## Learning Outcomes

If you work through this section you should be able to:

• Understand what is meant by a power or index.
• Evaluate expressions involving powers.
• Evaluate expressions using the rules of indices.
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A power or an exponent of a number states how many times to multiply the number by itself. It is written as a small superscripted number on the top right of a number, for example 25 = 2×2×2×2×2. The superscript 5 is called a power or index and the number 2 is called the base. It is pronounced “two raised to the power of five”.

Raising to the power of 2 is called squaring: “4 squared” is 42 = 4 × 4 = 16;

and to the power of 3 is called cubing: “5 cubed” is 53 = 5 × 5 × 5 = 125.

Note: Any number raised to the power of 1 is itself, e.g. 61 = 6.

Any number raised to the power of 0 has the value to 1, e.g. 90 = 1.

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According to the BEDMAS rule, exponentiation is done after brackets and before multiplication.

So: 3 × 23 = 3 × 8 = 24 — but (3 × 2)3 = 63 = 216.

#### Examples

1.   −42 = −4 × 4 = −16
2.   (−4)2 = (−4) × (−4) = 16
3.   2(3 × 52 − 8) = 2(3 × 25 − 8)
= 2 (75 − 8)
= 134
4.   3(6 + 2 × 32)2 = 3(6 + 2 × 9)2
= 3(6 + 18)2
= 3(24)2
= 3 × 576
= 1728

Negative powers denote the reciprocal. The positive powers are calculated first then the reciprocal is taken.

So:

If then $x$ is the ${a}^{\mathrm{th}}$ root of $b$. For example, if 62 = 36, then 6 is the second root of 36. The second root is usually called the square root. It is written as 6 = $\sqrt{36}$.

The third root is usually called the cube root. For example, 3 = $\sqrt[3]{27}$.

Note also that when −6 is squared we again obtain 36, that is (−6)2 = 36. This means that 36 has another square root, −6. Therefore, $\sqrt{36}$ = ±6

Fractional powers can be written as ${a}^{\frac{m}{n}}$. A fractional power can be broken into two parts: ${m}^{\mathrm{th}}$ power and ${n}^{\mathrm{th}}$ root. You can either do the power first then take the root, or alternatively take the root first and then do the power. Therefore:

${a}^{\frac{m}{n}}$ = $\sqrt[n]{{a}^{m}}$ =

So: ${81}^{\frac{3}{4}}$ = ${\left(\sqrt[4]{81}\right)}^{3}$ = 33 = 27

#### Examples

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The rules of indices can be used to manipulate powers related expressions. The rules of indices are given as below:

x

${a}^{n}÷{a}^{m}={a}^{n-m}$

${\left({a}^{n}\right)}^{m}={a}^{nm}$

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

${a}^{1}{n}}=\sqrt[n]{a}$

${a}^{n}{m}}={\left({a}^{1}{m}}\right)}^{n}={\left(\sqrt[m]{a}\right)}^{n}$

#### Examples

1.   43 x 45 = 43+5 = 48
2.   X2 x X7 ÷ X4 = X2+7-4
= X5
3.   (35)2 = 310
4.   s3 ÷ t−4 × (s−3t−2)3 = s3 ÷ t−4 × s−9 × t−6
= s3−9 × t4−6
= s−6t−2
5.   ${8}^{2}{3}}÷8$ = ${8}^{2}{3}-1}$
${8}^{-\frac{1}{3}}$
= $\frac{1}{\sqrt[3]{8}}$
= $\frac{1}{2}$
= 0.5