1.02a Indices: laws of indices for rational exponents

In this question you must show detailed reasoning. Find the smallest positive integers \(m\) and \(n\) such that \(\left(\frac{64}{49}\right)^{-\frac{3}{2}} = \frac{m}{n}\) [3]

Express \(\frac{a^{\frac{1}{2}} - a^{\frac{2}{3}}}{a^{\frac{1}{3}} - a}\) in the form \(a^m + \sqrt{a^n}\), where \(m\) and \(n\) are integers and \(a \neq 0\) or 1. [4]

Express each of the following in the form \(px^q\), where \(p\) and \(q\) are constants.

1. \(\frac{2}{\sqrt[3]{x}}\) [1]
2. \((5x\sqrt{x})^3\) [1]
3. \(\sqrt{2x^3} \times \sqrt{8x^5}\) [1]
4. \(x^5(27x^6)^{\frac{1}{3}}\) [2]

Do not use a calculator for this question

1. Find \(a\), given that \(a^3 = 64x^{12}y^3\). [2]
2. 1. Express \(\frac{81}{\sqrt{3}}\) in the form \(3^k\). [2]
   2. Express \(\frac{5 + \sqrt{3}}{5 - \sqrt{3}}\) in the form \(\frac{a + b\sqrt{3}}{c}\), where \(a\), \(b\) and \(c\) are integers. [3]

Simplify fully.

1. \(\sqrt{a^3} \times \sqrt{16a}\) [2]
2. \((4b^6)^{\frac{3}{2}}\) [2]