Exponent rules: products, quotients, powers of powers

About this lesson

Grade 8 › Exponents and scientific notation

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Exponent rules: products, quotients, powers of powers

Discover

$2^3 \cdot 2^4$ means $(2 \cdot 2 \cdot 2)(2 \cdot 2 \cdot 2 \cdot 2)$ . How many factors of 2 is that?

7
12

Key idea

Exponent rules come from counting factors.

For the same base:

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

(add),

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

(subtract),

(a^m)^n = a^{mn}

(multiply). Also

a^n \cdot b^n = (ab)^n

.

Worked example

Simplify $\dfrac{(5^2)^3 \cdot 5^4}{5^7}$ .

(5^2)^3 = 5^6

Power of a power.

5^6 \cdot 5^4 = 5^{10}

Product.

5^{10} \div 5^7 = 5^3

Quotient.

Your turn, guided

Your turn: $3^5 \cdot 3^2 = 3^{?}$ .

7

5 + 2.

Practice ★

$6^{ 2 } \cdot 6^{ 5 } = 6^{?}$ . What is the exponent?
$8^{ 5 } \cdot 8^{ 9 } = 8^{?}$ . What is the exponent?

$3^{ 12 } \div 3^{ 5 } = 3^{?}$ . What is the exponent?

Watch out!

Practice ★★

$\left(9^{ 4 }\right)^{ 4 } = 9^{?}$ . What is the exponent?
$\left(6^{ 3 }\right)^{ 3 } = 6^{?}$ . What is the exponent?

Exit ticket

Three questions to check you have got it.

$6^{ 2 } \cdot 6^{ 5 } = 6^{?}$ . What is the exponent?
$8^{ 5 } \cdot 8^{ 9 } = 8^{?}$ . What is the exponent?
$9^{ 3 } \cdot 9^{ 7 } = 9^{?}$ . What is the exponent?

Summary card

Exponent rules

Multiply: add. Divide: subtract. Power of a power: multiply.

Watch out: count the factors when in doubt.

End of the lesson

Next lesson: Zero and negative exponents Back to the chapter

Answers: Exponent rules: products, quotients, powers of powers

A

$7$
$14$

B

$7$

C

$16$
$9$

D

$7$
$14$
$10$

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