How Do You Simplify √45?
Simplifying square roots is an essential algebra skill. When you see √45, the goal is to express it in simplest radical form—where no perfect square factors remain under the radical. Here’s exactly how to simplify √45 step by step.
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Step-by-Step: How to Simplify √45
Step 1: Find the Prime Factorization of 45
Break 45 into prime factors: 45 = 9 × 5 = 3 × 3 × 5 = 3² × 5. The prime factorization is 3² × 5.
Step 2: Apply the Product Rule for Radicals
√45 = √(9 × 5) = √9 × √5. Since √9 = 3, we get √45 = 3√5.
Step 3: Verify Your Answer
Check: 3√5 ≈ 3 × 2.236 ≈ 6.708, and √45 ≈ 6.708. Both equal, so 3√5 is correct. For more practice with radicals and algebra, visit our math worksheets.
Why Does This Work?
The key property is √(a × b) = √a × √b when a and b are non-negative. We look for perfect square factors (1, 4, 9, 16, 25, 36, 49…) under the radical. Since 9 is a perfect square and 45 = 9 × 5, we can “pull out” the 9 as 3.
Quick Method: Factor Out Perfect Squares
List factors of 45: 1, 3, 5, 9, 15, 45. The largest perfect square factor is 9. So √45 = √(9×5) = √9·√5 = 3√5. This shortcut works for any radical. Explore more at Effortless Math.
Frequently Asked Questions
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Is 3√5 the same as √45?
Yes. 3√5 and √45 are equivalent. 3√5 is the simplified form.
Can √45 be simplified further?
No. Since 5 has no perfect square factors (other than 1), 3√5 is in simplest form.
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