# 9.6: Addition and Subtraction of Square Root Expressions

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## The Logic Behind The Process

Now we will study methods of simplifying radical expressions such as

$$4\sqrt{3} + 8\sqrt{3}$$ or $$5\sqrt{2x} - 11\sqrt{2x} + 4(\sqrt{2x} + 1)$$

The procedure for adding and subtracting square root expressions will become apparent if we think back to the procedure we used for simplifying polynomial expressions such as
$$4x + 8x$$ or $$5a - 11a + 4(a+1)$$

The variables $$x$$ and $$a$$ are letters representing some unknown quantities (perhaps $$x$$ represents $$\sqrt{3}$$ and $$a$$ represents $$\sqrt{2x}$$). Combining like terms gives us

$$\begin{array}{flushleft} 4x + 8x = 12x & \text{ or } & 4\sqrt{3} + 8\sqrt{3} = 12\sqrt{3}\\ \text{and}\\ 5a - 11a + 4(a + 1) & \text{ or } & 5\sqrt{2x} - 11\sqrt{2x} + 4(\sqrt{2x} + 1)\\ 5a - 11a + 4a + 4 && 5\sqrt{2x} - 11\sqrt{2x} + 4\sqrt{2x} + 4\\ -2a && -2\sqrt{2x} + 4 \end{array}$$

## The Process

Let's consider the expression $$4\sqrt{3} + 8\sqrt{3}$$. There are two ways to look at the simplification process.

##### Simplification Process

We are asking, "How many square roots of $$3$$ do we have?"

$$4 \sqrt{3}$$ means we have $$4$$ "square roots of $$3$$"

Thus, altogether we have $$12$$ "square roots of $$3$$."

We can also use the idea of combining like terms. If we recall, the process of combining like terms is based on the distributive property

$$4x + 8x = 12x$$ because $$4x + 8x = (4 + 8)x = 12x$$

We could simplify $$4\sqrt{3} + 8\sqrt{3}$$ using the distributive property.

4\sqrt{3} + 8\sqrt{3} = (4 + 8)\sqrt{3} = 12\sqrt{3}\)

Both methods will give us the same result. The first method is probably a bit quicker, but keep in mind, however, that the process works because it is based on one of the basic rules of algebra, the distributive property of real numbers.

## Sample Set A

##### Example $$\PageIndex{1}$$

$$-6\sqrt{10} + 11\sqrt{10} = 5\sqrt{10}$$

##### Example $$\PageIndex{2}$$

$$4\sqrt{32} + 5\sqrt{2} \text{ Simplify } \sqrt{32}$$

$$\begin{array}{flushleft} 4\sqrt{16 \cdot 2} + 5\sqrt{2} &= 4\sqrt{16}\sqrt{2} + 5\sqrt{2}\\ &= 4 \cdot 4\sqrt{2} + 5\sqrt{2}\\ &= 16\sqrt{2} + 5\sqrt{2}\\ &=21\sqrt{2} \end{array}$$

##### Example $$\PageIndex{3}$$

$$-3x\sqrt{75} + 2x\sqrt{48} - x\sqrt{27} \text{ Simplify each of the three radicals}$$

$$\begin{array}{flushleft} -3x\sqrt{75} + 2x\sqrt{48} - x\sqrt{27} &= -3x\sqrt{25 \cdot 3} + 2x\sqrt{16 \cdot 3} - x\sqrt{9 \cdot 2}\\ &= -15x\sqrt{3} + 8x\sqrt{3} - 3x\sqrt{3}\\ &=(-15x + 8x - 3x)\sqrt{3}\\ &=-10x\sqrt{3} \end{array}$$

##### Example $$\PageIndex{4}$$

$$5a\sqrt{24a^3} - 7\sqrt{54a^5} + a^2\sqrt{6a} + 6a \text{ Simplify each radical}$$

$$\begin{array}{flushleft} 5a\sqrt{24a^3} - 7\sqrt{54a^5} + a^2\sqrt{6a} + 6a &= 5a\sqrt{4 \cdot 6 \cdot a^2 \cdot a} - 7\sqrt{9 \cdot 6 \cdot a^4 \cdot a} + a^2\sqrt{6a} + 6a\\ &= 10a^2\sqrt{6a} - 21a^2\sqrt{6a} + a^2\sqrt{6a} + 6a\\ &= (10a^2 - 21a^2 + a^2) \sqrt{6a} + 6a\\ &= -10a^2\sqrt{6a} + 6a\\ &= -2a(5a\sqrt{6a} - 3) \end{array}$$

## Practice Set A

Find each sum or difference.

