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# GCSE Maths | Indices
- URL: https://www.a-level-maths-tutor.com/gcse-maths-indices/
- Published: 2026-04-18T11:06:25.000Z
- Updated: 2026-07-29T14:36:02.000Z
- Author: Jaisul Naik

$a^m \\times a^n = a^{m+n}$

When the base is the same, add the powers.

**Worked Example**

$m^4 \\times m^3 = m^{4+3} = m^7$

## Question 1

**a)** $m^2 \\times m^3$

**b)** $m^3 \\times m^3$

**c)** $m^6 \\times m^2$

**d)** $m^7 \\times m^3$

**e)** $m^6 \\times m^8$

**f)** $m^2 \\times m$

**g)** $m \\times m^3$

**h)** $m^7 \\times m^8$

**i)** $m^9 \\times m^2$

**j)** $m \\times m^8$

**k)** $m^6 \\times m^5$

**l)** $m^2 \\times m^2 \\times m^2 \\times m^2$

Show Answers 

**a)** $m^5$ **b)** $m^6$ **c)** $m^8$ **d)** $m^{10}$ **e)** $m^{14}$ **f)** $m^3$

**g)** $m^4$ **h)** $m^{15}$ **i)** $m^{11}$ **j)** $m^9$ **k)** $m^{11}$ **l)** $m^8$

$a^m \\div a^n = a^{m-n}$

When the base is the same, subtract the powers.

**Worked Example**

$n^7 \\div n^2 = n^{7-2} = n^5$

## Question 2

**a)** $n^5 \\div n^2$

**b)** $n^8 \\div n^3$

**c)** $n^9 \\div n^2$

**d)** $n^7 \\div n^5$

**e)** $n^3 \\div n$

**f)** $n^8 \\div n$

**g)** $n^7 \\div n^4$

**h)** $n^9 \\div n^3$

**i)** $n^4 \\div n^8$

**j)** $n \\div n^3$

**k)** $n^{45} \\div n^5$

**l)** $n^3 \\div n^3$

Show Answers 

**a)** $n^3$ **b)** $n^5$ **c)** $n^7$ **d)** $n^2$ **e)** $n^2$ **f)** $n^7$

**g)** $n^3$ **h)** $n^6$ **i)** $n^{-4}$ **j)** $n^{-2}$ **k)** $n^{40}$ **l)** $1$

$(a^m)^n = a^{m \\times n}$

When a power is raised to another power, multiply the powers.

**Worked Example**

$(y^4)^3 = y^{4 \\times 3} = y^{12}$

## Question 3

**a)** $(y^5)^2$

**b)** $(y^3)^2$

**c)** $(y^4)^3$

**d)** $(y^5)^4$

**e)** $(y^3)^6$

**f)** $(y^7)^3$

**g)** $(y^6)^6$

**h)** $(y^9)^2$

**i)** $(y^4)^8$

**j)** $(y^3)^{-5}$

**k)** $(y^{-5})^2$

Show Answers 

**a)** $y^{10}$ **b)** $y^6$ **c)** $y^{12}$ **d)** $y^{20}$ **e)** $y^{18}$ **f)** $y^{21}$

**g)** $y^{36}$ **h)** $y^{18}$ **i)** $y^{32}$ **j)** $y^{-15}$ **k)** $y^{-10}$

$(ka^m)^n = k^n \\times a^{m \\times n}$

Apply the outer power to both the number and the letter.

**Worked Example**

$(3x^2)^3 = 3^3 \\times x^{2 \\times 3} = 27x^6$

## Question 4

**a)** $(2x^3)^2$

**b)** $(5x^6)^2$

**c)** $(5x^5)^3$

**d)** $(2x^3)^4$

**e)** $(7x^5)^2$

**f)** $(4x^7)^3$

**g)** $(2x^6)^6$

**h)** $(10x^9)^3$

**i)** $(3x^4)^4$

Show Answers 

**a)** $4x^6$ **b)** $25x^{12}$ **c)** $125x^{15}$ **d)** $16x^{12}$ **e)** $49x^{10}$

**f)** $64x^{21}$ **g)** $64x^{36}$ **h)** $1000x^{27}$ **i)** $81x^{16}$

$a^0 = 1$ (for any non-zero $a$)

Anything to the power of $0$ equals $1$.

**Worked Example**

$6^0 = 1$

## Question 5

**a)** $5^0$

**b)** $12^0$

**c)** $x^0$

**d)** $100^0$

**e)** $(-3)^0$

**f)** $y^0$

**g)** $7^0$

**h)** $(2x)^0$

**i)** $\\left(\\dfrac{1}{2}\\right)^0$

**j)** $m^0$

**k)** $(4xy)^0$

**l)** $1^0$

Show Answers 

All answers $= 1$

$a^{-1} = \\dfrac{1}{a}$

A power of $-1$ means the reciprocal — flip the number.

**Worked Example**

$8^{-1} = \\dfrac{1}{8}$

## Question 6

**a)** $5^{-1}$

**b)** $2^{-1}$

**c)** $10^{-1}$

**d)** $3^{-1}$

**e)** $7^{-1}$

**f)** $4^{-1}$

**g)** $x^{-1}$

**h)** $12^{-1}$

**i)** $100^{-1}$

**j)** $\\left(\\dfrac{1}{2}\\right)^{-1}$

**k)** $\\left(\\dfrac{2}{3}\\right)^{-1}$

**l)** $\\left(\\dfrac{3}{5}\\right)^{-1}$

Show Answers 

**a)** $\\dfrac{1}{5}$ **b)** $\\dfrac{1}{2}$ **c)** $\\dfrac{1}{10}$ **d)** $\\dfrac{1}{3}$ **e)** $\\dfrac{1}{7}$ **f)** $\\dfrac{1}{4}$

**g)** $\\dfrac{1}{x}$ **h)** $\\dfrac{1}{12}$ **i)** $\\dfrac{1}{100}$ **j)** $2$ **k)** $\\dfrac{3}{2}$ **l)** $\\dfrac{5}{3}$

$a^{\\frac{1}{2}} = \\sqrt{a}$

A power of $\\frac{1}{2}$ means the square root.

