# Powers and Roots

## Powers and Roots Revision

**Powers and Roots**

**Powers** are a shorthand way of expressing repeated multiplication. **Roots** are ways of reversing this. There are a total of **10 indices rules**. This page will give you the **7** easy rules to remember; there are 3 further more complex rules which can be found in the laws of indices page.

Make sure you are happy with the following topics before continuing.

**Indices Rule 1: The Multiplication Law**

The **multiplication law** states that when you **multiply similar terms**, you **add the powers** as shown,

a^\textcolor{red}{b} \times a^\textcolor{blue}{c} = a^{\textcolor{red}{b} + \textcolor{blue}{c}}

This **multiplication law** applies to all terms with powers (positive or negative): e.g.

x^{\textcolor{red}{-m}}\times x^\textcolor{blue}{n}=x^{({\textcolor{red}{-m})}\textcolor{blue}{+n}}=x^{\textcolor{blue}{n}\textcolor{red}{-m}}

**Indices Rule 2: The Division Law**

The division law is when you divide similar terms and in doing so, you subtract the powers:

a^\textcolor{red}{b} \div a^\textcolor{blue}{c} = a^{\textcolor{red}{b} - \textcolor{blue}{c}}

The **division law** applies to all numbers, negative numbers and fractional powers,

x^\textcolor{red}{6}\div x^\textcolor{blue}{2}=\dfrac{x^\textcolor{red}{6}}{x^\textcolor{blue}{2}}=x^{\textcolor{red}{6} - \textcolor{blue}{2}} = x^{4}

**Indices Rule 3: Multiple Powers Law**

The **multiple powers law** is when you **raise one power to another**, i.e. the power of a power. When this happens the **powers are multiplied**:

\left(a^\textcolor{red}{b}\right)^\textcolor{limegreen}{c}=a^{\textcolor{red}{b}\textcolor{limegreen}{c}}

A basic example shows how the **multiple powers law** works with numbers:

\left(x^\textcolor{red}{3}\right)^\textcolor{limegreen}{2}=x^{\textcolor{red}{3}\times\textcolor{limegreen}{2}}=x^{6}

**Indices Rule 4: Power 0 Law**

Anything to the **power** 0 = 1

a^\textcolor{blue}{0} = \textcolor{red}{1}

The **power 0 law** applies to everything: 100^\textcolor{blue}{0}=\textcolor{red}{1}, \quad x^\textcolor{blue}{0}=\textcolor{red}{1} \quad \pi^\textcolor{blue}{0}=\textcolor{red}{1}

**Indices Rule 5: Power 1 Law**

Anything to the power 1 is just itself.

\textcolor{red}{a}^\textcolor{blue}{1} = \textcolor{red}{a}

The power 1 law applies to everything: \textcolor{red}{100}^\textcolor{blue}{1}=\textcolor{red}{100}, \quad \textcolor{red}{x}^\textcolor{blue}{1}=\textcolor{red}{x}, \quad \textcolor{red}{\pi}^\textcolor{blue}{1}=\textcolor{red}{\pi}

**Indices Rule 6: The 1 Law**

1 to the **power anything** = 1 e.g.

\textcolor{red}{1}^\textcolor{blue}{x} =\textcolor{red}{1}

This works for any power: \textcolor{red}{1}^\textcolor{blue}{100} =\textcolor{red}{1}, \quad \textcolor{red}{1}^\textcolor{blue}{-5} =\textcolor{red}{1}

**Indices Rule 7: The Fraction Law**

The power of a fraction applies to both the **top and bottom** of the fraction.

