ACT® · Math: Number and Quantity
Rational Exponents and Radicals
A rational exponent like 1/2 or 3/4 is just another way to write a radical: the denominator is the root, the numerator is the power. The same exponent rules you already know still apply.
Raising a number to the 1/2 power means square root because the exponent rules demand it. Power of a power multiplies exponents, so (x1/2)2 = x1/2 times 2 = x1 = x. The only number that gives you back x when squared, by definition of a square root, is √x itself. So x1/2 and √x have to be the same value for the whole system of exponent rules to stay consistent. Extend that logic and x1/q is the qth root of x, while xp/q is the qth root of x raised to the p, or equivalently the qth root of xp.
This lets you move freely between radical form and exponent form, which matters because the test often disguises a radical problem as an exponent problem or the reverse. 163/4 means take the 4th root of 16 first, getting 2, then cube it, getting 8 — working in that order keeps the numbers small. Cube root of x5 becomes x5/3, already a single power with no radical in sight. When the index and the inside exponent share a common factor, you can reduce that fraction just like any other, which simplifies the radical's index.
This topic sits right next to complex numbers and matrices in the number-and-quantity items, because all three test whether you track structure carefully rather than compute by instinct. Radicals and rational exponents are where that discipline starts: one wrong assumption about signs or roots breaks the rest of the arithmetic.
Key terms
- Rational exponent
- An exponent written as a fraction, where the denominator gives the root and the numerator gives the power.
- Principal root
- The single nonnegative root that a radical symbol by itself always represents.
- Index
- The small number on a radical sign telling you which root to take, such as the 3 in a cube root.
- Power of a power rule
- The exponent rule stating that (xa)b equals x raised to the product a times b.
Watch it explained
- How it worksHow does the rule that a power of a power multiplies exponents force x1/2 to be a square root?
- Worked exampleFind the value of 163/4 without touching a calculator.
- Worked exampleWalk me through rewriting the cube root of x5 as one power of x.
- Common mistakeWhy isn't the square root of 16 equal to both 4 and -4 when the test writes the radical alone?
- Going furtherWhat happens to a radical when the index and the exponent inside share a common factor?
Check yourself
End of topic.