(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 \(128 = 2^7\), \(x = 7\).
(ii) Ans: 4
Combine exponents: \(2^{x+9} = 2^{13}\). Thus, \(x + 9 = 13\) and \(x = 4\).
(iii) Ans: 7
Rewrite as \(2^{9-x} = 2^2\). Then \(9 – x = 2\) and \(x = 7\).
(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, find the \(x\)-value where \(y = 5\). The solution is approximately \(x \approx 2.3\).
(iv) Ans: \(y = 3x – 1\) oe
Slope \(m = \frac{8-2}{3-1} = 3\). Using point \((1, 2)\), the equation is \(y – 2 = 3(x – 1)\) or \(y = 3x – 1\).
(v) Ans: \(-1.7\) to \(-1.5\) and \(2\)
Rewrite as \(2^x = x + 2\). Draw \(y = x + 2\) and find intersections with \(y = 2^x\). Solutions are \(x \approx -1.6\) and \(x = 2\).
