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}}
This works for fractional powers too. Remember when adding fractions, they must share a common denominator.
x^{\textcolor{red}{\frac{1}{3}}} \times x^{\textcolor{blue}{\frac{1}{6}}}=x^{\textcolor{red}{\frac{2}{6}}+\textcolor{blue}{\frac{1}{6}}} = x^{\textcolor{black}{\frac{3}{6}}} = x^{\textcolor{black}{\frac{1}{2}}}
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.
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
Worksheets and Exam Questions
(NEW) Powers and Roots Exam Style Questions - MME
Level 1-3 New Official MMEDrill Questions
Powers And Square roots - Drill Questions
Learning resources you may be interested in
We have a range of learning resources to compliment our website content perfectly. Check them out below.
