Powers and Index Notation
A power is a short way to write repeated multiplication: \(6^4\) means \(6 \times 6 \times 6 \times 6\). In NSW Year 7 maths (NESA outcome MA4‑IND‑C‑01) you learn the words base and index and work with powers of whole numbers.
You will write products in index form and evaluate powers, list powers of 2 and 10, write numbers in expanded form with powers of 10, follow the order of operations when powers appear, find a missing base or index, and see why \((-3)^2\) is positive but \((-3)^3\) is negative. Bacteria doubling every 20 minutes shows how fast powers grow. It is part of Year 7 Whole Numbers.
Every question with a fully worked solution.
- Powers and Index Notation - Video - Intro To Exponents (aka Indices) Watch
Theory
A power is a short way to write a number multiplied by itself: \(6^4\) means four \(6\)s multiplied together. In \(5^3\), \(5\) is the base and \(3\) is the index. This Year 7 maths page shows index form and expanded form, powers of \(10\), negative bases and powers in the order of operations.
Base and index. In \(5^3\), the base is \(5\), the number being multiplied. The small raised \(3\) is the index (also called the exponent or power). It tells you how many times the base is used as a factor: \(5^3 = 5 \times 5 \times 5\). Say “\(5\) to the power of \(3\)”. A power of \(2\) is read squared and a power of \(3\) is read cubed.
Index form and expanded form. \(5^3\) is in index form; \(5 \times 5 \times 5\) is the expanded form. To evaluate, multiply one step at a time: \(4^4 = 16 \times 4 \times 4 = 64 \times 4 = 256\).
Powers of 10. \(10^n\) is \(1\) followed by \(n\) zeros: \(10^3 = 1000\). Place value uses them: \(6450 = 6 \times 10^3 + 4 \times 10^2 + 5 \times 10^1\).
A negative base. Brackets make the whole negative number the base: \((-2)^2 = (-2) \times (-2) = 4\). An even index gives a positive answer and an odd index gives a negative answer.
Order of operations. Brackets first, then powers, then \(\times\) and \(\div\), then \(+\) and \(-\).
Powers to know:
| Index | \(1\) | \(2\) | \(3\) | \(4\) |
|---|---|---|---|---|
| Powers of 2 | \(2\) | \(4\) | \(8\) | \(16\) |
| Powers of 4 | \(4\) | \(16\) | \(64\) | \(256\) |
| Powers of 8 | \(8\) | \(64\) | \(512\) | \(4096\) |
| Powers of 10 | \(10\) | \(100\) | \(1000\) | \(10\,000\) |
How to evaluate an expression with powers
- Work out any brackets.
- Write each power in expanded form and evaluate it.
- Then multiply and divide, then add and subtract.
(i) \(4^3 + 3^4\)
(ii) \((4 + 3)^2\)
| \(4^3 + 3^4\) | \(=\) | \(64 + 81\) |
| \(\) | \(=\) | \(145\) |
| \(7 \times 7\) | \(=\) | \(49\) |
(i) \(145\)
(ii) \((4 + 3)^2 = 7^2 = 49\) (brackets first)
| \(2460\) | \(=\) | \(2000 + 400 + 60\) |
| \(\) | \(=\) | \(2 \times 10^3 + 4 \times 10^2 + 6 \times 10\) |
\(2460 = 2 \times 10^3 + 4 \times 10^2 + 6 \times 10^1\)
(ii) Write \(9 \times 9 \times 9 \times 9\) in index form.
| \(7 \times 7 \times 7\) | \(=\) | \(49 \times 7 = 343\) |
(i) base \(7\), index \(3\), \(7^3 = 343\)
(ii) four \(9\)s: \(9^4\)
(ii) Write \(5080\) in expanded form using powers of \(10\).
| \(5 \times 5 \times 5 \times 5\) | \(=\) | \(625\) |
| \(5080\) | \(=\) | \(5 \times 1000 + 8 \times 10\) |
(i) Powers of \(5\): \(5, 25, 125, 625\), so \(625 = 5^4\).
(ii) No hundreds: \(5080 = 5 \times 10^3 + 8 \times 10^1\)
(i) \(10^2 - 2 \times 3^3\)
(ii) \((5 - 2)^3 + 4^2\)
| \(100 - 2 \times 27\) | \(=\) | \(100 - 54 = 46\) |
| \(27 + 16\) | \(=\) | \(43\) |
(i) \(46\)
(ii) \(43\)
(i) \((-2)^3\)
(ii) \((-2)^4\)
(iii) Is \((-2)^9\) positive or negative?
| \((-2)^3\) | \(=\) | \(4 \times (-2) = -8\) |
| \((-2)^4\) | \(=\) | \(4 \times 4 = 16\) |
(i) \(-8\)
(ii) \(16\)
(iii) \(9\) is odd, so negative
Common pitfalls
Frequently asked questions
What is an index?
The small raised number that says how many times the base is multiplied.
What does 5 cubed mean?
\(5^3 = 5 \times 5 \times 5 = 125\).
What is 10 to the power of 4?
\(10\,000\): \(1\) followed by four zeros.
Is a negative number to an even power positive?
Yes: \((-3)^2 = 9\).
Do powers come before multiplication?
Yes. Brackets, then powers, then × and ÷, then + and −.