If (0.25)$^y$ = 32, find the value of y.

Analyze the question: We need y such that (0.25)^y = 32. The trick is to write both sides as powers of the same base, 2.

Concept: Convert the decimal to a fraction, then to a power of 2, and use the laws of indices:

• (a^m)^n = a^(mn)
• 1/a^n = a^(-n)

Solution:

Step 1: Write 0.25 as a power of 2. 0.25 = 1/4 = 1/2² = 2⁻²

Step 2: Write 32 as a power of 2. 32 = 2⁵

Step 3: Substitute into the equation. (2⁻²)^y = 2⁵ 2^(-2y) = 2⁵

Step 4: Since the bases are equal, equate the powers. -2y = 5 y = -5/2

Check: (0.25)^(-5/2) = (1/4)^(-5/2) = 4^(5/2) = (√4)⁵ = 2⁵ = 32 ✓

Answer: y = -5/2, which is option A.

Tip: In index equations, always reduce both sides to the same base first. Remember that a fraction like 1/4 becomes a negative power (2⁻²), and watch the sign, since options C and D are tempting if you forget it.