10
Powers with a rational exponent and their properties
Textbook: pp. 42–48
GoalKnow the definition and properties of a power with a rational exponent and use them in calculations.
New words
rational exponent · ratsional ko‘rsatkichbase of a power · darajaning asosinegative exponent · manfiy ko‘rsatkichwriting a power as a root · darajani ildiz ko‘rinishida yozish
Explanation
For a > 0 and a rational number m/n (m an integer, n ≥ 2 natural) we define a^(m/n) = ⁿ√(aᵐ); for example 8^(2/3) = ³√(8²) = (³√8)² = 4. For a negative exponent, a^(-r) = 1/a^r. Reducing the fraction in the exponent does not change the power: a^(m/n) = a^(mk/(nk)). All properties of powers with a natural exponent remain true for rational exponents: aᵖ · aᵠ = aᵖ⁺ᵠ, aᵖ : aᵠ = aᵖ⁻ᵠ, (aᵖ)ᵠ = aᵖᵠ, (ab)ᵖ = aᵖbᵖ, (a/b)ᵖ = aᵖ/bᵖ (a > 0, b > 0). With these properties we can write root expressions as powers and simplify the calculation.
Worked examples
8^(2/3) = (³√8)² = 2² = 4; 27^(-2/3) = 1/27^(2/3) = 1/(³√27)² = 1/9.
4^(1/2) · 4^(3/2) = 4^(1/2 + 3/2) = 4² = 16; 16^(3/4) = (⁴√16)³ = 2³ = 8.
Class activity
“Power ↔ root”: teams match cards: a^(1/2) with √a, a^(2/3) with ³√(a²), a^(-1/2) with 1/√a and so on.
Practice
1
Compute 81^(3/4).
27
2
Compute 8^(4/3).
16
3
Simplify 7^(2/5) · 7^(3/5).
7
4
Why is a^(-r) = 1/a^r?
Because a^r · a^(-r) = a⁰ = 1, so a^(-r) is the reciprocal of a^r.