If 1.41_x = \frac{46}{25}_{10}, where x is a positive integer base, find the value of x. — JAMB UTME

If $1.41_x = \frac{46}{25}_{10}$ , where $x$ is a positive integer base, find the value of $x$ .

Cevap: 5

The value of the base

x

is 5.

Expanding

1.41_x

yields

1 + \frac{4}{x} + \frac{1}{x^2}

. Setting this equal to

\frac{46}{25}

gives

\frac{4x + 1}{x^2} = \frac{21}{25}

. Cross-multiplying results in the quadratic equation

21x^2 - 100x - 25 = 0

, which factors as

(21x + 5)(x - 5) = 0

. Since a number base must be a positive integer greater than 4,

x = 5

is the only valid solution.

Expand

1.41_x

using place value powers of

x

1 + \frac{4}{x} + \frac{1}{x^2}

Fractional digits to the right of the radix point represent negative powers of the base (

x^{-1}, x^{-2}, \dots

Equate the expanded form to

\frac{46}{25}

and simplify.

\frac{4x + 1}{x^2} = \frac{21}{25}

Subtracting 1 from both sides isolates the fractional place values.

Cross-multiply and solve the quadratic equation

21x^2 - 100x - 25 = 0

(21x + 5)(x - 5) = 0 \implies x = 5

A base must be a positive integer strictly greater than any individual digit in the number (digits present are 1 and 4).

Conversion of fractional numbers in non-decimal bases to base 10 and solving polynomial equations in unknown bases.

Soru