Given that 220.32 = x, 220.89 = y, and xz = y8, the value of z is closest to:

Concept

For the same positive base a, if am = u and an = v, then up = vq can be rewritten as amp = anq.

When a ≠ 1, equal powers with the same base have equal exponents; therefore mp = nq.

Application

1. Substitute x = 220.32 and y = 220.89 into xz = y8:

   (220.32)z = (220.89)8

2. Apply the power-of-a-power rule, (am)n = amn:

   220.32z = 220.89 × 8 = 227.12

3. Because the bases are identical and 22 ≠ 1, equate the exponents:

   0.32z = 7.12

4. Solve for z:

   z = 7.12 ÷ 0.32 = 22.25

Cross-check

Substitution confirms the result: 0.32 × 22.25 = 7.12, while 0.89 × 8 = 7.12. Hence both sides of xz = y8 have the same exponent of 22. Therefore, z is closest to 22.25.