To simplify 256^(5/8), express 256 as a power of 2. Since 256 = 2^8, we have: 256^(5/8) = (2^8)^(5/8) = 2^(8 Ć 5/8) = 2^5 = 32. The answer is 32.
256^(5/8) = 32.
256 = 2^8 (key step).
(2^8)^(5/8) = 2^(8Ć5/8) = 2^5 = 32.
Law used: (a^m)^n = a^(mn).
Fractional exponent p/q means: take the q-th root, then raise to power p.
Step 1: Express 256 as a power of 2. 256 = 2 Ć 2 Ć 2 Ć 2 Ć 2 Ć 2 Ć 2 Ć 2 = 2^8
Step 2: Substitute. 256^(5/8) = (2^8)^(5/8)
Step 3: Apply law of exponents: (a^m)^n = a^(mĆn) = 2^(8 Ć 5/8) = 2^(40/8) = 2^5
Step 4: Calculate 2^5. 2^5 = 32
Answer: 256^(5/8) = 32
A fractional exponent a^(p/q) means: a^(p/q) = (a^(1/q))^p = (q-th root of a)^p
For 256^(5/8): ā¢ 256^(1/8) = 8th root of 256 = 2 (since 2^8 = 256) ā¢ (256^(1/8))^5 = 2^5 = 32
More examples: ā¢ 64^(2/3) = (4Ā³)^(2/3) = 4Ā² = 16 Or: 64^(1/3) = 4; 4Ā² = 16
ā¢ 81^(3/4) = (3^4)^(3/4) = 3^3 = 27 Or: 81^(1/4) = 3; 3Ā³ = 27
ā¢ 32^(3/5) = (2^5)^(3/5) = 2^3 = 8
Key law of exponents used: (a^m)^n = a^(mn) a^(p/q) = (a^p)^(1/q) = (a^(1/q))^p
256^(5/8) = 32. Since 256 = 2^8: (2^8)^(5/8) = 2^(8Ć5/8) = 2^5 = 32.
Use the law (a^m)^n = a^(mn). Express the base as a perfect power, then simplify. For 256^(5/8): write 256 = 2^8, then (2^8)^(5/8) = 2^5 = 32.
a^(p/q) = (q-th root of a)^p. For 256^(5/8): take the 8th root of 256 (= 2), then raise to power 5: 2^5 = 32.
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