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Radicals and roots — ACT practice questions

Radicals and roots on the ACT Mathematics test covers products of square roots, conjugate pairs, and binomials that mix integers with roots. A student has to multiply radicals, apply the difference of squares to a conjugate, expand a squared two-term radical expression, and then simplify until the result is an integer or a fully reduced radical. Items are usually a compact expression set equal to a question mark, with five answer choices that mix integers and leftover roots. Common traps include dropping the middle term when squaring, keeping an unsimplified radical, or flipping a sign on a conjugate.

  • Section: Mathematics 45 questions on the paper
  • 36 practice questions in the app

Sample questions

Question 1

50⋅18=\sqrt{50}\cdot\sqrt{18}={} ?

  1. A68\sqrt{68}
  2. B15215\sqrt{2}
  3. C30
  4. D900
Show answer

Answer: C — 30

50=52\sqrt{50}=5\sqrt{2} and 18=32\sqrt{18}=3\sqrt{2}, so the product is 15⋅2=3015\cdot2=30.

Question 2

(7+2)(7−2)=(\sqrt{7}+\sqrt{2})(\sqrt{7}-\sqrt{2})={} ?

  1. A9−2149-2\sqrt{14}
  2. B5\sqrt{5}
  3. C5
  4. D9
Show answer

Answer: C — 5

The factors are conjugates, so their product is the difference of squares: (7)2−(2)2=7−2=5(\sqrt{7})^2-(\sqrt{2})^2=7-2=5.

Question 3

(23−2)2=(2\sqrt{3}-\sqrt{2})^2={} ?

  1. A14−4614-4\sqrt{6}
  2. B14−2614-2\sqrt{6}
  3. C10
  4. D14+4614+4\sqrt{6}
Show answer

Answer: A — 14−4614-4\sqrt{6}

Using (a−b)2=a2−2ab+b2(a-b)^2=a^2-2ab+b^2 gives x=12−46+2=14−46x=12-4\sqrt{6}+2=14-4\sqrt{6}.

Practice 36 Radicals and roots questions in the app

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