how to simplify square roots

How to Simplify Square Roots

Simplifying square roots means rewriting the square root in its simplest radical form. Here’s a step-by-step guide to help you simplify square roots effectively:

Step 1: Factor the Number Inside the Square Root

• Find the prime factorization of the number under the square root.
• For example, to simplify √72, first find the prime factors of 72:
  • 72 = 2 × 2 × 2 × 3 × 3

Step 2: Pair the Factors

• Group the prime factors into pairs because √(a²) = a.
• In the example of √72, the factors can be paired as (2 × 2) and (3 × 3), with one 2 left unpaired.

Step 3: Take Out the Pairs

• Each pair of identical factors comes out of the square root as a single factor.
• From √72, the pairs (2 × 2) and (3 × 3) come out as 2 and 3, respectively.
• The leftover 2 stays inside the square root.

Step 4: Multiply Outside Factors and Write the Simplified Form

• Multiply the factors outside the root: 2 × 3 = 6.
• Write the leftover factor inside the root: √2.
• So, √72 simplifies to 6√2.

Additional Tips:

• If the number inside the root is a perfect square (like 4, 9, 16, 25...), the square root is just the whole number.
• For example, √49 = 7 because 7 × 7 = 49.
• If the number is a fraction, simplify numerator and denominator separately.

Example:

Simplify √50.

1. Factor 50: 50 = 2 × 5 × 5
2. Pair factors: (5 × 5)
3. Take out pairs: 5
4. Leftover inside root: 2
5. Final simplified form: 5√2

If you want, I can provide practice problems or explain how to simplify cube roots or other radicals!