Powers, Roots & Indices (Edexcel GCSE Maths: Foundation)

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Jamie W

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30 April 2024

Powers & Roots

What are powers (indices)?

Powers (or indices) are the small 'floating' values that are used when a number is multiplied by itself repeatedly
- 6¹ means 6
- 6² means 6 × 6
- 6³ means 6 × 6 × 6
The big number at the bottom is called the base
The small number that is raised is called the index, power, or exponent
Any non-zero number to the power of 0 is equal to 1
- 3⁰ = 1
Any number to the power of 1 is equal to itself
- 3¹=3

What are square roots?

Roots are the reverse of powers
A square root of 25 is a number that when squared equals 25
- The two square roots are 5 and -5
Every positive number has two square roots
- One is positive and one is negative
- Negative numbers do not have a square root
The notation $\sqrt{}$ refers to the positive square root of a number
- $\sqrt{25} = 5$
- You can show both roots at once using the plus or minus symbol ±
- Square roots of 25 are $\pm \sqrt{25} = \pm 5$

What are cube roots?

A cube root of 125 is a number that when cubed equals 125
- The cube root of 125 is 5
  - Unlike square roots, each number only has one cube root
- Every positive and negative number has a cube root
- The notation $\sqrt[3]{}$ refers to the cube root of a number
  - $\sqrt[3]{125} = 5$

What are n^th roots?

An nth root of a number is a value that when raised to the power n equals the original number
- 3⁵=243 therefore 3 is a 5th root of 243
If n is even, there will be a positive and negative nth root
- The 6th roots of 64 are 2 and -2
- The notation $\sqrt[n]{}$ refers to the positive nth root of a number
  - $\sqrt[6]{64} = 2$
- Negative numbers do not have an nth root if n is even
If n is odd then there will only be one nth root
- Every positive and negative number will have an nth root
  - The 5th root of -32 is -2

How do I estimate a root?

You can estimate roots by finding the closest integer roots
- To estimate
  - We know that $\sqrt{16} = 4$ and $\sqrt{25} = 5$
  - So $\sqrt{20}$ must be between 4 and 5

What are reciprocals?

The reciprocal of a number is the number that you multiply it by to get 1
- The reciprocal of 2 is $\frac{1}{2}$
- The reciprocal of $\frac{1}{4}$ is 4
- The reciprocal of $\frac{3}{2}$ is $\frac{2}{3}$
The reciprocal of a number can be written as an index of -1
- 5^-1 is the reciprocal of 5, so $\frac{1}{5}$
This can be extended to other negative indices
- 5^-2 means the reciprocal of 5², so $\frac{1}{5^{2}}$ or $\frac{1}{25}$

Examiner Tip

If your calculator shows "Math Error" or similar when finding a square root, this is probably because you have accidentally entered a negative number!

Laws of Indices

What are the laws of indices?

Index laws are rules you can use when doing operations with powers
- They work with both numbers and algebra

Law	Description	How it works
$a^{1} = a$	Anything to the power of 1 is itself	$6^{1} = 6$
$a^{0} = 1$	Anything to the power of 0 is 1	$8^{0} = 1$
$a^{m} \times a^{n} = a^{m + n}$	To multiply indices with the same base, add their powers	$4^{3} \times 4^{2} = (4 \times 4 \times 4) \times (4 \times 4) = 4^{5}$
$a^{m} \div a^{n} = \frac{a^{m}}{a^{n}} = a^{m - n}$	To divide indices with the same base, subtract their powers	$7^{5} \div 7^{2} = \frac{7 \times 7 \times 7 \times 7 \times 7}{7 \times 7} = 7^{3}$
${(a^{m})}^{n} = a^{m n}$	To raise indices to a new power, multiply their powers	${(14^{3})}^{2} = (14 \times 14 \times 14) \times (14 \times 14 \times 14) = 14^{6}$
${(a b)}^{n} = a^{n} b^{n}$	To raise a product to a power, apply the power to both numbers, and multiply	${(3 \times 4)}^{2} = 3^{2} \times 4^{2}$
$a^{- 1} = \frac{1}{a}$ $a^{- n} = \frac{1}{a^{n}}$	A negative power is the reciprocal	$6^{- 1} = \frac{1}{6}$ $11^{- 3} = \frac{1}{11^{3}}$

How do I deal with different bases?

Index laws only work with terms that have the same base
- $2^{3} \times 5^{2}$ cannot be simplified using index laws
Sometimes expressions involve different base values, but one is related to the other by a power
- e.g. $2^{5} \times 4^{3}$
You can use powers to rewrite one of the bases
- $2^{5} \times 4^{3} = 2^{5} \times {(2^{2})}^{3}$
- This can then be simplified more easily, as the two bases are now the same
- $2^{5} \times {(2^{2})}^{3} = 2^{5} \times 2^{6} = 2^{11}$

Worked example

(a)

Find the value of

x

when

6^{10} \times 6^{x} = 6^{2}

Using the law of indices

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

we can rewrite the left hand side

6^{10} \times 6^{x} = 6^{10 + x}

So the equation is now

6^{10 + x} = 6^{2}

Comparing both sides, the bases are the same, so we can say that

10 + x = 2

Subtract 10 from both sides

x = - 8

(b)

Find the value of

n

when

5^{n} \div 5^{4} = 5^{6}

Using the law of indices

a^{m} \div a^{n} = a^{m - n}

we can rewrite the left hand side

5^{n} \div 5^{4} = 5^{n - 4}

So the equation is now

5^{n - 4} = 5^{6}

Comparing both sides, the bases are the same, so we can say that

n - 4 = 6

Add 4 to both sides

n = 10

(c)

Without using a calculator, find the value of

2^{- 4}

Using the law of indices

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

we can rewrite the expression

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

2^{4} = 2 \times 2 \times 2 \times 2 = 16

so we can rewrite the expression

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

\frac{1}{16}

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Powers, Roots & Indices (Edexcel GCSE Maths: Foundation)

Revision Note

What are powers (indices)?

What are square roots?

What are cube roots?

What are nth roots?

How do I estimate a root?

What are reciprocals?

What are the laws of indices?

How do I deal with different bases?

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Author: Jamie W

What are n^th roots?