Year 8AC9M8N02

Exponent Laws

Establish and apply the exponent laws with positive integer exponents and the zero-exponent, using exponent notation with numbers.

Every exponent law comes from writing the powers out in full. If you ever forget one, expand it and the rule reappears.

Builds onYr 7 · Square numbers and square roots

Why this works

Do not memorise these. Derive them — it takes seconds and it never fails you.

Multiplying: write both out and count the factors.

2^{3} \times 2^{4} = (2 \cdot 2 \cdot 2)(2 \cdot 2 \cdot 2 \cdot 2) = 2^{7}

Seven factors of 2 — and

3 + 4 = 7

. That is why you add exponents when multiplying.

a^{m} \times a^{n} = a^{m+n}

Dividing: write it out and cancel.

\frac{2^{5}}{2^{2}} = \frac{2 \cdot 2 \cdot 2 \cdot 2 \cdot 2}{2 \cdot 2} = 2^{3}

Two factors cancel, leaving three. And

5 - 2 = 3

, so you subtract when dividing.

\frac{a^{m}}{a^{n}} = a^{m-n}

The zero exponent follows for free. Any number divided by itself is 1.

\frac{2^{3}}{2^{3}} = 2^{0} \ \text{ and } \ \frac{8}{8} = 1 \ \therefore \ a^{0} = 1

That last line is worth pausing on. $a^{0} = 1$ is not an arbitrary decision — it is forced by the division law. Any other value would break the pattern.

Worked example

e.g.

\frac{3^{5} \times 3^{2}}{3^{4}}

Deal with the multiplication on top. Same base, so add the exponents. $\frac{3^{5+2}}{3^{4}} = \frac{3^{7}}{3^{4}}$
Now divide. Same base, so subtract the exponents. $3^{7-4}$
Simplify. $3^{3} = 27$

Common mistakes

Wrong

2^{3} \times 2^{4} = 4^{7}

Right

2^{3} \times 2^{4} = 2^{7}

Multiplying the bases as well. The base does not change — you are only counting how many factors there are altogether.

Wrong

2^{3} \times 2^{4} = 2^{12}

Right

2^{3} \times 2^{4} = 2^{7}

Multiplying the exponents instead of adding. Multiplying exponents is for a power of a power, which is a different situation.

Wrong

5^{0} = 0

Right

5^{0} = 1

Anything to the power of zero is 1, not 0. It follows from the division law:

\tfrac{5^{2}}{5^{2}} = 5^{0}

, and any number over itself is 1.

Wrong

3^{2} + 3^{4} = 3^{6}

Right

3^{2} + 3^{4} = 9 + 81 = 90

The laws apply to multiplication and division only. When powers are added, work each one out separately and then add.

Practice

FluencyGet quick and accurate at the method.

2^{4} \times 2^{3}

Answer

2^{7} = 128

\frac{5^{6}}{5^{4}}

Answer

5^{2} = 25

\left(3^{2}\right)^{3}

Answer

3^{6} = 729

7^{0}

Answer

1

\frac{4^{5} \times 4^{2}}{4^{6}}

Answer

4^{1} = 4

10^{3} \times 10^{2}

Answer

10^{5} = 100\,000

\frac{2^{8}}{2^{5}}

Answer

2^{3} = 8

6^{2} \times 6^{0}

Answer

6^{2} = 36

\left(2^{3}\right)^{2} \div 2^{4}

Answer

2^{2} = 4

3^{2} + 3^{3}

Answer

9 + 27 = 36

ReasoningExplain why. Say it in your own words.

Explain, without quoting the rule, why

2^{3} \times 2^{4} = 2^{7}

AnswerWrite both out:

(2\cdot2\cdot2)

and

(2\cdot2\cdot2\cdot2)

. Multiplying places all seven factors of 2 side by side, which is

2^{7}

. Adding the exponents is just counting the factors.

Why must

a^{0}

equal 1 rather than 0?

AnswerFrom the division law,

\tfrac{a^{3}}{a^{3}} = a^{3-3} = a^{0}

. But any non-zero number divided by itself is 1. For the laws to stay consistent,

a^{0}

has to be 1.

Why does

3^{2} + 3^{4}

not simplify to

3^{6}

AnswerThe exponent laws apply to multiplying and dividing powers, not adding them.

3^{2}+3^{4}

9+81=90

, whereas

3^{6}=729

. Expanding shows immediately that they are unrelated.

Someone says

\left(2^{3}\right)^{2} = 2^{5}

. What have they confused, and what is the correct answer?

AnswerThey have added when they should have multiplied.

\left(2^{3}\right)^{2}

means

2^{3} \times 2^{3}

, which is six factors of 2, so

2^{6} = 64

. Adding exponents is for multiplying powers; multiplying exponents is for a power of a power.