L4 · Rational Exponents: Roots Written as Powers
“I can explain why a^(1/n) must mean the nth root of a, convert between rational exponents and radicals in both directions, evaluate unit-fraction powers of perfect powers by hand, and state - with the reason - when an even index forbids a negative base.”
Ask: what must 9^(1/2) mean? Step 1: Suppose the power-of-a-power law (L2) is going to keep working for fractional exponents. Then (9^(1/2))^2 must equal 9^((1/2) * 2) = 9^1 = 9. Step 2: So 9^(1/2) is a number whose square is 9. That number is 3. Therefore 9^(1/2) = 3 - and 'the number whose square is 9' is exactly what the square root of 9 means. Step 3: Same argument with thirds: (8^(1/3))^3 must equal 8^1 = 8, and 2^3 = 8, so 8^(1/3) = 2 - the cube root of 8. The definition the laws force: a^(1/n) = the nth root of a. The mapping to memorize is the one the proof labels: the denominator of the exponent is the index of the root. Read it both ways, every time: power form to radical form and radical form to power form.
Why must 16^(1/4) mean the fourth root of 16, rather than 16 / 4?
Evaluate 27^(1/3) -> 3 (the number whose cube is 27)
"32^(1/5): you need a number whose ___ power is 32; since 2^___ = 32, the value is ___" -> fifth; 5; 2
10 scored questions with a 3-rung hint ladder. Mastery needs 8/10 plus both gate questions.