Powers & Roots

Spec reference: Number - Powers and roots
Key idea: Calculate and use powers, roots and index laws.

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Powers (indices)

$a^n$ means $a$ multiplied by itself $n$ times.

$$3^4=3\times 3\times 3\times 3=81$$

• $a$ is the base
• $n$ is the index (or power or exponent)

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Square roots and cube roots

$$\sqrt{a}\text{ is the square root of }a\sqrt{81}=9$$$$\sqrt[3]{a}\text{ is the cube root of }a\sqrt[3]{64}=4$$

WARNING

Every positive number has two square roots: $\sqrt{25}=\pm 5$
In most GCSE questions, only the positive root is required unless stated otherwise.

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Laws of indices

Law	Rule	Example
Multiplication	$a^m\times a^n=a^{m+n}$	$2^3\times 2^4=2^7$
Division	$a^m÷a^n=a^{m-n}$	$5^6÷5^2=5^4$
Power of a power	$(a^m{)}^n=a^{mn}$	$(3^2{)}^4=3^8$
Zero index	$a^0=1$	$7^0=1$
Negative index	$a^{-n}={\displaystyle \frac{1}{a^n}}$	$2^{-3}={\displaystyle \frac{1}{8}}$
Fractional index	$a^{\frac{1}{n}}=\sqrt[n]{a}$	${25}^{\frac{1}{2}}=5$
Fractional index	$a^{\frac{m}{n}}=(\sqrt[n]{a}{)}^m$	$8^{\frac{2}{3}}=(\sqrt[3]{8}{)}^2=4$

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Working with fractional indices

For $a^{\frac{m}{n}}$: the denominator is the root, the numerator is the power.

Remember

Denominator = Down (root goes down below, i.e. you take the root first)

Example

Evaluate ${27}^{\frac{2}{3}}$

$${27}^{\frac{2}{3}}={\left(\sqrt[3]{27}\right)}^2=3^2=\boxed{{\displaystyle 9}}$$

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Negative indices

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

Example

Evaluate $4^{-2}$

$$4^{-2}=\frac{1}{4^2}=\frac{1}{16}$$

Example

Evaluate ${\left({\displaystyle \frac{2}{3}}\right)}^{-2}$

$${\left(\frac{2}{3}\right)}^{-2}={\left(\frac{3}{2}\right)}^2=\frac{9}{4}=2\frac{1}{4}$$

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Surds (introduction)

A surd is an irrational root that cannot be simplified to a whole number.

$$\sqrt{2},\sqrt{3},\sqrt{5},\sqrt{7}\text{ are surds}$$$$\sqrt{4}=2,\sqrt{9}=3\text{ are NOT surds}$$

Simplifying surds

$$\sqrt{ab}=\sqrt{a}\times \sqrt{b}$$

Example

Simplify $\sqrt{48}$

$$\sqrt{48}=\sqrt{16\times 3}=\sqrt{16}\times \sqrt{3}=\boxed{{\displaystyle 4\sqrt{3}}}$$

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Exam tips

Watch out for

• $a^0=1$ for any non-zero value of $a$
• $(-3{)}^2=9$ but $-3^2=-9$ (the negative is not squared unless inside brackets)
• For fractional indices, always take the root first to keep numbers manageable

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Test Yourself

Question 1 of 5

Evaluate $27^{\frac{2}{3}}$