If 9x – 9x – 1 = 648, then find the value of xx

Concept

For two exponential terms sharing the same base, a factor of the smaller power can be pulled out: ax − ax−1 = ax−1 · (a − 1). Once a single power term is isolated and matched to another power of the same base, the exponents can be equated directly, since am = an implies m = n for a > 0, a ≠ 1.

Applying it here

1. Factor the left-hand side using the identity above with a = 9: 9x − 9x−1 = 9x−1 · 9 − 9x−1 · 1 = 9x−1(9 − 1).

2. So the equation becomes 9x−1(9 − 1) = 648, i.e., 9x−1 × 8 = 648.

3. Divide both sides by 8: 9x−1 = 81.

4. Express 81 as a power of 9: 81 = 92.

5. Equate the exponents (same base 9 on both sides): x − 1 = 2, so x = 3.

6. The question asks for xx, not x — substitute x = 3: xx = 33 = 27.

Cross-check

Substitute x = 3 back into the original equation: 93 − 92 = 729 − 81 = 648, which matches the given value — confirming x = 3 and xx = 27.