##### Practice Problem $$\PageIndex{1}$$

$$4\sqrt{18} - 5\sqrt{8}$$

$$2\sqrt{2}$$

##### Practice Problem $$\PageIndex{2}$$

$$6x\sqrt{48} + 8x\sqrt{75}$$

$$64x\sqrt{3}$$

##### Practice Problem $$\PageIndex{3}$$

$$-7\sqrt{84x} - 12\sqrt{189x} + 2\sqrt{21x}$$

$$-48\sqrt{21x}$$

##### Practice Problem $$\PageIndex{4}$$

$$9\sqrt{6} - 8\sqrt{6} + 3$$

$$\sqrt{6} + 3$$

##### Practice Problem $$\PageIndex{5}$$

$$\sqrt{a^3} + 4a\sqrt{a}$$

$$5a\sqrt{a}$$

##### Practice Problem $$\PageIndex{6}$$

$$4x\sqrt{54x^3} + \sqrt{36x^2} + 3\sqrt{24x^5} - 3x$$

$$18x^2\sqrt{6x} + 3x$$

## Sample Set B

##### Example $$\PageIndex{8}$$

$$\dfrac{3 + \sqrt{8}}{3 - \sqrt{8}}$$ We'll rationalize the denominator by multiplying this fraction by $$1$$ in the form $$\dfrac{3 + \sqrt{8}}{3 + \sqrt{8}}$$.

$$\begin{array}{flushleft} \dfrac{3+\sqrt{8}}{3-\sqrt{8}} \cdot \frac{3+\sqrt{8}}{3+\sqrt{8}} &=\dfrac{(3+\sqrt{8})(3+\sqrt{8})}{3^{2}-(\sqrt{8})^{2}} \\ &=\dfrac{9+3 \sqrt{8}+3 \sqrt{8}+\sqrt{8} \sqrt{8}}{9-8} \\ &=\dfrac{9+6 \sqrt{8}+8}{1} \\ &=17+6 \sqrt{8} \\ &=17+6 \sqrt{4 \cdot 2} \\ &=17+12 \sqrt{2} \end{array}$$

##### Example $$\PageIndex{9}$$

$$\dfrac{2 + \sqrt{7}}{4 - \sqrt{3}}$$. Rationalize the denominator by multiplying this fraction by $$1$$ in the form $$\dfrac{4 + \sqrt{3}}{4 + \sqrt{3}}$$.

$$\begin{array}{flushleft} \dfrac{2+\sqrt{7}}{4-\sqrt{3}} \cdot \dfrac{4+\sqrt{3}}{4+\sqrt{3}} &=\dfrac{(2+\sqrt{7})(4+\sqrt{3})}{4^{2}-(\sqrt{3})^{2}} \\ &=\dfrac{8+2 \sqrt{3}+4 \sqrt{7}+\sqrt{21}}{16-3} \\ &=\dfrac{8+2 \sqrt{3}+4 \sqrt{7}+\sqrt{21}}{13} \end{array}$$

## Practice Set B

Simplify each by performing the indicated operation.

##### Practice Problem $$\PageIndex{7}$$

$$\sqrt{5}(\sqrt{6} - 4)$$

$$\sqrt{30} - 4\sqrt{5}$$

##### Practice Problem $$\PageIndex{8}$$

$$(\sqrt{5} + \sqrt{7})(\sqrt{2} + \sqrt{8})$$

$$3\sqrt{10} + 3\sqrt{14}$$

##### Practice Problem $$\PageIndex{9}$$

$$(3\sqrt{2} - 2\sqrt{3})(4\sqrt{3} + \sqrt{8})$$

$$8\sqrt{6} - 12$$

##### Practice Problem $$\PageIndex{10}$$

$$\dfrac{4 + \sqrt{5}}{3 - \sqrt{8}}$$

$$12 + 8\sqrt{2} + 3\sqrt{5} + 2\sqrt{10}$$

## Exercises

For the following problems, simplify each expression by performing the indicated operation.

