Law 2: Division Law

\( \scriptsize a^m \div a^n = a^{m \: – \: n} \)

Similar to the Multiplication Law, the Division Law applies only when we have numbers with the same base numbers. 

e.g.

2⁷ ÷ 2² 

= 2^7-2

= 2⁵

By expansion:

= \(\scriptsize 2^7 \: \div \:2^2 \\= \frac{2^7}{2^2} \\ = \frac{2 \: \times \: 2 \: \times \: 2\: \times \: 2\: \times \: 2\: \times \: 2\: \times \: 2}{2 \: \times \: 2}\\ = \frac{2 \: \times \: 2 \: \times \: 2\: \times \: 2\: \times \: 2\: \times \: \not{2}\: \times \: \not{2}}{\not{2}\: \times \: \not{2}}\\ \scriptsize = 2^5\)

Worked Example 5.2.1:

Simplify the following by expansion

a. 7⁶ ÷ 7²

b. m⁹ ÷ m⁴

c. x⁷ ÷ x⁴

d. h¹⁰ ÷ h⁴

e. 13¹² ÷13⁹

f. 18h⁵ ÷ 9h⁴

g. 3x² ÷ 21x⁷

Solution

a. \(\scriptsize 7^6 \: \div \:7^2 \\= \frac{7^6}{7^2} \\ = \frac{7 \: \times \: 7 \: \times \: 7\: \times \: 7\: \times \: \not{7}\: \times \: \not{7}}{\not{7} \: \times \: \not{7}}\\ \scriptsize = 7^4\)