What is the value of (7-14 − 7-15)?

For a common base a, when subtracting two powers that have different exponents, factor out the power with the smaller (more negative) exponent: am − an = an × (am−n − 1) for m > n. Equivalently, a-p = 1/ap lets any negative exponent be rewritten as a multiple of another power of the same base.

1. Rewrite 7-14 in terms of 7-15: since −14 = −15 + 1, 7-14 = 71 × 7-15 = 7 × 7-15.

2. Substitute into the original expression: 7 × 7-15 − 7-15.

3. Factor out the common power 7-15: (7 − 1) × 7-15.

4. Simplify the coefficient: 6 × 7-15.

Cross-check by factoring out the larger power instead: 7-14 − 7-15 = 7-14 × (1 − 7-1) = 7-14 × (6/7) = 6 × 7-15 (since 7-14 × 7-1 = 7-15). Both factoring choices agree on the same value: 6 × 7-15.