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

b. \(\scriptsize m^9 \: \div \:m^4 \\= \frac{m^9}{m^4} \\ =\scriptsize \frac{m \: \times \: m \: \times \: m\: \times \: m\: \times \: m\: \times \: \not{m}\: \times \: \not{m}\: \times \: \not{m}\: \times \: \not{m}}{\not{m} \: \times \: \not{m}\: \times \: \not{m}\: \times \: \not{m}}\\ \scriptsize = m^5\)

c. \(\scriptsize x^7 \: \div \:x^4 \\= \frac{x^7}{x^4} \\ = \frac{x \: \times \: x \: \times \: x\; \times \: \not{x}\: \times \:\not{ x}\: \times \: \not{x}\: \times \: \not{x}}{\not{x} \: \times \: \not{x}\: \times \: \not{x}\: \times \: \not{x}}\\ \scriptsize = x^3\)

d. \(\scriptsize h^{10} \: \div \:h^4 \\= \frac{h^{10}}{h^4} \\ = \frac{h \: \times \: h \: \times \: h \: \times \: h \: \times \: h \: \times \:h\: \times \: \not{h} \: \times \:\not{ h}\: \times \: \not{h}\: \times \: \not{h}}{\not{h} \: \times \: \not{h}\: \times \: \not{h}\: \times \: \not{h}}\\ \scriptsize = h^6\)

e. \(\scriptsize 13^{12} \: \div \:13^9 \\= \frac{13^{12}}{13^9} \\ = \frac{13 \: \times \: 13 \: \times \: 13 \: \times \: \not{13}\\ \: \times \: \not{13} \: \times \:\not{13} \: \times \: \not{13} \: \times \: \not{13}\\\: \times \: \not{13}\: \times \: \not{13} \: \times \: \not{13} \: \times \: \not{13}}{\not{13} \: \times \: \not{13}\: \times \: \not{13}\: \times \: \not{13}\\ \: \times \: \not{13} \: \times \: \not{13}\: \times \: \not{13}\: \times \: \not{13} \: \times \: \not{13}}\\ \scriptsize = 13^3\)

f. \(\scriptsize 18 h^{5} \: \div \: 9 h^4 \\= \frac{18h^{5}}{9h^4} \\ = \frac{18 \: \times \: h \: \times \: \not{h} \: \times \: \not{h} \: \times \: \not{h} \: \times \: \not{h} }{9 \: \times \: \not{h} \: \times \: \not{h}\: \times \: \not{h}\: \times \: \not{h}}\\ \scriptsize = 2h\)

g. \( \scriptsize 3x^2 \: \div \: 21x^7 \\ = \frac{3x^2}{21x^7} \\ = \frac{3 \: \times \: \not{x} \: \times \: \not{x}}{21 \: \times \: x \: \times \: x \: \times \: x \: \times \: x \: \times \: x\: \times \: \not{x} \: \times \: \not{x}} \\ = \frac{1}{7x^5} \)

Example 2

Simplify the following by the law of indices

a. 7⁶ ÷ 7² 

b. m⁹ ÷ m

c. x⁷ ÷ x⁴ 

d. h¹⁰ ÷ h⁴  

e. 13¹² ÷ 13⁹

f. 18h⁵ ÷ 9h⁴  

g. 3x² ÷ 21x⁷

Solution

a. 7⁶ ÷ 7² 

= 7^6-2 

= 7⁴

same base

b. m⁹ ÷ m⁴

= m^9-4

=  m⁵

same base

c. x⁷ ÷ x⁴ 

= x^7-4 

= x³

same base

d. h¹⁰ ÷ h⁴  

=  h^10-4  

=  h⁶

same base

e. 13¹² ÷ 13⁹ 

=  13^12-9  

=  13³

same base

f. 18h⁵ ÷ 9h⁴  

\(= \frac{18h^8}{9h^4}\\ = \scriptsize 2h^{5\:-\:4}\\ = \scriptsize 2h^1 \\= \scriptsize 2h  \)

g. 3x² ÷ 21x⁷

=   \( \frac{3}{21} \scriptsize x^2 \: \div \: x^7 \\=  \frac{3}{21} \scriptsize x^{2 \: -\: 7}\\ = \frac{1}{7} \scriptsize x^{-5} \)