1. **State the problem:** Solve the equation $6^a - 6^b = 1260$ for $a$ and $b$. 2. **Recall the properties of exponents:** For any real numbers $x$, $y$, and base $c > 0$, $c^x - c^y$ can sometimes be factored or simplified if $x$ and $y$ are related. 3. **Rewrite the equation:** Assume without loss of generality that $a > b$. Then we can write: $$6^a - 6^b = 6^b(6^{a-b} - 1) = 1260$$ 4. **Set $m = a - b$ and $n = b$:** Then the equation becomes: $$6^n(6^m - 1) = 1260$$ 5. **Factor 1260 into prime factors:** $$1260 = 2^2 \times 3^2 \times 5 \times 7$$ 6. **Try possible values for $6^n$:** Since $6^n$ must divide 1260, possible values are powers of 6 that divide 1260: - $6^0 = 1$ - $6^1 = 6$ - $6^2 = 36$ - $6^3 = 216$ (does not divide 1260) 7. **Test $6^n = 1$:** $$6^m - 1 = 1260 \implies 6^m = 1261$$ 1261 is not a power of 6, so discard. 8. **Test $6^n = 6$:** $$6^m - 1 = \frac{1260}{6} = 210 \implies 6^m = 211$$ 211 is not a power of 6, discard. 9. **Test $6^n = 36$:** $$6^m - 1 = \frac{1260}{36} = 35 \implies 6^m = 36$$ Since $6^2 = 36$, we have $m = 2$. 10. **Find $a$ and $b$:** Recall $m = a - b = 2$ and $n = b$ with $6^n = 36$, so $b = 2$. Therefore, $a = b + m = 2 + 2 = 4$. 11. **Check the solution:** $$6^4 - 6^2 = 1296 - 36 = 1260$$ Correct. **Final answer:** $$a = 4, \quad b = 2$$