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By the end of this section, you will be able to:
  • Add and subtract like square roots
  • Add and subtract square roots that need simplification

Before you get started, take this readiness quiz.

  1. Add: 3 x + 9 x 5 m + 5 n .
    If you missed this problem, review [link] .
  2. Simplify: 50 x 3 .
    If you missed this problem, review [link] .

We know that we must follow the order of operations to simplify expressions with square roots. The radical is a grouping symbol, so we work inside the radical first. We simplify 2 + 7 in this way:

2 + 7 Add inside the radical. 9 Simplify. 3

So if we have to add 2 + 7 , we must not combine them into one radical.

2 + 7 2 + 7

Trying to add square roots with different radicands is like trying to add unlike terms.

But, just like we can add x + x , we can add 3 + 3 . x + x = 2 x 3 + 3 = 2 3

Adding square roots with the same radicand is just like adding like terms. We call square roots with the same radicand like square roots to remind us they work the same as like terms.

Like square roots

Square roots with the same radicand are called like square roots    .

We add and subtract like square roots in the same way we add and subtract like terms. We know that 3 x + 8 x is 11 x . Similarly we add 3 x + 8 x and the result is 11 x .

Add and subtract like square roots

Think about adding like terms with variables as you do the next few examples. When you have like radicands, you just add or subtract the coefficients. When the radicands are not like, you cannot combine the terms.

Simplify: 2 2 7 2 .

Solution

2 2 7 2 Since the radicals are like, we subtract the coefficients. −5 2

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Simplify: 8 2 9 2 .

2

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Simplify: 5 3 9 3 .

−4 3

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Simplify: 3 y + 4 y .

Solution

3 y + 4 y Since the radicals are like, we add the coefficients. 7 y

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Simplify: 2 x + 7 x .

9 x

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Simplify: 5 u + 3 u .

8 u

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Simplify: 4 x 2 y .

Solution

4 x 2 y Since the radicals are not like, we cannot subtract them. We leave the expression as is. 4 x 2 y

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Simplify: 7 p 6 q .

7 p 6 q

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Simplify: 6 a 3 b .

6 a 3 b

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Simplify: 5 13 + 4 13 + 2 13 .

Solution

5 13 + 4 13 + 2 13 Since the radicals are like, we add the coefficients. 11 13

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Simplify: 4 11 + 2 11 + 3 11 .

9 11

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Simplify: 6 10 + 2 10 + 3 10 .

11 10

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Simplify: 2 6 6 6 + 3 3 .

Solution

2 6 6 6 + 3 3 Since the first two radicals are like, we subtract their coefficients. 4 6 + 3 3

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Simplify: 5 5 4 5 + 2 6 .

5 + 2 6

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Simplify: 3 7 8 7 + 2 5 .

−5 7 + 2 5

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Simplify: 2 5 n 6 5 n + 4 5 n .

Solution

2 5 n 6 5 n + 4 5 n Since the radicals are like, we combine them. 0 5 n Simplify. 0

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Simplify: 7 x 7 7 x + 4 7 x .

−2 7 x

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Simplify: 4 3 y 7 3 y + 2 3 y .

3 y

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When radicals contain more than one variable, as long as all the variables and their exponents are identical, the radicals are like.

Simplify: 3 x y + 5 3 x y 4 3 x y .

Solution

3 x y + 5 3 x y 4 3 x y Since the radicals are like, we combine them. 2 3 x y

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Simplify: 5 x y + 4 5 x y 7 5 x y .

−2 5 x y

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Simplify: 3 7 m n + 7 m n 4 7 m n .

0

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Add and subtract square roots that need simplification

Remember that we always simplify square roots by removing the largest perfect-square factor. Sometimes when we have to add or subtract square roots that do not appear to have like radicals , we find like radicals after simplifying the square roots.

Simplify: 20 + 3 5 .

Solution

20 + 3 5 Simplify the radicals, when possible. 4 · 5 + 3 5 2 5 + 3 5 Combine the like radicals. 5 5

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Simplify: 18 + 6 2 .

9 2

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Simplify: 27 + 4 3 .

7 3

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Simplify: 48 75 .

Solution

48 75 Simplify the radicals. 16 · 3 25 · 3 4 3 5 3 Combine the like radicals. 3

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Practice Key Terms 1

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Source:  OpenStax, Elementary algebra. OpenStax CNX. Jan 18, 2017 Download for free at http://cnx.org/content/col12116/1.2
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