Find the value of x that satisfies the exponential equation \sqrt{\frac{125^{x+1}}{5^{x-1}… — JAMB UTME

Find the value of

x

that satisfies the exponential equation

\sqrt{\frac{125^{x+1}}{5^{x-1}}} = 25^{x-1}

Answer: 4

The value of

x

is 4.

Converting all terms to base 5 yields

5^{x+2}

on the left-hand side and

5^{2x-2}

on the right-hand side. Setting the exponents equal gives

x + 2 = 2x - 2

, which solves to

x = 4

Express all terms with a common base of 5

125 = 5^3

and

25 = 5^2

Converting terms to prime base 5 allows the application of standard laws of indices.

Simplify the fraction inside the square root

\frac{5^{3x+3}}{5^{x-1}} = 5^{(3x+3) - (x-1)} = 5^{2x+4}

Subtract the denominator exponent from the numerator exponent when dividing like bases.

Apply the square root as a fractional exponent

\sqrt{5^{2x+4}} = (5^{2x+4})^{1/2} = 5^{x+2}

Taking the square root of a power is equivalent to multiplying the exponent by 1/2.

Equate the simplified exponents of both sides

x + 2 = 2x - 2

With identical bases of 5 on both sides, the exponents must be equal.

Solve the linear equation for x

x = 4

Rearranging terms gives

2x - x = 2 + 2

, which yields

x = 4

Indices and Laws of Indices

Question