Law 1: Multiplication Law

\( \scriptsize a^m \: \times \: a^n = a^{m \:+ \:n} \)

This law applies only when numbers have the same base number.

Note: 2³ and 2⁴ have the same base number, 2, while 5⁴ and 6¹ do not have the same base number because 5 and 6 are different.

e.g.

2³ × 2⁴   =  2³⁺⁴   =  2⁷ ✅

2³ × 3⁴   =   2 × 3³⁺⁴  ❌ (incorrect)

Worked Example 5.1.1 (Using the Multiplication Law):

Simplify the following 

a. 2⁵ × 2⁴

b. c⁵  × c⁴

c. a² × a⁷

d. 5² × 5³ × 5⁴

e. 7² × 7² × 7³

Solution

a. 2⁵ × 2⁴   

= \( \scriptsize 2^{5\:+\:4} \)

= 2⁹

(same base ‘2’)

b. c⁵ × c⁴    

=  \( \scriptsize c^{5\:+\:4} \)

=  c⁹

(same base ‘c’)

c. a² × a⁷    

=  \( \scriptsize a^{2\:+\:7} \)

=  a⁹

(same base ‘a’)

d. 5² × 5³ × 5⁴ 

=  \( \scriptsize 5^{2\:+\:3\:+\:4} \)

=  5⁹

(same base ‘5’)

e. 7² × 7² × 7³   

=  \( \scriptsize 7^{2\:+\:2\:+\:3} \)

= \( \scriptsize 7^{7} \)

(same base ‘7’)

Worked Example 5.1.2

Simplify the following by expanding the terms

a. 2⁵ × 2⁴

b. 5² × 5³ × 5⁴

c. c⁵ × c⁴

d. a² × a⁷

e. 7² × 7² × 7³

Solution

a. 2⁵ × 2⁴   

= (2 × 2 × 2 × 2 × 2)  × (2 × 2 × 2 × 2)  

=  2⁹

b. 5² × 5³ × 5⁴

= (5 × 5) × (5 × 5 × 5 ) × (5 × 5 × 5 × 5)

=  5⁹

c. c⁵ × c⁴ 

= (c × c × c × c × c) × (c × c × c × c)

=  c⁹

d. a² × a⁷   

= (a × a) × (a × a × a × a × a × a × a)

=  a⁹

e. 7² × 7² × 7³ 

= (7 × 7) × (7 × 7) × (7 × 7 × 7)

= 7 × 7 × 7 × 7 × 7 × 7 × 7 

= \( \scriptsize 7^{7} \)

Worked Example 5.1.3:

Simplify the following;

a. \( \scriptsize 2f^2 \: \times \: 3f^3 \)

b. \( \frac{1}{2} \scriptsize y^2 \: \times \: 4y^2 \: \times \: y^3 \)

c. \( \scriptsize 10k^2 \: \times \: 2k^3 \)

Solution

a. \( \scriptsize 2f^2 \: \times \: 3f^3 \\ \scriptsize = 2 \: \times \: 3 \: \times \: f^2 \: \times \: f^3 \\ \scriptsize = 6 \: \times \: f^{2\:+\:3} \\ \scriptsize= 6 \: \times \: f^5 \\ \scriptsize = 6f^5 \)

b. \( \frac{1}{2} \scriptsize y^2 \: \times \: 4y^2 \: \times \: y^3 \\ = \frac{1}{2} \scriptsize \: \times \: y^2 \: \times \: 4 \: \times \:y^2 \: \times \: y^3 \\ = \frac{1}{2} \scriptsize \: \times \: 4 \: \times \: y^2 \: \times \:y^2 \: \times \: y^3 \\ =\scriptsize 2 \: \times \: y^{2+2+3} \\ = \scriptsize 2 \: \times \: y^7 \\ \scriptsize = 2y^7\)

c. \( \scriptsize 10k^2 \: \times \: 2k^3 \\ \scriptsize = 10 \: \times \: 2 \: \times \: k^2 \: \times \: k^3 \\ \scriptsize = 20 \: \times \: k^{2+3} \\ \scriptsize = 20k^5 \)

d. \(\scriptsize y \: \times \: \frac{1}{2}y^2 \: \times \: y^3 \\ \scriptsize = \frac{1}{2} \: \times \: y \: \times \: y^2 \: \times \: y^3 \\ \scriptsize = \frac{1}{2} \: \times \: y^{1+2+3} \\ \scriptsize = \frac{1}{2}y^6\)