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Lesson 2, Topic 6
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# Equations Involving Number Bases

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Example 1

If 132four = Xeight  find X

First convert the left hand side (132four ) to base 10

132four  = 1 x 42 + 3 x 41 + 2 x 40

16 + 12 + 2 = 30ten

Since 30ten = X eight ,  Find the value of X.

Convert 30ten to base eight

30ten = 36eight

30ten = X eight

36eight = X eight

Example 2

43x =  11111two , Find x

Convert both sides to base 10.

(4 × x1 + 3 × x0)ten   = 1 × 24 + 1 × 23+ 1 × 22 + 1 × 21 + 1 × 20

(4x + 3) ten = (16 + 8 + 4 + 2 + 1)ten

(4x + 3) ten  =  (31)ten

4x + 3  =  31

4x = 31 – 3

4x = 28

Divide both sides by 4

$$\frac{4x}{4} = \frac{28}{4}$$

$$\frac{\not{4}x}{\not{4}} = \frac{\not{\!28^{7}}}{\not{4}}$$

Example 3

Find the square root of 1100100two

N.B  First convert to base ten.

Find the square root and convert it to the req. root

1100100two  = 1 x 26 + 1 x 25 + 0 x 24 + 0 x 23 + 0 x 21 + 0 x 20

=  64 + 32 + 4 = 100

$$\scriptsize \sqrt{100_{ten}} = 10_{ten}$$

Convert 10ten to base 2

= 1010two

Answer: Square root of 1100100two = 1010two

Example 4

Given that 11y = 6, find y.

First, convert the left-hand side to base 10. (Note 6 is already in base 10)

11y = 1 x y1 + 1 x y0

11y = y + 1 x 1

11y = y + 1

Since 11y = 6

y + 1 = 6

y = 6 – 1

y = 5

Example 5

Given that 101x = 5, find x

First, convert the left-hand side to base 10. (Note 5 is already in base 10)

1 × x2 + 0 × x1 + 1 × x0  =  5

x2 + 1 = 5

x2 = 5 – 1

x2 = 4

Find the square root of both sides

$$\scriptsize \sqrt{x^2} = \sqrt{4}$$

$$\scriptsize x = \sqrt{4}$$

$$\scriptsize x = 2$$ 