Square Roots - How to Add and Subtract Square Roots

How to Add and Subtract Square Roots - 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 square number so we can cancel the inverses.

We have the options to rewrite 20 as 1*20 or 2*10 or 4*5. Only, one has a square number greater than one ==> 4*5.

We have the options to rewrite 40 as 1*40 or 2*20 or 4*10 or 5*8. Only, one has a square number greater than one ==> 4*10.

√20 + √40

• • √4*5 + √4*10 We rewrite as a multiplication of two numbers.

  • √4*√5 + √4*√10 We separate the two numbers as two square roots.

  • √2² *√5 + √2² *√10 We rewrite both 4s as a numbered squared.

  • 2 *√5 + 2 *√10 The inverses (squares and square roots) cancel.

  • 2√5 + 2 √10 We drop the multiplication symbol.

So 2√5 + 2 √10 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 20 as a factor pair where one number is a square number so we can cancel the inverses.

We have the options to rewrite 48 as 1*48 or 2*24 or 3*16 or 4*12 or 6*8. The square number 16 is bigger than 4 so the one we pick is ==> 3*16.

We have the options to rewrite 75 as 1*75 or 3*25. Only, one has a square number greater than one ==> 3*35.

√48 + √75

• • √(16*3) + √(25*3) We rewrite as a multiplication of two numbers.

  • √16*√3 + √25*√3 We separate the two numbers as two square roots.

  • √4² *√3 + √5² *√3 We rewrite 16 and 25 as a numbered squared.

  • 2 *√3 + 5 *√3 The inverses (squares and square roots) cancel.

  • 7√3 We add the numbers in front of the square root symbols.

So 7√3 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 20 as a factor pair where one number is a square number so we can cancel the inverses.

We have the options to rewrite 50 as 1*50 or 2*25 or 5*10. Only, one has a square number greater than one ==> 2*25.

We have the options to rewrite 72 as 1*72 or 2*36 or 3*24 or 4*18 or 6*12 or 8*9. The square number 36 is bigger than 4 and 9 so the one we pick is ==> 2*36.

√50 - √72

• • √(25*2) - √(36*2) We rewrite as a multiplication of two numbers.

  • √25*√2 - √36*√2 We separate the two numbers as two square roots.

  • √5² *√2 - √6² *√2 We rewrite 25 and 36 as a numbered squared.

  • 5 *√2 - 6 *√2 The inverses (squares and square roots) cancel.

  • -√2 We subtract the numbers in front of the square root symbols.

So -√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 20 as a factor pair where one number is a square number so we can cancel the inverses.

We have the options to rewrite 54 as 1*54 or 2*27 or 3*18 or 6*9. Only, one has a square number greater than one ==> 6*9.

We have the options to rewrite 45 as 1*45 or 5*9. Only, one has a square number greater than one ==> 5*9.

We have the options to rewrite 150 as 1*150 or 2*75 or 3*50 or 6*25 or 10*15. Only, one has a square number greater than one ==> 6*25.

√54 - √45 + √150

• • √(9*6) - √(9*5) + - √(25*6) We rewrite as a multiplication of two numbers.

  • √9*√6 - √9*√5 + √25*√6 We separate the two numbers as two square roots.

  • √3² *√6 - √3² *√5 + √5² *√6 We rewrite 25 and 36 as a numbered squared.

  • 3 *√6 - 3 *√5 + 5 *√6 The inverses (squares and square roots) cancel.

  • 3 *√6 + 5 *√6 - 3 *√5 We arrange the terms.

  • 8 *√6 - 3 *√5 We add the numbers in front of the square roots.

  • 8 √6 - 3 √5 We drop the multiplication symbol.

So 8 √6 - 3 √5 is the final answer.

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