Multiplication & Division of Square Roots

1. For Equations Involving Multiplication:

i. Split the square roots if both numbers are perfect squares e.g.

\(\scriptsize \sqrt{4 \: \times \: 16} \\ \scriptsize = \sqrt{4} \: \times \: \sqrt{16} \\ \scriptsize = 2 \: \times \: 4 \\ \scriptsize = 8 \)

ii.  Multiply directly, if one or both numbers are not perfect squares e.g.

\(\scriptsize \sqrt{2 \: \times \: 50}\\ \scriptsize = \sqrt{100}\\ \scriptsize = 10\)

This is because 2 and 50 are not perfect squares, but their product 100 is.

2. For Equations Involving Division:

i. Split the square roots, if both numbers are perfect squares e.g.

\(\sqrt{ \frac{144 }{81}}\\= \frac{\sqrt{144}}{\sqrt{81}} \\ = \frac{12}{9} \\ \ = \frac{4}{3} \\ \scriptsize = 1 \normalsize \frac{1}{3} \)

ii. Divide directly if one or both numbers are not perfect squares e.g.

\(\scriptsize \sqrt{128 \: \div \: 2}\\ \scriptsize =\sqrt{64} \\ \scriptsize = 8 \)

Worked Example 3.1.1:

Evaluate the following;

a. \(\scriptsize \sqrt{24 + 1}\)

b. \(\scriptsize \sqrt{50 \: – \: 25}\)

c. \(\scriptsize \sqrt{200 \: – \: 4}\)

d. \(\scriptsize \sqrt{10 + 20 + 30 + 50 \: – \: 10}\)

e. \(\scriptsize \sqrt{64 \: \times \: 81}\)

f. \(\sqrt{ \frac{400}{4}}\)

g. \(\scriptsize \sqrt{4 \: \times \: 6 \: \times \: 6}\)

h. \(\scriptsize \sqrt{ 9 \: \times \: 8 \: \div \: 2}\)

i. \(\scriptsize \sqrt{ 27 \: \times \: 3 \: \div \: 9}\)

Solution

a. \(\scriptsize \sqrt{24 \:+\:1} \\ \scriptsize = \sqrt{25}\\ \scriptsize = 5\)

b. \(\scriptsize \sqrt{50 \: – \: 25} \\ \scriptsize = \sqrt{25}\\ \scriptsize = 5\)

c. \(\scriptsize \sqrt{200 \: – \: 4} \\ \scriptsize = \sqrt{196} \\ \scriptsize = 14 \)

d. \(\scriptsize \sqrt{10\: + \: 20\: +\: 30 \:+ \: 50 \: – \: 10}\\ \scriptsize =\sqrt{110\: – \:10}\\ \scriptsize = \sqrt{100} \\ \scriptsize= 10 \)

e. \(\scriptsize \sqrt{64 \: \times \: 81} \\ \scriptsize = \sqrt{64} \: \times \sqrt{81} \\ \scriptsize = 8 \: \times \: 9 \\ \scriptsize = 72\)

f. \(\sqrt{ \frac{400}{4}}\\ \scriptsize = \sqrt{100} \\ \scriptsize = 10 \)

or

\(\sqrt{ \frac{400}{4}}\\ = \frac{\sqrt{400}}{\sqrt{4}}\\ = \frac{20}{2} \\ \scriptsize = 10\)

g. \(\scriptsize \sqrt{4 \: \times \: 6 \: \times \: 6}\)

= \(\scriptsize \sqrt{4 \: \times \: 36}\\ \scriptsize = \sqrt{4} \: \times \: \sqrt{36} \)

= \( \scriptsize 2 \: \times \: 6 = 12 \)

h. \(\scriptsize \sqrt{ 9 \: \times \: 8 \: \div \: 2}\\ \scriptsize = \sqrt{9 \: \times \: 4} \)

= \(\scriptsize \sqrt{9} \: \times \: \sqrt{4}\)

= \( \scriptsize 3 \: \times \: 2 = 6 \)

i. In this case apply BODMAS (Do the division first)

\(\scriptsize \sqrt{ 27 \: \times \: 3 \: \div \: 9}\\ \scriptsize = \sqrt{ \normalsize \frac{27 \: \times \: 3 }{9}} \\ \scriptsize = \sqrt{3 \: \times \: 3} \)

\( \scriptsize = \sqrt{9} = 3 \)