\bigg(\dfrac{\textcolor{red}{a}}{\textcolor{blue}{b}}\bigg)^\textcolor{limegreen}{c}= \dfrac{\textcolor{red}{a}^\textcolor{limegreen}{c}}{\textcolor{blue}{b}^\textcolor{limegreen}{c}}

This also applies to mixed factions

\bigg(2\dfrac{\textcolor{red}{3}}{\textcolor{blue}{4}}\bigg)^\textcolor{limegreen}{5} = \bigg(\dfrac{\textcolor{red}{11}}{\textcolor{blue}{4}}\bigg)^\textcolor{limegreen}{5} = \bigg(\dfrac{\textcolor{red}{11}^\textcolor{limegreen}{5}}{\textcolor{blue}{4}^\textcolor{limegreen}{5}}\bigg)

**Roots**

The opposite to taking a power of some number is to take a **root**. Let’s consider **square roots** – these do the opposite of squaring. e.g.

\textcolor{blue}{4}^\textcolor{red}{2} = \textcolor{limegreen}{16}

\sqrt[\textcolor{red}{2}]{\textcolor{limegreen}{16}} = \textcolor{blue}{4}

We also have** cube roots**, 4th roots, 5th roots, etc, for when the powers are higher. e.g.

\textcolor{blue}{2}^\textcolor{red}{3} = \textcolor{limegreen}{8}

\sqrt[\textcolor{red}{3}]{\textcolor{limegreen}{8}} = \textcolor{blue}{2}

These **roots** use the same symbol, just with a different number in the top left to show the power, e.g. \sqrt[3]{8}=2

A **4th root** would be shown by \sqrt[4]{}, and so on.

**Example 1: Multiplication**

Write 5p^2q^3\times3pq^4 in its simplest form.

**[2 marks]**

To simplify this expression, we must recognise that it can be broken up into parts, i.e. we can write

5p^2q^3\times3pq^4=5\times p^2\times q^3\times3\times p\times q^4

Then, we can rearrange the terms, putting like terms together.

5\times 3\times p^2\times p\times q^3\times q^4

Finally using **rule 1** we can multiply the following,

5\times3=15

p^2\times p=p^3

q^3\times q^4=q^7

This gives the final answer to be,

15p^3q^7

**Example 2: Multiplication and Division**

Work out the value of \dfrac{3^4\times3^7}{3^8}.

**[2 marks]**

First we must multiply out the top of the fraction,

3^4\times3^7=3^{4+7}=3^{11}

So, the calculation becomes

\dfrac{3^{11}}{3^8}

Next calculating the division we get,

\dfrac{3^{11}}{3^8}=3^{11-8}=3^3

This gives the final answer to be,

3^3=27.

## Powers and Roots Example Questions

**Question 1:** Work out a^2\times a^3

(Non-calculator)

**[1 mark]**

we know that:

a^a \times a^c = a^{b + c}

so,

a^2 \times a^3 = a^{2+3}

a^2 \times a^3 = a^5

**Question 2:** Work out \sqrt{144}+\sqrt{196}

(Non-calculator)

**[2 marks]**

It is helpful to be able to recognise the first 15 square numbers.

In this case, we can recognise,

12^2=144 and 14^2=196

Hence the calculation is simply,

\sqrt{144}+\sqrt{196}=12+14=26

**Question 3: **Work out (3^2)^3\div3^4

(Non-calculator)

**[2 marks]**

We can rewrite the first term of the expression as,

(3^2)^3=3^2\times3^2\times3^2

The multiplication law tells us that,

3^2\times3^2\times3^2=3^{2+2+2}=3^6

This is the same result as the power-law gives,

(3^2)^3=3^{2\times3}=3^6

Hence, the expression now looks like,

3^6\div3^4

Using the division law we find,

3^6\div3^4=3^{6-4}=3^2=9

**Question 4: **Work out \dfrac{7^5\times7^3}{7^6}

(Non-calculator)

**[2 marks]**

First considering the numerator, the laws of indices tell us,

7^5\times7^3=7^{5+3}=7^8

Thus the expression now is,

\dfrac{7^8}{7^6}

This can be simplified to,

\dfrac{7^8}{7^6}=7^{8-6}=7^2

Hence we are left with a simple calculation of,

7^2=7\times7=49

**Question 5:** Work out the value of 20^1+100^0

(Non-calculator)

**[2 marks]**

We know that,

20^1 = 20

and

100^0 = 1

So we can calculate

20 + 1 = 21