Lesson 4, Topic 2
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# Fractional Base Number & Decimal Base Number

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Whenever there is an index with a fractional base, or decimal base number, multiply the base number the same number of times as the index
e.g

$$\left ( \frac{2}{5} \right) ^5 \\ = \left ( \frac{2}{5} \: \times \: \frac{2}{5}\: \times \: \frac{2}{5}\: \times \: \frac{2}{5}\: \times \: \frac{2}{5} \right)$$

(0.2) =  0.2 x 0.2 x 0.2 x 0.2

### Example 1

Change the following to ordinary form expanded

a. $$\left ( \frac{1}{4} \right) ^6$$

b. $$\left ( \frac{1}{2} \right) ^3$$

c. $$\left ( \frac{3}{5} \right) ^2$$

d. (1.5)3

e. (2.7)4

Solution

a. $$\left ( \frac{1}{4} \right) ^6 \\ = \frac{1}{4} \: \times \: \frac{1}{4}\: \times \: \frac{1}{4}\: \times \: \frac{1}{4}\\ \: \times \: \frac{1}{4}\: \times \: \frac{1}{4}$$

b. $$\left ( \frac{1}{2} \right) ^3 \\ = \left ( \frac{1}{2} \: \times \: \frac{1}{2}\: \times \: \frac{1}{2} \right)$$

c. $$\left ( \frac{3}{5} \right) ^2 \\ = \left ( \frac{3}{5} \: \times \: \frac{3}{5} \right)$$

d. (1.5)3 = $$\scriptsize 1.5 \: \times \: 1.5\: \times \: 1.5$$

e. (2.7)4 =  $$\scriptsize 2.7 \: \times \: 2.7\: \times \: 2.7 \: \times \: 2.7$$

### Example 2

Evaluate the following

a. $$\left (\frac{1}{4} \right) ^2$$

b. $$\left ( \frac{1}{2} \right) ^4$$

c. $$\left (\frac{3}{5} \right) ^4$$

d. $$\scriptsize (1.5)^2$$

e. $$\scriptsize (0.2)^3$$

Solution

a. $$\left (\frac{1}{4} \right) ^2 \\ = \frac{1}{4} \: \times \: \frac{1}{4}\\ = \frac{1 \: \times \: 1}{4 \: \times \: 4} \\ = \frac{1}{16}$$

b. $$\left ( \frac{1}{2} \right) ^4 \\ = \frac{1}{2} \: \times \: \frac{1}{2} \: \times \: \frac{1}{2}\: \times \: \frac{1}{2} \\= \frac{1 \: \times \: 1\: \times \: 1\: \times \: 1}{2 \: \times \: 2\: \times \: 2\: \times \: 2}\\ = \frac{1}{16}$$

c. $$\left (\frac{3}{5} \right) ^4 \\= \frac{3}{5} \: \times \: \frac{3}{5} \: \times \: \frac{3}{5}\: \times \: \frac{3}{5}\\= \frac{3 \: \times \: 3 \: \times \: 3\: \times \: 3}{5 \: \times \: 5\: \times \: 5\: \times \: 5} \\ = \frac{81}{625}$$

d. $$\scriptsize (1.5)^2\\ \scriptsize = 1.5 \: \times \: 1.5 \\ \frac{15}{10} \: \times \: \frac{15}{10}\\ = \frac{225}{100}\\ \scriptsize = 2.25$$

e. $$\scriptsize (0.2)^3 \\ \scriptsize = 0.2 \: \times \: 0.2 \: \times \: 0.2 \\ = \frac{2}{10} \: \times \: \frac{2}{10} \: \times \: \frac{2}{10}\\ = \frac{8}{1000}\\ \scriptsize = 0.008$$

Evaluation

1. Write the following in index forms.

a. 4 x 4 x 4 x 4
b. 2 x 2 x 2 x 2
c. 9 x 9 x 9 x 9
d. 11 x 11 x 11 x 11 x 11 x 11 x 11 x 11 x 11
e. 13 x 13 x 13 x 13 x 13

2. Express the following in index form

a. 64
b. 343
c. 243
d. 27 x 8
e. 49 x 343

3. Find the value of the following

a. 4 x 4 x 4
b. 2 x 2 x 2 x 2 x 2
c. 6 x 6 x 6 x 6 x 6
d. 10 x 10 x 10 x 10 x 10 x 10

4. Change the following to ordinary expanded form.

a. $$\left(\frac{2}{5} \right)^5$$

b. $$\left(\frac{1}{6} \right)^3$$

c. $$\left(\frac{2}{7} \right)^7$$

d. $$\left(\scriptsize 0.25 \right)^4$$

e. $$\left(\scriptsize 1.2 \right)^2$$

f. $$\left(\scriptsize 1.3 \right)^3$$

5. Evaluate the following:

a. $$\left(\frac{2}{5} \right)^2$$

b. $$\left(\frac{1}{6} \right)^3$$

c. $$\left(\frac{2}{7} \right)^3$$

d. $$\left(\scriptsize 0.25 \right)^4$$

e. $$\left(\scriptsize 1.2 \right)^5$$

f. $$\left(\scriptsize 1.3 \right)^3$$

error: