How to Add and Subtract Radical Expressions

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Adding and Subtracting Radical Expressions

To add or subtract radicals, they should be, like radicals, radicals with the same radicand and index. Radicand is the number inside the radical.
"Like radicals" can be added to or subtracted by adding or subtracting the coefficients.

Steps of Adding and Subtracting Radical Expressions

Factorize the given radicals and simplify each phrase.
Determine the like radicals.
By adding or subtracting their coefficients, add or subtract like radicals.

Example1

Simplify: $5 \ \sqrt{2} \ + \ 7 \ \sqrt{32}$

Solution:

First, we must simplify the radicals to get "like radicals":
$5 \ \sqrt{2} \ + \ 7 \ \sqrt{32} \ = \ 5 \ \sqrt{2} \ + \ 7 \times 4 \ \sqrt{2} \ = \ 5 \ \sqrt{2} \ + \ 28 \ \sqrt{2}$
Then, add like terms:
$5 \ \sqrt{2} \ + \ 28 \ \sqrt{2} \ = \ (5 \ + \ 28) \ \sqrt{2} \ = \ 33 \ \sqrt{2}$

Example2

Simplify: $6 \ \sqrt{27} \ - \ 11 \ \sqrt{3}$

Solution:

First, we must simplify the radicals to get "like radicals":
$6 \ \sqrt{27} \ - \ 11 \ \sqrt{3} \ = \ 6 \times 3 \ \sqrt{3} \ - \ 11 \ \sqrt{3} \ = \ 18 \ \sqrt{3} \ - \ 11 \ \sqrt{3}$
Then, combine like terms:
$18 \ \sqrt{3} \ - \ 11 \ \sqrt{3} \ = \ (18 \ - \ 11) \ \sqrt{3} \ = \ 7 \ \sqrt{3}$

Free printable Worksheets

Exercises for Adding and Subtracting Radical Expressions

1) Find the answer: $\sqrt{5 \ n^3} \ - \ \sqrt{5 \ n} \ =$

2) Find the answer: $\sqrt{14 \ q^3} \ + \ \sqrt{56 \ q^5} \ =$

3) Find the answer: $\sqrt{8 \ p^7} \ + \ \sqrt{72 \ p^9} \ =$

4) Find the answer: $\sqrt{180 \ x^3} \ - \ \sqrt{45 \ x} \ =$

5) Find the answer: $\sqrt{6 \ z^5} \ + \ \sqrt{96 \ z^3} \ =$

6) Find the answer: $\sqrt{75 \ t^{13}} \ - \ \sqrt{3 \ t^3} \ =$

7) Find the answer: $\sqrt{117 \ y^{23}} \ - \ \sqrt{52 \ t^{11}} \ =$

8) Find the answer: $\sqrt{175 \ n^{17}} \ + \ \sqrt{112 \ n^3} \ =$

9) Find the answer: $-2 \ \sqrt{60 \ q^9} \ + \ q^4 \ \sqrt{135 \ q} \ =$

10) Find the answer: $6 \ p^7 \sqrt{44 \ p^3} \ - \ 5 \ p^5 \sqrt{99 \ p^7} \ =$

Answers

1) Find the answer: $\sqrt{5 \ n^3} \ - \ \sqrt{5 \ n} \ =$

$\color{red}{\sqrt{5 \ n^3} \times \sqrt{5 \ n} \ = \ n \ \sqrt{5 \ n} \ - \ \sqrt{5 \ n} \ = \ (n \ - \ 1) \ \sqrt{5 \ n}}$

2) Find the answer: $\sqrt{14 \ q^3} \ + \ \sqrt{56 \ q^5} \ =$

$\color{red}{\sqrt{14 \ q^3} \ + \ \sqrt{56 \ q^5} \ = \ q \ \sqrt{14 \ q} \ + \ 2 \ q^2 \ \sqrt{14 \ q} \ = \ (q \ + \ 2 \ q^2) \ \sqrt{14 \ q}}$

