Square Roots - How to Simplify Square Roots - Part 1

How to Simplify Square Roots with Prime Numbers- How it Works - Video

Example 1

Example 1:

When simplifying square roots or radicals, we need to look for square numbers, since squares are the inverse of square roots. We need to rewrite 20 as a factor pair where one number is a prime number. We have the options to rewrite 20 as 1*20 or 2*10 or 4*5. Since we want prime numbers, we can choose 2*10 or 4*5. Both will give you the same answer, and in this case we will work with 2*10.

√20

• • √(2 * 10) We rewrite as a multiplication of two numbers.

  • √(2*2*5) We rewrite 10 as a multiplication of two numbers.

  • √2² * 5 We rewrite 2 * 2 as 2².

  • 2 * √5 We cancel the inverses of square and square root.

  • 2√5 We drop the multiplication symbol.

So 2√5 is the final answer.

Example 2

Example 2:

When simplifying square roots or radicals, we need to look for square numbers, since squares are the inverse of square roots. We need to rewrite 105 as a factor pair where one number is a prime number. We have the options to rewrite 105 as 1*105 or 3*35 or 5*21. Since we want prime numbers, we can choose 3*35 or 5*21. Both will give you the same answer, and in this case we will work with 3*35.

-√105

• • -√(3 * 35) We rewrite as a multiplication of two numbers.

  • -√(3*5*7) We rewrite 35 as a multiplication of two numbers.

  • -√105 We multiply the numbers because we don't have two of the same number.

So -√105 is the final answer.

Example 3

Example 3:

When simplifying square roots or radicals, we need to look for square numbers, since squares are the inverse of square roots. We need to rewrite 72 as a factor pair where one number is a prime number. We have the options to rewrite 72 as 1*72 or 2*36 or 3*24. Since we want prime numbers, we can choose 2*36 or 3*24. Both will give you the same answer, and in this case we will work with 2*36.

√72

• • √(2 * 36) We rewrite as a multiplication of two numbers.

  • √(2 * 2 * 18) We rewrite 36 as a multiplication of two numbers.

  • √(2 * 2 * 2 * 9) We rewrite 18 as a multiplication of two numbers.

  • √(2 * 2 * 2 * 3 * 3) We rewrite 9 as a multiplication of two numbers.

  • √(2² * 3² * 2) We rewrite 2 * 2 as 2² and 3 * 3 as 3².

  • 2 * 3 * √2 We cancel the inverses of square and square root.

  • 6*√2 We multiply the numbers in front of the square root.

  • 6√2 We drop the multiplication symbol.

So 6√2 is the final answer.

Example 4

Example 4:

When simplifying square roots or radicals, we need to look for square numbers, since squares are the inverse of square roots. We need to rewrite 41 as a factor pair where one number is a prime number. We only the option to rewrite 41 as 1*41. None of those factor pairs are greater than one so we can't simplify anymore.

-√41

• • -√1 * 41 We rewrite as a multiplication of two numbers.

  • -√41 We can't simplify any further.

So -√41 is the final answer.

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