Fix the mistake: Exponential Functions (Extension of Exponents, Exponential Equations, and Graphs)
Extension of Exponents
Extend exponents to rational numbers using a⁰=1、a⁻ⁿ=1/aⁿ、a^(m/n)=(ⁿ√a)ᵐ。
Laws of Exponents (a>0, b>0)
- aˢ×aᵗ=aˢ⁺ᵗ / aˢ÷aᵗ=aˢ⁻ᵗ / (aˢ)ᵗ=aˢᵗ
Example ①
(1) 8^(2/3) (2) 27^(−1/3) (3) (1/4)^(−3/2)
Solution:(1) 4 (2) 1/3 (3) 8
(1) 8^(2/3) (2) 27^(−1/3) (3) (1/4)^(−3/2)
Solution:(1) 4 (2) 1/3 (3) 8
Example ②
Simplify a^(5/6) × a^(−1/3) ÷ a^(1/2).
Solution:Sum of exponents: 5/6−1/3−1/2 = 0 → 1
Simplify a^(5/6) × a^(−1/3) ÷ a^(1/2).
Solution:Sum of exponents: 5/6−1/3−1/2 = 0 → 1
Key Point
- a^(m/n)=(ⁿ√a)ᵐ: taking the nth root first makes the calculation easier
Practice Problems
- Calculate (1) 16^(3/4) (2) 32^(−2/5) (3) (9/4)^(3/2).
- Simplify (a^(1/2)×b^(2/3))^6.
- Calculate 2^(1/3)×4^(1/3)×8^(−1/2).
Solutions and Explanations
- Answer:(1) 8 (2) 1/4 (3) 27/8
- Answer:a³b⁴
- Answer:2^(1/3+2/3−3/2)=2^(1−3/2)=2^(−1/2)=1/√2=√2/2
Explanation:4^(1/3)=2^(2/3)、8^(−1/2)=2^(−3/2)。Sum of exponents=1/3+2/3−3/2=1−3/2=−1/2。
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