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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