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:

  • Laws of indices
  • Multiplying indices
  • Dividing indices
  • Fractional indices
  • Negative indices

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

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 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, 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}

Show answer

1

2) Simplify

8 x ^ {0}

Show answer

8

3) Simplify

2 ^ {2}  x ^ {0}

Show answer

4

To the power of 0 GCSE questions

1. Work out the value of

8 ^ {0}

(1 mark)

Show answer

1

(1)

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

(1 mark)

Show answer

1

(1)

To the power of 0 worksheets

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

Learning checklist

You have now learned how to:

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

The next lessons are

  • Laws of indices
  • Multiplying indices
  • Dividing indices
  • Negative indices
  • Fractional indices

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