##### Exercise $$\PageIndex{1}$$

$$4\sqrt{5} - 2\sqrt{5}$$

$$2\sqrt{5}$$

##### Exercise $$\PageIndex{2}$$

$$10 \sqrt{2} + 8\sqrt{2}$$

##### Exercise $$\PageIndex{3}$$

$$-3\sqrt{6} - 12\sqrt{6}$$

$$-15 \sqrt{6}$$

##### Exercise $$\PageIndex{4}$$

$$-\sqrt{10} - 2\sqrt{10}$$

##### Exercise $$\PageIndex{5}$$

$$3\sqrt{7x} + 2\sqrt{7x}$$

$$5\sqrt{7x}$$

##### Exercise $$\PageIndex{6}$$

$$6\sqrt{3a} + \sqrt{3a}$$

##### Exercise $$\PageIndex{7}$$

$$2\sqrt{18} + 5\sqrt{32}$$

$$26\sqrt{2}$$

##### Exercise $$\PageIndex{8}$$

$$4\sqrt{27} - 3\sqrt{48}$$

##### Exercise $$\PageIndex{9}$$

$$\sqrt{200} - \sqrt{128}$$

$$2\sqrt{2}$$

##### Exercise $$\PageIndex{10}$$

$$4\sqrt{300} + 2\sqrt{500}$$

##### Exercise $$\PageIndex{11}$$

$$6\sqrt{40} + 8\sqrt{80}$$

$$12\sqrt{10} + 32\sqrt{5}$$

##### Exercise $$\PageIndex{12}$$

$$2\sqrt{120} - 5\sqrt{30}$$

##### Exercise $$\PageIndex{13}$$

$$8\sqrt{60} - 3\sqrt{15}$$

$$13\sqrt{15}$$

##### Exercise $$\PageIndex{14}$$

$$\sqrt{a^3} - 3a\sqrt{a}$$

##### Exercise $$\PageIndex{15}$$

$$\sqrt{4x^3} + x\sqrt{x}$$

$$3x\sqrt{x}$$

##### Exercise $$\PageIndex{16}$$

$$2b\sqrt{a^3b^5} + 6a\sqrt{ab^7}$$

##### Exercise $$\PageIndex{17}$$

$$5xy\sqrt{2xy^3} - 3y^2\sqrt{2x^3y}$$

$$2xy^2\sqrt{2xy}$$

##### Exercise $$\PageIndex{18}$$

$$5\sqrt{20} + 3\sqrt{45} - 3\sqrt{40}$$

##### Exercise $$\PageIndex{19}$$

$$\sqrt{24} - 2\sqrt{54} - 4\sqrt{12}$$

$$-4\sqrt{6} - 8\sqrt{3}$$

##### Exercise $$\PageIndex{20}$$

$$6\sqrt{18} + 5\sqrt{32} + 4\sqrt{50}$$

##### Exercise $$\PageIndex{21}$$

$$-8\sqrt{20} - 9\sqrt{125} + 10\sqrt{180}$$

$$-\sqrt{5}$$

##### Exercise $$\PageIndex{22}$$

$$2\sqrt{27} + 4\sqrt{3} - 6\sqrt{12}$$

##### Exercise $$\PageIndex{23}$$

$$\sqrt{14} + 2\sqrt{56} - 3\sqrt{136}$$

$$5\sqrt{14} - 6\sqrt{34}$$

##### Exercise $$\PageIndex{24}$$

$$3\sqrt{2} + 2\sqrt{63} + 5\sqrt{7}$$

##### Exercise $$\PageIndex{25}$$

$$4ax\sqrt{3x} + 2\sqrt{3a^2x^3} + 7\sqrt{3a^2x^3}$$

$$13ax\sqrt{3x}$$

##### Exercise $$\PageIndex{26}$$

$$3by\sqrt{5y} + 4\sqrt{5b^2y^3} - 2\sqrt{5b^2y^3}$$

##### Exercise $$\PageIndex{27}$$

$$\sqrt{2}(\sqrt{3} + 1)$$

$$\sqrt{6} + \sqrt{2}$$

##### Exercise $$\PageIndex{28}$$

$$\sqrt{3}(\sqrt{5} - 3)$$

##### Exercise $$\PageIndex{29}$$

$$\sqrt{5}(\sqrt{3} - \sqrt{2})$$

$$\sqrt{15} - \sqrt{10}$$

##### Exercise $$\PageIndex{30}$$

$$\sqrt{7}(\sqrt{6} - \sqrt{3})$$

##### Exercise $$\PageIndex{31}$$

$$\sqrt{8}(\sqrt{3} + \sqrt{2})$$

$$2(\sqrt{6} + 2)$$

##### Exercise $$\PageIndex{32}$$

$$\sqrt{10}(\sqrt{10} - \sqrt{5})$$

##### Exercise $$\PageIndex{33}$$

$$(1 + \sqrt{3})(2 - \sqrt{3})$$

$$-1 + \sqrt{3}$$

##### Exercise $$\PageIndex{34}$$

$$(5 + \sqrt{6})(4 - \sqrt{6})$$

##### Exercise $$\PageIndex{35}$$

$$(3 - \sqrt{2})(4 - \sqrt{2})$$

$$7(2 - \sqrt{2})$$

