10. Exponents and roots
Rewrite with prime bases. Factor out the smaller power when adding. Power towers resolve top-down.
Core ideas
- Break every base into primes first; only matching prime bases let you combine exponents.
- Adding or subtracting powers? Factor out the SMALLER power: 5^12 - 5^10 = 5^10(5^2 - 1).
- Exponent towers resolve top-down: 2^(3^2) = 2^9, not (2^3)^2.
- Roots are fractional exponents; sqrt(x^2) = |x|, and comparing roots means comparing under a common power.
Worked example 1
Simplify 6^10 / (2^8 * 3^9).
Show solution
Convert to prime bases: 6^10 = (2 * 3)^10 = 2^10 * 3^10. Now subtract exponents on matching bases: 2^(10-8) * 3^(10-9) = 2^2 * 3^1 = 4 * 3 = 12.
Worked example 2
(5^12 - 5^10) / 24 = ?
Show solution
Never subtract giant powers directly; factor out the smaller one. 5^12 - 5^10 = 5^10(5^2 - 1) = 5^10 * 24. Dividing by 24 leaves 5^10. The pattern is always the same: factor out, and the parenthetical becomes a small clean number.
Practice set
Question 1
If 9^x = 27^(x - 2), what is the value of x ?
Get both sides onto the common prime base 3, then equate exponents.
Rewrite with base 3: 9^x = 3^(2x) and 27^(x-2) = 3^(3x-6). Equating exponents: 2x = 3x - 6, so x = 6. Check: 9^6 = 3^12 and 27^4 = 3^12.
Question 2
(2^12 + 2^12) / 2^10 =
Adding two equal powers of 2 raises the exponent by exactly one.
2^12 + 2^12 = 2 * 2^12 = 2^13. Then 2^13 / 2^10 = 2^3 = 8. A common error is to add the exponents and get 2^24 in the numerator.
Question 3
What is the value of the square root of 0.0081 ?
Write 0.0081 as 81 times a power of 10, then take the root of each factor.
0.0081 = 81 * 10^-4. The square root is 9 * 10^-2 = 0.09. Check: 0.09 * 0.09 = 0.0081.
Question 4
What is the value of 2^(3^2) - (2^3)^2 ?
A power tower without parentheses resolves from the top down, so the two terms are not equal.
The tower 2^(3^2) resolves top-down: 3^2 = 9, so it equals 2^9 = 512. The parenthesized version (2^3)^2 = 8^2 = 64. The difference is 512 - 64 = 448. Choosing 0 comes from treating the two expressions as identical.