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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.

Any non-zero value raised to the power of 0 is equal to 1.

Get your free to the power of 0 worksheet of 20+ laws of indices questions and answers. Includes reasoning and applied questions.

DOWNLOAD FREEGet your free to the power of 0 worksheet of 20+ laws of indices questions and answers. Includes reasoning and applied questions.

DOWNLOAD FREE**Power of 0** is part of our series of lessons to support revision on **laws of indices**. You may find it helpful to start with the main laws of indices lesson for a summary of what to expect, or use the step by step guides below for further detail on individual topics. Other lessons in this series include:

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**

Another way to think about this is we can write:

\[2^{3}=2 \times 2 \times 2\]

Which is exactly the same as

\[2^{3}=1 \times 2 \times 2 \times 2\]

In other words we can think of 2^{3} 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 2^{0} 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.

**Note** – Raising 0 to the power of 0 is a complicated problem and its not as easy as saying the answer is 1.

There are many different arguments to this problem and some conclude that the answer should be 1 and others say it should be 0. A lot of mathematicians like to classify 0^{0} as an undefined value and its something that gets explored using much more complicated mathematical techniques.

Simplify:

\[a^{0}\]

= 1

*Note: Anything to the power of zero is 1.*

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.

Simplify:

\[\frac{2 x^{0}}{4^{2}}\]

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

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:

\[2.5 x^{0} \div 2^{-2}\]

\[=2.5(1) \div \frac{1}{2^{2}} \]

\[=2.5 \div \frac{1}{4} \]

\[=2.5 \times 4 \]

\[=10\]

**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 unless the base value is 1.

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 .

1. Work out the value of

8 ^ {0}

**(1 mark)**

Show answer

1

**(1)**

2. Simplify

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

**(2 marks)**

Show answer

x^{0}=1 \text \quad { and } \quad x^{5} \times x^{-5}=x^{0}=1 \quad(1)

\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)**

Show answer

1

**(1)**

You have now learned how to:

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

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