Radicals on ACT Math mean simplifying square roots, combining like radicals, and rationalizing denominators—using the rule that the square root of ab equals the square root of a times the square root of b, and its partner rule for division. They rarely appear alone; they hide inside algebra and geometry questions where a clean radical form is the answer.
Simplify square roots
Factor out perfect squares. The square root of 72 is the square root of 36 times 2, which is 6 times the square root of 2. The square root of 50 is 5 times the square root of 2. If a factor is a perfect square, pull its root outside. That is the whole game for most ACT items.
Multiply and divide radicals
Square root of a times square root of b equals square root of ab when both are nonnegative. Square root of 3 times square root of 12 equals square root of 36, which is 6. For division, square root of a divided by square root of b equals square root of a/b. Simplify after you combine. These connect directly to exponent rules because a square root is the same as an exponent of 1/2.
Add and subtract only like radicals
3 times the square root of 2 plus 5 times the square root of 2 equals 8 times the square root of 2. You cannot add the square root of 2 and the square root of 3—they are different terms. Sometimes you simplify first to reveal like radicals: the square root of 8 plus the square root of 18 becomes 2 times the square root of 2 plus 3 times the square root of 2, which is 5 times the square root of 2.
Rationalize the denominator
The ACT prefers no radicals in the bottom of a fraction. For 5 divided by the square root of 2, multiply top and bottom by the square root of 2 to get 5 times the square root of 2, all over 2. For 3 divided by (the square root of 5 minus 1), multiply by the conjugate—the square root of 5 plus 1—to clear the radical in the denominator. One multiplication step, then simplify.
Common traps
- Treating the square root of (a + b) as the square root of a plus the square root of b. Test with numbers: the square root of 9 is 3, but the square root of 4 plus the square root of 5 is not 3.
- Forgetting to simplify fully. The square root of 32 should become 4 times the square root of 2, not stay as the square root of 32.
- Adding unlike radicals as if the radicands add. The square root of 2 plus the square root of 3 is not the square root of 5.
How much to prep
Radicals are a small slice of the algebra on the test. For where they sit among the rest, see how much algebra is on the ACT. Know simplify, combine like terms, and rationalize, and you are covered for what the ACT actually asks.
Start practicing
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This article offers general ACT prep guidance. The ACT can change from year to year, including its format, scoring, policies, test dates, and fees, so always confirm the latest details on the official ACT website at act.org before you make decisions. ACT® is a registered trademark of ACT, Inc. thirty-six is not affiliated with, endorsed by, or sponsored by ACT.