A fraction in the exponent means a root
What should b^(1/2) mean? Demand that the power rule still hold: (b^(1/2))^2 = b^(1/2 · 2) = b^1 = b. So b^(1/2) is a number that squares to b — it’s the square root. In general a rational exponent 1/n is the nth root: b^(1/n) = the nth root of b.
A general fraction m/n combines both: b^(m/n) means take the nth root and raise to the m, in either order. The denominator is the index (the root); the numerator is the power. Reading b^(2/3) as “cube root, then square” keeps the arithmetic small.
9^(1/2) = sqrt(9) = 3 8^(1/3) = cube root of 8 = 2 8^(2/3) = (8^(1/3))^2 = 2^2 = 4 (root first, then square) 16^(3/4) = (16^(1/4))^3 = 2^3 = 8 Negative + fractional: 4^(−1/2) = 1 / 4^(1/2) = 1/2
Rationalizing a one-term denominator
By convention we don’t leave a radical in the denominator. Rationalizing removes it by multiplying the fraction by a clever form of 1. For a single square-root denominator, multiply top and bottom by that same root: the denominator becomes a whole number and the value is unchanged.
3 3 sqrt(2) 3·sqrt(2) ------ = ------ · ------- = --------- sqrt(2) sqrt(2) sqrt(2) 2 5 5 sqrt(3) 5·sqrt(3) ------- = ------ · ------- = --------- 2·sqrt(3) 2·sqrt(3) sqrt(3) 6
Two-term denominators: the conjugate
When the denominator has two terms like 1 + sqrt(2), multiplying by the same root won’t clear it. Instead multiply by the conjugate — the same two terms with the middle sign flipped. The product follows (a+b)(a−b) = a^2 − b^2, and squaring kills the radical.
1 1 (1 − sqrt(2))
--------- = --------------- · -------------
1 + sqrt(2) 1 + sqrt(2) 1 − sqrt(2)
denominator: (1)^2 − (sqrt(2))^2 = 1 − 2 = −1
1 − sqrt(2)
= ----------- = −(1 − sqrt(2)) = sqrt(2) − 1
−1