rightarrow y = 3x – 1\).
(v) Ans: \(-1.7\) to \(-1.5\) and \(2\)
Rewrite as \(2^{x} = x + 2\). The intersections of \(y = 2^{x}\) and \(y = x + 2\) give \(x \approx -1.6\) and \(x = 2\).
