Square Roots and Simplification — ACT Math Guide

Q: What You Need to Know

• Perfect squares: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225 • √(a²) = |a| (absolute value matters for variables) • √(ab) = √a × √b (product property) • √(a/b) = √a / √b (quotient property) • Rationalize denominators by multiplying by the conjugate • Like radicals can be added/su

Square roots ACT problems appear regularly in the elementary algebra section, testing your ability to simplify radical expressions and perform calculations. Understanding square roots means finding which number, when multiplied by itself, gives you the original number. These questions typically show up 3-4 times across the 60 questions in 60 minutes, making them worth mastering for your ACT math score.

What You Need to Know

Perfect squares: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225

√(a²) = |a| (absolute value matters for variables)

√(ab) = √a × √b (product property)

√(a/b) = √a / √b (quotient property)

Rationalize denominators by multiplying by the conjugate

Like radicals can be added/subtracted (same index and radicand)

📐 KEY FORMULA: √(a²b) = a√b when a ≥ 0

⏱️ ACT TIME TIP: Memorize perfect squares up to 15² = 225 to save precious seconds on the 60-question test

How to Solve Square Roots on the ACT

Example Question 1 — Easy/Medium Difficulty

Which of the following is equivalent to √72?

A) 6√2

B) 8√3

C) 6√3

D) 9√2

E) 12√6

Solution:

Step 1: Factor 72 into perfect square factors: 72 = 36 × 2

Step 2: Apply the product property: √72 = √(36 × 2) = √36 × √2

Step 3: Simplify the perfect square: √36 × √2 = 6√2

✅Answer: A — We extracted the perfect square factor 36 to get 6√2.

Example Question 2 — Hard Difficulty

If x > 0, what is the simplified form of (√18x³)/(√2x)?

A) 3x

B) 3√x

C) 9x

D) x√9

E) 3x√x

Solution:

Step 1: Use the quotient property: (√18x³)/(√2x) = √(18x³/2x)

Step 2: Simplify the fraction under the radical: 18x³/2x = 9x²

Step 3: Take the square root: √(9x²) = 3x (since x > 0)

✅Answer: A — The quotient simplifies to 3x when x is positive.

Common ACT Math Mistakes to Avoid

❌Mistake: Forgetting absolute value signs when simplifying √(x²)

✅Fix: Remember √(x²) = |x|, not just x

❌Mistake: Adding radicals with different radicands like √2 + √3 = √5

✅Fix: Only combine like radicals — different radicands cannot be simplified together

❌Mistake: Leaving radicals in denominators like 1/√3

✅Fix: Always rationalize by multiplying by √3/√3 to get √3/3

❌Mistake: Rushing through perfect square identification

✅Fix: Your calculator is allowed throughout the ACT math section — use it to verify perfect squares

Practice Question — Try It Yourself

What is the value of √50 + √32?

A) √82

B) 9√2

C) 6√2

D) 7√2

E) 8√3

Show Answer

Answer: B — √50 = √(25×2) = 5√2 and √32 = √(16×2) = 4√2, so 5√2 + 4√2 = 9√2

Key Takeaways for the ACT

Master perfect squares 1-225 to quickly identify simplification opportunities

Use your calculator freely — ACT math allows calculators throughout the entire section

Factor out perfect squares first, then simplify the remaining radical

Watch for absolute value requirements when variables are involved

ACT math questions test simplification more than complex radical operations