# Power Of 0

Here you will learn everything you need to know about raising terms to the power of 0 for GCSE & iGCSE maths (Edexcel, AQA and OCR).

Look out for the laws of indices worksheets and exam questions at the end.

## What is raising a value to the power of 0?

Anything raised to the power of 0 is equal to 1.

This lesson is part of our series on the laws of indices. You may find it helpful to start with the main laws of indices lesson for a summary of what to expect and then also work through the following:

## How to raise something to the power of 0

Raising a term to the zeroth power means multiplying the term by itself zero times. This will give 1.

Let’s look at this in three different ways:

1 Division

When we divide something by itself we get 1.

E.g.

\begin{aligned} &5 \div 5=1 \\\\ &\frac{5}{6} \div \frac{5}{6}=1 \\\\ &2 x \div 2 x=1 \end{aligned}

So,

$x^{2} \div x^{2}=1$

Using the division power rule (exponent rule) when we divide two terms with the same base we subtract the powers.

$x^{2} \div x^{2}=x^{2-2}=x^{0}$

So this means that

$x^{0}=1$

21 x the base

$2^{3}=2 \times 2 \times 2$

Which is exactly the same as

$2^{3}=1 \times 2 \times 2 \times 2$

In other words we can think of 23 as telling us to do 1 multiplied by 2 however many times the power tells us, in this case 3 times.

If we continue this pattern we get the following:

\begin{array}{l} 2^{3}=1 \times 2 \times 2 \times 2 \\ 2^{2}=1 \times 2 \times 2 \\ 2^{1}=1 \times 2 \\ 2^{0}=1 \end{array}

In other words we can think of 20 as telling us to do 1 multiplied by 2 however many times the power tells us, in this case 0 times.

So,

$2^{0}=1$

3Comparing other powers of the same base

We could also think about it like this

\begin{aligned} 2^{2}&=1 \times 2 \times 2=4 \\\\ 2^{1}&=1 \times 2=2 \\\\ 2^{0}&=1 \\\\ 2^{-1}&=1 \times \frac{1}{2}=\frac{1}{2} \\\\ 2^{-2}&=1 \times \frac{1}{2} \times \frac{1}{2}=\frac{1}{4} \end{aligned}

Each time the index power decreases by 1, we divide the value by whatever the base is.

In this case we divide by 2.

So,

$2^{0}=1$

Whichever way we look at it, if a term has a zero exponent (it is raised to the power of zero) its value is 1.

## To the power of 0 examples

### Example 1: no coefficient in front of base

Simplify:

$a^{0}$

= 1

Note: Anything to the power of zero is 1.

### Example 2: coefficient in front of base

Simplify:

$6a^{0}$

= 6(1) = 6

We can see that

$6 \times 1=6$

Multiplying anything by 1 leaves it unchanged; this is called the multiplicative identity.

### Example 3: coefficient in front of base and positive powers

Simplify:

$\frac{2 x^{0}}{4^{2}}$

$=\frac{2(1)}{4^{2}}=\frac{2}{16}=\frac{1}{8}$

### Example 4: with negative exponents and decimals

This example uses negative numbers as the indices. It is a good idea to check out our negative indices page to see how to change a negative index on the numerator into a positive number on the denominator.

Simplify:

$2.5 x^{0} \div 2^{-2}$

$=2.5(1) \div \frac{1}{2^{2}}$
$=2.5 \div \frac{1}{4}$
$=2.5 \times 4$
$=10$

### Common misconceptions

• Confusing integer and fractional powers

Raising a term to the power of 2 means we square it

E.g.

$a^{2}=a \times a$

Raising a term to the power of

$\frac{1}{2}$

means we find the square root of it

E.g.

$a^{\frac{1}{2}}=\pm \sqrt{a}$

Raising a term to the power of 3 means we cube it

E.g

$a^{3}=a \times a \times a$

Raising a term to the power of

$\frac{1}{3}$

means we find the cube root of it

E.g.

$a^{\frac{1}{3}}=\sqrt [3] {a}$

• Indices, powers or exponents

Indices can also be called powers or exponents.

• Raising any term or real number to the power of 0 is 1.

Regardless of whether it is a whole number or a decimal or a fraction, or a positive or a negative number, or a rational number (e.g. 4, 0.25, ½ etc.), or an irrational number (e.g. π, √5,  e (Euler’s number) etc.) raising a base number or a base variable to the power 0 will give a value of 1.

Raising algebraic polynomials to the power of 0 is also 1.

Any index that is a non-zero number will not give 1.

### Practice to the power of 0 questions

1. Simplify

x^{0}

0

x

1

x^{0}

This is because any constant or variable raised to the power zero is equal to 1

2. Simplify

8 x^{0}

8

80

1

8x

This is because the variable, x , raised to the power zero equals 1. Therefore, we have 8 lots of 1 , which is 8 .

3. Simplify

2^{2} x^{0}

1

4x

4

x^{4}

This is because the variable, x , raised to the power zero equals 1. The coefficient, 2^2 is equal to 4 . Therefore, we have 4 lots of 1 , which is 4 .

### To the power of 0 GCSE questions

1. Work out the value of

8 ^ {0}

(1 mark)

1

(1)

2. Simplify

\frac { x ^ {0} \times x ^ {5} }{ x ^ {5} \times x ^ {-5} }

(2 marks)

\frac {1 \times x ^ {5} } {1} = x ^ {5}\quad(1)

(2)

3. Simplify

\left(\left(4^{\frac{1}{2}}\right)^{3}\right)^{0}

(1 mark)

1

(1)

## Learning checklist

You have now learned how to:

• Simplify expressions involving the laws of indices
• Calculate with roots, and with integer and fractional indices

## Still stuck?

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