If 2x × 8(1/4) = 2(1/4), then find the value of x.

Concept

When two exponential expressions with the same base are equal, their exponents must be equal: if am = an (with a ≠ 0, 1, -1), then m = n. Two other laws of indices are used here: the power-of-a-power rule (am)n = amn, and the product rule am × an = a(m+n). Before combining terms, every term must first be rewritten with the same base.

Application

1. Start from the given equation: 2x × 8(1/4) = 2(1/4).

2. Express 8 as a power of 2, since 2 × 2 × 2 = 8: 8 = 23.

3. Substitute this into the equation: 2x × (23)(1/4) = 2(1/4).

4. Apply the power-of-a-power rule to simplify the left-hand term: (23)(1/4) = 2(3/4).

5. The equation now reads: 2x × 2(3/4) = 2(1/4).

6. Apply the product rule to combine the exponents on the left, since the bases are the same: 2[x + (3/4)] = 2(1/4).

7. As the bases on both sides are equal, equate the exponents: x + 3/4 = 1/4.

8. Solve for x: x = 1/4 − 3/4 = −2/4 = −1/2.

Cross-check

Substitute x = -1/2 back into the original equation: 2(-1/2) × 8(1/4) = 2(-1/2) × 2(3/4) = 2(-1/2 + 3/4) = 2(1/4), which matches the right-hand side, confirming the result.

Hence x = -1/2.