Lesson 5, Topic 2
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# Law 2: Division Law

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

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

e.g.

27 Ã· 22

= 27-2

= 25

By expansion:

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

### Example 1

Simplify the following by expansion

a. 76 Ã· 72

b. m9 Ã· m4

c. x7 Ã· x4

d. h10 Ã· h4

e. 1312 Ã·139

f. 18h5 Ã· 9h4

g. 3x2 Ã· 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. $$\scriptsize m^9 \: \div \:m^4 \\= \frac{m^9}{m^4} \\ =\scriptsize \frac{m \: \times \: m \: \times \: m\: \times \: m\: \times \: m\: \times \: m\: \times \: m\: \times \: \not{m}\: \times \: \not{m}}{\not{m} \: \times \: \not{m}\: \times \: \not{m}\: \times \: \not{m}}\\ \scriptsize = m^5$$

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

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 Ã· 72Â

b. m9 Ã· m

c. x7 Ã· x4Â

d. h10 Ã· h4 Â

e. 1312 Ã· 139

f. 18h5 Ã· 9h4 Â

g. 3x2 Ã· 21x7

Solution

a. 76 Ã· 72

= 76-2

= 74

same base

b. m9 Ã· m4

= m9-4

=  m5

same base

c. x7 Ã· x4

= x7-4

= x3

same base

d. h10 Ã· h4

=  h10-4

=  h6

same base

e. 1312 Ã· 13

=  1312-9

=  133

same base

f. 18h5 Ã· 9h4

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

same base

g. 3x2 Ã· 21x7

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

same base

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