15
Raising numerical inequalities to a power
Textbook: pp. 80–84
GoalKnow how to raise an inequality with positive sides to a natural or rational power and use it to compare numbers.
New words
raising to a power · darajaga ko‘tarishpositive sides · musbat qismlarnegative exponent · manfiy ko‘rsatkichcomparing numbers · sonlarni taqqoslash
Explanation
If a > b > 0 and n is a natural number, then aⁿ > bⁿ; this follows from multiplying the inequality by itself n times. The property extends to positive rational exponents: if a > b > 0 and r > 0, then aʳ > bʳ; for example 5^(1/2) > 3^(1/2) and 7^(2/3) > 5^(2/3). If r < 0 the sign reverses: aʳ < bʳ. For negative bases the rule fails: -3 < -2 but (-3)² > (-2)². To compare roots we raise them to a common power.
Worked examples
Compare √2 and ³√3: raise both to the 6th power: (√2)⁶ = 2³ = 8, (³√3)⁶ = 3² = 9. Since 9 > 8, ³√3 > √2.
0.8 > 0.5 and r = -2 < 0: (0.8)^(-2) < (0.5)^(-2). Indeed 1/0.64 ≈ 1.56 and 1/0.25 = 4.
Class activity
“Power arena”: two root numbers are given; teams find a common exponent, raise both to it and name the greater number.
Practice
1
Compare (2/3)⁴ and (3/4)⁴.
(2/3)⁴ < (3/4)⁴
2
Compare √3 and ⁴√8.
√3 > ⁴√8
3
Compare 7^(-1/2) and 5^(-1/2).
7^(-1/2) < 5^(-1/2)
4
Why can’t we conclude (-3)² < (-2)² from -3 < -2?
Because the rule works only for inequalities with positive sides; (-3)² = 9 and (-2)² = 4, and 9 > 4.