Math 9 · Powers and exponent laws
Counting copies, and the laws that follow
A power is shorthand for multiplying the same number over and over. Every exponent law in this unit comes from writing the copies out once and counting them — which is why none of them has to be memorised blind, and why they sit on the no-calculator Part A of the PAT.
- 1. The words, first
- 2. What a power means
- 3. The exponent laws
- 4. Order of operations with powers
- 5. What costs marks
The words, first
The idea: Base and exponent are the two words everything else is built on.
| Word | What it means |
|---|---|
| Power | A number written with an exponent, like 2⁵. It stands for a repeated multiplication. |
| Base | The number being multiplied: the 2 in 2⁵. |
| Exponent | How many copies of the base are multiplied: the 5 in 2⁵. So 2⁵ = 2 × 2 × 2 × 2 × 2 = 32. |
| Zero exponent | Any power with exponent 0 equals 1, as long as the base is not 0: 7⁰ = 1. |
| Factor, product, quotient | Each number being multiplied is a factor; the answer is the product. The answer to a division is the quotient. |
| Exponent law | A shortcut for combining powers with the same base, such as adding the exponents when multiplying. |
| Order of operations | Brackets, then exponents, then division and multiplication from left to right, then addition and subtraction from left to right. Often remembered as BEDMAS. |
What a power means
The idea: The exponent counts copies of the base, and the base is whatever the exponent is touching. Brackets decide that.
2⁴ = 2 × 2 × 2 × 2 = 16. It is not 2 × 4 = 8; multiplying the base by the exponent is the most common slip in the unit.
Negative bases. The brackets decide what is being multiplied:
(−2)⁴ = (−2)(−2)(−2)(−2) = 16 · −2⁴ = −(2 × 2 × 2 × 2) = −16
In −2⁴ the exponent touches only the 2, and the negative is applied afterwards. An even number of negative factors gives a positive answer and an odd number a negative one, so (−2)³ = −8.
Why 5⁰ = 1. Follow the pattern down: 5³ = 125, 5² = 25, 5¹ = 5. Each step divides by 5, so the next one is 5 ÷ 5 = 1. The same works for any base except 0. Careful with signs again: (−3)⁰ = 1, but −3⁰ = −1.
In a problem. Bacteria that double every hour, starting from one cell, number 2¹⁰ = 1024 after 10 hours.
The exponent laws
The idea: Each law is what you get by writing the copies out and counting them. They only work when the bases are the same.
| Law | Example | Why it works |
|---|---|---|
| Product: aᵐ × aⁿ = aᵐ⁺ⁿ | 2³ × 2⁴ = 2⁷ | three 2s, then four more, is seven 2s |
| Quotient: aᵐ ÷ aⁿ = aᵐ⁻ⁿ | 5⁶ ÷ 5² = 5⁴ | two of the six 5s cancel |
| Power of a power: (aᵐ)ⁿ = aᵐⁿ | (3²)⁴ = 3⁸ | four copies of 3², each holding two 3s |
| Power of a product: (ab)ᵐ = aᵐbᵐ | (2 × 5)³ = 2³ × 5³ | every factor inside gets the exponent |
| Power of a quotient: (a/b)ⁿ = aⁿ/bⁿ | (2/3)³ = 8/27 | top and bottom are each multiplied three times |
Worked example. Evaluate (3² × 3⁴) ÷ 3³.
3² × 3⁴ = 3⁶ → 3⁶ ÷ 3³ = 3³ → 3³ = 27
Simplify first, evaluate last. That is exactly how Part A questions like 7⁹ ÷ 7⁷ are built: simplifying leaves 7² = 49, and nobody needs to know what 7⁹ is.
With a negative base. (−2)⁵ × (−2)² ÷ (−2)⁴ = (−2)³ = −8. The laws work the same; the sign comes from whether the final exponent is odd or even.
Where the laws do not apply. 2³ × 3² has different bases, so evaluate each: 8 × 9 = 72. And the laws are for multiplying and dividing only: 2³ + 2³ = 16, which is 2⁴, not 2⁶. Math 10C keeps all of these laws and lets the exponent be negative or a fraction.
Order of operations with powers
The idea: Powers come straight after brackets. Multiplying, dividing, adding and subtracting all wait for them.
Worked example. Evaluate 3 + 2 × 4².
- Exponent first: 4² = 16.
- Then multiply: 2 × 16 = 32.
- Then add: 3 + 32 = 35.
Adding first gives 5 × 16 = 80; squaring the 2 × 4 gives 3 + 64 = 67. Both are common, and both are wrong.
(−2)³ + 4 × 3² − 5⁰ = −8 + 4 × 9 − 1 = −8 + 36 − 1 = 27
Brackets are finished first. (6² − 4²) ÷ (2³ − 3) = (36 − 16) ÷ (8 − 3) = 20 ÷ 5 = 4.
Two classic traps. 2 × 3² is 2 × 9 = 18, not 6² = 36: the exponent belongs to the 3 alone. And −3² + (−3)² = −9 + 9 = 0: the first power has no brackets, so only the 3 is squared.
What costs marks
The idea: Nearly all of these are about what the exponent is touching.
- Base times exponent. 3⁴ is 81, not 12.
- −2⁴ read as (−2)⁴. Without brackets the negative is not part of the base.
- 5⁰ = 0. It is 1.
- Adding exponents across different bases, or when the powers are added: 2³ + 2³ is not 2⁶.
- Adding instead of multiplying for a power of a power. (3²)⁴ = 3⁸, not 3⁶.
- Giving the exponent to one factor only. (2 × 5)³ is 1000, not 2 × 125 = 250.