3) Find the answer: $\sqrt{8 \ p^7} \ + \ \sqrt{72 \ p^9} \ =$

$\color{red}{\sqrt{8 \ p^7} \ + \ \sqrt{72 \ p^9} \ = \ 2 \ p^3 \ \sqrt{2 \ p} \ + \ 6 \ p^4 \ \sqrt{2 \ p} \ = \ (2 \ p^3 \ + \ 6 \ p^4) \ \sqrt{2 \ p}}$

4) Find the answer: $\sqrt{180 \ x^3} \ - \ \sqrt{45 \ x} \ =$

$\color{red}{\sqrt{180 \ x^3} \ - \ \sqrt{45 \ x} \ = \ 6 \ x \ \sqrt{5 \ x} \ - \ 3 \ \sqrt{5 \ x} \ = \ (6 \ x \ - \ 3) \ \sqrt{5 \ x}}$

5) Find the answer: $\sqrt{6 \ z^5} \ + \ \sqrt{96 \ z^3} \ =$

$\color{red}{\sqrt{6 \ z^5} \ + \ \sqrt{96 \ z^3} \ = \ z^2 \ \sqrt{6 \ z} \ + \ 4 \ z \ \sqrt{6 \ z} \ = \ (z^2 \ + \ 4 \ z) \ \sqrt{6 \ z}}$

6) Find the answer: $\sqrt{75 \ t^{13}} \ - \ \sqrt{3 \ t^3} \ =$

$\color{red}{\sqrt{75 \ t^{13}} \ - \ \sqrt{3 \ t^3} \ = \ 5 \ t^6 \ \sqrt{3 \ t} \ - \ t \ \sqrt{3 \ t} \ = \ (5 \ t^6 \ - \ t) \ \sqrt{3 \ t}}$

7) Find the answer: $\sqrt{117 \ y^{23}} \ - \ \sqrt{52 \ t^{11}} \ =$

$\color{red}{\sqrt{117 \ y^{23}} \ - \ \sqrt{52 \ y^{11}} \ = \ 3 \ y^{11} \ \sqrt{13 \ y} \ - \ 2 \ y^5 \ \sqrt{13 \ y} \ = \ (3 \ y^{11} \ - \ 2 \ y^5) \ \sqrt{13 \ y}}$

8) Find the answer: $\sqrt{175 \ n^{17}} \ + \ \sqrt{112 \ n^3} \ =$

$\color{red}{\sqrt{175 \ n^{17}} \ + \ \sqrt{112 \ n^3} \ = \ 5 \ n^8 \ \sqrt{7 \ n} \ + \ 4 \ n \ \sqrt{7 \ n} \ = \ (5 \ n^8 \ + \ 4 \ n) \ \sqrt{7 \ n}}$

9) Find the answer: $-2 \ \sqrt{60 \ q^9} \ + \ q^4 \ \sqrt{135 \ q} \ =$

$\color{red}{-2 \ \sqrt{60 \ q^9} \ + \ q^4 \ \sqrt{135 \ q} \ = \ 2 \ q^4 \ \sqrt{15 \ q} \ + \ 3 \ q^4 \ \sqrt{15 \ q} \ = \ (2 \ q^4 \ + \ 3 \ q^4) \ \sqrt{15 \ q}}$ $\color{red}{ \ = \ 5 \ q^4 \ \sqrt{15 \ q}}$

10) Find the answer: $6 \ p^7 \sqrt{44 \ p^3} \ - \ 5 \ p^5 \sqrt{99 \ p^7} \ =$

$\color{red}{6 \ p^7 \sqrt{44 \ p^3} \ - \ 5 \ p^5 \sqrt{99 \ p^7} \ = \ 12 \ p^8 \sqrt{11 \ p} \ - \ 15 \ p^8 \sqrt{11 \ p} \ = \ (12 \ p^8 \ - \ 15 \ p^8) \ \sqrt{11 \ p}}$

How to Add and Subtract Radical Expressions