Zero Index
Any number except zero raised to the power 0 equals 1. In NSW Year 8 maths you see why by dividing a power by itself in two ways: \(4^3 \div 4^3 = 1\), and the division law also gives \(4^{3-3} = 4^0\) (NESA outcome MA4‑IND‑C‑01).
You will evaluate expressions such as \(5 \times 8^0 \times 6 \times 3^0\) and \((7 \times 4)^0 + 7 \times 4^0\), noticing that a zero index belongs only to the base it sits on. You will cancel equal powers in fractions, combine the zero index with the other laws and simplify terms like \(2m^0\), a first look at Stage 5 work. Place value uses it too, because the ones column is \(10^0\). It is part of Year 8 Algebra.
Every question with a fully worked solution.
- Zero Index - Video - Zero Index - Example 2 Watch
Theory
Any number or pronumeral (except \(0\)) raised to the power \(0\) equals \(1\): \(a^0 = 1\). The index belongs only to the base it is written on: \(4x^0 = 4 \times 1 = 4\), but \((3x)^0 = 1\). This Year 8 maths page shows why, and how to use it.
The zero index. \(a^0 = 1\) for every base \(a\) except \(0\).
Why. Divide a power by itself two ways: \(9^2 \div 9^2 = 81 \div 81 = 1\), and by the division law \(9^2 \div 9^2 = 9^{2-2} = 9^0\). So \(9^0 = 1\). The base can be a fraction, a negative number or a pronumeral.
Where the index sits. A zero index belongs only to its own base: \(8 \times 9^0 = 8\), and \(4x^0 \times 2y^0 = 4 \times 1 \times 2 \times 1 = 8\). Brackets make the whole product the base: \((8 \times 9)^0 = 1\) and \((3x)^0 = 3^0 \times x^0 = 1\).
In a fraction. The same power on the top and the bottom cancels to \(1\), since \(a^k \div a^k = a^0\).
Place value. The ones column is \(10^0 = 1\): \(6 \times 10^2 + 5 \times 10^1 + 3 \times 10^0 = 653\).
The zero index (a not zero):
How to simplify with a zero index
- Find what each zero index sits on (one base, or a whole bracket).
- Replace that power by \(1\).
- Finish with the other index laws and the order of operations.
| \((3x)^0\) | \(=\) | \(3^0 \times x^0\) |
| \(\) | \(=\) | \(1 \times 1 = 1\) |
\(1\)
| \(4x^0 \times 2y^0\) | \(=\) | \((4 \times 2) \times x^0 \times y^0\) |
| \(\) | \(=\) | \(8 \times 1 \times 1 = 8\) |
\(8\)
(i) \(9 \times 12^0\)
(ii) \((9 \times 12)^0\)
| \(9 \times 12^0\) | \(=\) | \(9 \times 1 = 9\) |
| \((9 \times 12)^0\) | \(=\) | \(108^0 = 1\) |
(i) \(9\)
(ii) \(1\)
| \(18 \div 6\) | \(=\) | \(3\) |
| \(5^3 \div 5^3\) | \(=\) | \(5^0 = 1\) |
\(3\). Check: \(2250 \div 750 = 3\).
| \((10^3)^0\) | \(=\) | \(1\) |
| \(5 \times 10^0\) | \(=\) | \(5\) |
| \(1 \times 100 \div 5\) | \(=\) | \(20\) |
\(20\)
(i) \(3m^0 + 2m - 5n^0\)
(ii) \((2x)^3 \times 5x^0 \div 4x^3\)
| \(3m^0 + 2m - 5n^0\) | \(=\) | \(3 + 2m - 5\) |
| \(\) | \(=\) | \(2m - 2\) |
| \(8x^3 \times 5 \div 4x^3\) | \(=\) | \(40x^3 \div 4x^3\) |
| \(\) | \(=\) | \(10\) |
(i) \(2m - 2\)
(ii) \(10\)
Common pitfalls
Frequently asked questions
What is any number to the power 0?
\(1\), as long as the number is not \(0\): \(25^0 = 1\).
Why is a number to the power 0 equal to 1?
Because \(a^n \div a^n = 1\), and the division law gives \(a^{n-n} = a^0\).
What is 5x to the power 0?
It depends on brackets: \((5x)^0 = 1\), but \(5x^0 = 5\).
Does the zero index work for fractions and negatives?
Yes: a fraction such as three eighths to the power \(0\) is \(1\), and \((-7)^0 = 1\).
Where is the zero index used?
In place value: the ones column is \(10^0 = 1\).