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In order to access this I need to be confident with:

Negative numbers

Multiplication

Division

Converting between fractions and decimals

This topic is relevant for:

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.

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

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

COMING SOONGet your free to the power of 0 worksheet of 20+ questions and answers. Includes reasoning and applied questions.

COMING SOONThis 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:

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.

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.

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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