**Worked Example**

$25^{\\frac{1}{2}} = \\sqrt{25} = 5$

## Question 7

**a)** $64^{\\frac{1}{2}}$

**b)** $9^{\\frac{1}{2}}$

**c)** $144^{\\frac{1}{2}}$

**d)** $25^{\\frac{1}{2}}$

**e)** $4^{\\frac{1}{2}}$

**f)** $100^{\\frac{1}{2}}$

**g)** $49^{\\frac{1}{2}}$

**h)** $16^{\\frac{1}{2}}$

**i)** $400^{\\frac{1}{2}}$

**j)** $81^{\\frac{1}{2}}$

**k)** $36^{\\frac{1}{2}}$

**l)** $121^{\\frac{1}{2}}$

Show Answers 

**a)** $8$ **b)** $3$ **c)** $12$ **d)** $5$ **e)** $2$ **f)** $10$

**g)** $7$ **h)** $4$ **i)** $20$ **j)** $9$ **k)** $6$ **l)** $11$

$a^{\\frac{1}{3}} = \\sqrt\[3\]{a}$

A power of $\\frac{1}{3}$ means the cube root.

**Worked Example**

$64^{\\frac{1}{3}} = \\sqrt\[3\]{64} = 4$

## Question 8

**a)** $216^{\\frac{1}{3}}$

**b)** $8^{\\frac{1}{3}}$

**c)** $729^{\\frac{1}{3}}$

**d)** $1^{\\frac{1}{3}}$

**e)** $64^{\\frac{1}{3}}$

**f)** $\\left(\\dfrac{1}{8}\\right)^{\\frac{1}{3}}$

**g)** $343^{\\frac{1}{3}}$

**h)** $27^{\\frac{1}{3}}$

**i)** $1000^{\\frac{1}{3}}$

**j)** $512^{\\frac{1}{3}}$

**k)** $\\left(\\dfrac{1}{27}\\right)^{\\frac{1}{3}}$

**l)** $125^{\\frac{1}{3}}$

Show Answers 

**a)** $6$ **b)** $2$ **c)** $9$ **d)** $1$ **e)** $4$ **f)** $\\dfrac{1}{2}$

**g)** $7$ **h)** $3$ **i)** $10$ **j)** $8$ **k)** $\\dfrac{1}{3}$ **l)** $5$

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

Do the root first (from the denominator), then the power (from the numerator). Rooting first keeps the numbers small.

**Worked Example**

$8^{2/3} \\rightarrow 2$ (cube root) $\\rightarrow 4$ (squared)

## Question 9

Part 1 — powers of $\\frac{3}{2}$:

**a)** $25^{\\frac{3}{2}}$

**b)** $4^{\\frac{3}{2}}$

**c)** $100^{\\frac{3}{2}}$

**d)** $9^{\\frac{3}{2}}$

**e)** $36^{\\frac{3}{2}}$

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

Part 2 — powers of $\\frac{2}{3}$:

**g)** $125^{\\frac{2}{3}}$

**h)** $8^{\\frac{2}{3}}$

**i)** $1000^{\\frac{2}{3}}$

**j)** $27^{\\frac{2}{3}}$

**k)** $216^{\\frac{2}{3}}$

**l)** $64^{\\frac{2}{3}}$

Show Answers 

**a)** $125$ **b)** $8$ **c)** $1000$ **d)** $27$ **e)** $216$ **f)** $64$

**g)** $25$ **h)** $4$ **i)** $100$ **j)** $9$ **k)** $36$ **l)** $16$

$a^{-\\frac{m}{n}} = \\dfrac{1}{a^{\\frac{m}{n}}}$

The minus sign makes it a reciprocal. Then evaluate the fractional power as before: root first, then power.

**Worked Example**

$9^{-3/2} \\rightarrow \\dfrac{1}{9^{3/2}} \\rightarrow \\dfrac{1}{3}$ (square root) $\\rightarrow \\dfrac{1}{27}$ (cubed)

## Question 10

**a)** $4^{-\\frac{3}{2}}$

**b)** $9^{-\\frac{3}{2}}$

**c)** $8^{-\\frac{2}{3}}$

**d)** $27^{-\\frac{2}{3}}$

**e)** $16^{-\\frac{3}{2}}$

**f)** $4^{-\\frac{5}{2}}$

**g)** $8^{-\\frac{4}{3}}$

**h)** $27^{-\\frac{4}{3}}$

**i)** $16^{-\\frac{3}{4}}$

**j)** $32^{-\\frac{2}{5}}$

Show Answers 

**a)** $\\dfrac{1}{8}$ **b)** $\\dfrac{1}{27}$ **c)** $\\dfrac{1}{4}$ **d)** $\\dfrac{1}{9}$ **e)** $\\dfrac{1}{64}$

**f)** $\\dfrac{1}{32}$ **g)** $\\dfrac{1}{16}$ **h)** $\\dfrac{1}{81}$ **i)** $\\dfrac{1}{8}$ **j)** $\\dfrac{1}{4}$

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