GCSE Maths | Indices

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

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) $n^3 \div n^3$

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

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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) $(3x^4)^4$

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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) $1^0$

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

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

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

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

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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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