If 4^{2x + 1} \times 8^{1 - x} = 32^{x - 1}, find the value of x. — JAMB UTME

If $4^{2x + 1} \times 8^{1 - x} = 32^{x - 1}$ , find the value of $x$ .

The value of

x

\frac{5}{2}

Converting all terms to base

2

yields

4^{2x+1} = 2^{4x+2}

8^{1-x} = 2^{3-3x}

, and

32^{x-1} = 2^{5x-5}

. Applying the law

a^m \times a^n = a^{m+n}

gives

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

. Equating exponents gives

x+5 = 5x-5

, which solves cleanly to

x = \frac{5}{2}

Express all terms with a common prime base (base 2)

4 = 2^2

8 = 2^3

, and

32 = 2^5

To apply the laws of indices, all terms must share the same base.

Apply the power of a power law

(a^m)^n = a^{mn}

to each term

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

Multiplying the inner exponent by the outer exponent simplifies nested powers.

Apply the multiplication law

a^m \times a^n = a^{m+n}

on the left-hand side

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

Powers with the same base being multiplied require adding their exponents.

Equate the exponents and solve for

x

x + 5 = 5x - 5 \implies 5 + 5 = 5x - x \implies 10 = 4x \implies x = \frac{10}{4} = \frac{5}{2}

Since the bases are equal and non-zero, their exponents must be equal.

Solving exponential equations by expressing numbers in terms of a common prime base and applying laws of indices.

Question