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# A-Level Maths | Indices
- URL: https://www.a-level-maths-tutor.com/a-level-maths-indices/
- Published: 2026-06-27T14:47:31.000Z
- Updated: 2026-06-27T14:47:31.000Z
- Author: Jaisul Naik

**Rules of Indices**

$x^{\\frac{1}{n}} = \\sqrt\[n\]{x}$

$x^{\\frac{m}{n}} = \\left(\\sqrt\[n\]{x}\\right)^m$

$x^{-n} = \\dfrac{1}{x^n}$

## Question 1

Solve each equation.

**a)** $x^{\\frac{1}{2}} = 6$

**b)** $x^{\\frac{1}{3}} = 5$

**c)** $x^{-\\frac{1}{2}} = 2$

**d)** $x^{-\\frac{1}{4}} = \\tfrac{1}{3}$

**e)** $x^{\\frac{3}{2}} = 8$

**f)** $x^{\\frac{2}{3}} = 16$

**g)** $x^{\\frac{4}{3}} = 81$

**h)** $x^{-\\frac{3}{2}} = 27$

Show Answers 

**a)** $x = 36$ **b)** $x = 125$ **c)** $x = \\tfrac{1}{4}$ **d)** $x = 81$

**e)** $x = 4$ **f)** $x = 64$ **g)** $x = 27$ **h)** $x = \\tfrac{1}{9}$
  
  
## Question 2

Express in the form $x^k$.

**a)** $\\sqrt{x}$

**b)** $\\dfrac{1}{\\sqrt\[3\]{x}}$

**c)** $x^2 \\times \\sqrt{x}$

**d)** $\\dfrac{\\sqrt\[4\]{x}}{x}$

**e)** $\\sqrt{x^3}$

**f)** $\\sqrt{x} \\times \\sqrt\[3\]{x}$

**g)** $(\\sqrt{x})^5$

**h)** $\\sqrt\[3\]{x^2} \\times (\\sqrt{x})^3$

Show Answers 

**a)** $x^{\\frac{1}{2}}$ **b)** $x^{-\\frac{1}{3}}$ **c)** $x^{\\frac{5}{2}}$ **d)** $x^{-\\frac{3}{4}}$

**e)** $x^{\\frac{3}{2}}$ **f)** $x^{\\frac{5}{6}}$ **g)** $x^{\\frac{5}{2}}$ **h)** $x^{\\frac{13}{6}}$
  
  
## Question 3

Express each of the following in the form $ax^b$, where $a$ and $b$ are rational constants.

**a)** $\\dfrac{4}{\\sqrt{x}}$

**b)** $\\dfrac{1}{2x}$

**c)** $\\dfrac{3}{4x^3}$

**d)** $\\dfrac{1}{(3x)^2}$

**e)** $\\dfrac{2}{5\\sqrt\[3\]{x}}$

**f)** $\\dfrac{1}{\\sqrt{9x^3}}$

show Answers 

**a)** $4x^{-\\frac{1}{2}}$ **b)** $\\tfrac{1}{2}x^{-1}$ **c)** $\\tfrac{3}{4}x^{-3}$

**d)** $\\tfrac{1}{9}x^{-2}$ **e)** $\\tfrac{2}{5}x^{-\\frac{1}{3}}$ **f)** $\\tfrac{1}{3}x^{-\\frac{3}{2}}$
  
  
## Question 4

Simplify each of the following.

**a)** $\\dfrac{x^3 + 2x}{x}$

**b)** $\\dfrac{4t^5 - 6t^3}{2t^2}$

**c)** $\\dfrac{x^{\\frac{3}{2}} - 3x}{x^{\\frac{1}{2}}}$

**d)** $\\dfrac{y^2(y^3 - 6)}{3y}$

**e)** $\\dfrac{p + p^{\\frac{3}{2}}}{p^{\\frac{3}{4}}}$

**f)** $\\dfrac{8w - 2w^{\\frac{1}{2}}}{4w^{-\\frac{1}{2}}}$

**g)** $\\dfrac{x + 1}{x^{\\frac{1}{2}} + x^{-\\frac{1}{2}}}$

**h)** $\\dfrac{2t^3 - 4t}{t^{\\frac{3}{2}} - 2t^{-\\frac{1}{2}}}$

Show Answers 

**a)** $x^2 + 2$ **b)** $2t^3 - 3t$ **c)** $x - 3x^{\\frac{1}{2}}$ **d)** $\\tfrac{1}{3}y^4 - 2y$

**e)** $p^{\\frac{1}{4}} + p^{\\frac{3}{4}}$ **f)** $2w^{\\frac{3}{2}} - \\tfrac{1}{2}w$ **g)** $x^{\\frac{1}{2}}$ **h)** $2t^{\\frac{3}{2}}$

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