Equations Involving Number Bases

In this topic we will be guiding you through the necessary steps required to solve problems related to Number Bases.

Worked Example 2.6.1:

If 132_four = X_eight  find X

Solution

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

132_four  = 1 × 4² + 3 × 4¹ + 2 × 4⁰

16 + 12 + 2 = 30_ten

Since 30_ten = X _{eight ,}  Find the value of X.

convert 30_ten to base eight

∴ 30_ten = 36_eight

30_ten = X_eight

∴ 36_eight = X_eight

Answer: X = 36

Worked Example 2.6.2:

43_x =  11111_two , Find x

Solution

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

\( \frac{4x}{4} = \frac{28}{4} \)

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

Answer: x = 7

Worked Example 2.6.3:

Find the square root of 1100100_two

Solution

N.B  First convert to base ten.

Find the square root and convert it to the required root

1100100_two  = 1 × 2⁶ + 1 × 2⁵ + 0 × 2⁴ + 0 × 2³ + 0 × 2¹ + 0 × 2⁰

=  64 + 32 + 4 = 100

\( \scriptsize \sqrt{100_{ten}} = 10_{ten} \)

convert 10_ten to base 2

= 1010_two

Answer: Square root of 1100100_two = 1010_two

Worked Example 2.6.4:

Given that 11_y = 6, find y.

Solution

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

11_y = 1 × y¹ + 1 × y⁰

11_y = y + 1 × 1

11_y = y + 1

Since 11_y = 6

∴ y + 1 = 6

y = 6 – 1

y = 5

Worked Example 2.6.5:

Given that 101_x = 5, find x

Solution

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

\( \scriptsize \sqrt{x^2} = \sqrt{4}\)

\( \scriptsize x = \sqrt{4}\)

\( \scriptsize x = 2 \)

Answer: x = 2