##### Exercise $$\PageIndex{36}$$

$$(5 + \sqrt{7})(4 - \sqrt{7})$$

##### Exercise $$\PageIndex{37}$$

$$(\sqrt{2} + \sqrt{5})(\sqrt{2} + 3\sqrt{5})$$

$$17 + 4\sqrt{10}$$

##### Exercise $$\PageIndex{38}$$

$$(2\sqrt{6} - \sqrt{3})(3\sqrt{6} + 2\sqrt{3})$$

##### Exercise $$\PageIndex{39}$$

$$(4\sqrt{5} - 2\sqrt{3})(3\sqrt{5} + \sqrt{3})$$

$$54 - 2\sqrt{15}$$

##### Exercise $$\PageIndex{40}$$

$$(3\sqrt{8} - 2\sqrt{2})(4\sqrt{2} - 5\sqrt{8})$$

##### Exercise $$\PageIndex{41}$$

$$(\sqrt{12} + 5\sqrt{3})(2\sqrt{3} - 2\sqrt{12})$$

$$-42$$

##### Exercise $$\PageIndex{42}$$

$$(1 + \sqrt{3})^2$$

##### Exercise $$\PageIndex{43}$$

$$(3 + \sqrt{5})^2$$

$$14 + 6\sqrt{5}$$

##### Exercise $$\PageIndex{44}$$

$$(2 - \sqrt{6})^2$$

##### Exercise $$\PageIndex{45}$$

$$(2 - \sqrt{7})^2$$

$$11 - 4\sqrt{7}$$

##### Exercise $$\PageIndex{46}$$

$$(1 + \sqrt{3x})^2$$

##### Exercise $$\PageIndex{47}$$

$$(2 + \sqrt{5x})^2$$

$$4 + 4\sqrt{5x} + 5x$$

##### Exercise $$\PageIndex{48}$$

$$(3 - \sqrt{3x})^2$$

##### Exercise $$\PageIndex{49}$$

$$(8 - \sqrt{6b})^2$$

$$64 - 16\sqrt{6b} + 6b$$

##### Exercise $$\PageIndex{50}$$

$$(2a + \sqrt{5a})^2$$

##### Exercise $$\PageIndex{51}$$

$$(3y - \sqrt{7y})^2$$

$$9y^2 - 6y\sqrt{7y} + 7y$$

##### Exercise $$\PageIndex{52}$$

$$(3 + \sqrt{3})(3 - \sqrt{3})$$

##### Exercise $$\PageIndex{53}$$

$$(2 + \sqrt{5})(2 - \sqrt{5})$$

$$-1$$

##### Exercise $$\PageIndex{54}$$

$$(8 + \sqrt{10})(8 - \sqrt{10})$$

##### Exercise $$\PageIndex{55}$$

$$(6 + \sqrt{7})(6 - \sqrt{7})$$

$$29$$

##### Exercise $$\PageIndex{56}$$

$$(\sqrt{2} + \sqrt{3})(\sqrt{2} - \sqrt{3}$$

##### Exercise $$\PageIndex{57}$$

$$(\sqrt{5} + \sqrt{2})(\sqrt{5} - \sqrt{2})$$

$$3$$

##### Exercise $$\PageIndex{58}$$

$$(\sqrt{a} + \sqrt{b})(\sqrt{a} - \sqrt{b})$$

##### Exercise $$\PageIndex{59}$$

$$(\sqrt{x} + \sqrt{y})(\sqrt{x} - \sqrt{y})$$

$$x - y$$

##### Exercise $$\PageIndex{60}$$

$$\dfrac{2}{5 + \sqrt{3}}$$

##### Exercise $$\PageIndex{61}$$

$$\dfrac{4}{6 + \sqrt{2}}$$

$$\dfrac{2(6 - \sqrt{2})}{17}$$

##### Exercise $$\PageIndex{62}$$

$$\dfrac{1}{3 - \sqrt{2}}$$

##### Exercise $$\PageIndex{63}$$

$$\dfrac{1}{4 - \sqrt{3}}$$

$$\dfrac{4 + \sqrt{3}}{13}$$

##### Exercise $$\PageIndex{64}$$

$$\dfrac{8}{2 - \sqrt{6}}$$

##### Exercise $$\PageIndex{65}$$

$$\dfrac{2}{3 - \sqrt{7}}$$

$$3 + \sqrt{7}$$

##### Exercise $$\PageIndex{66}$$

$$\dfrac{\sqrt{5}}{3 + \sqrt{3}}$$

##### Exercise $$\PageIndex{67}$$

$$\dfrac{\sqrt{3}}{6 + \sqrt{6}}$$

$$\dfrac{2\sqrt{3} - \sqrt{2}}{10}$$

##### Exercise $$\PageIndex{68}$$

$$\dfrac{2 - \sqrt{8}}{2 + \sqrt{8}}$$

##### Exercise $$\PageIndex{69}$$

$$\dfrac{4 + \sqrt{5}}{4 - \sqrt{5}}$$

$$\dfrac{21 + 8\sqrt{5}}{11}$$

##### Exercise $$\PageIndex{70}$$

$$\dfrac{1 + \sqrt{6}}{1 - \sqrt{6}}$$

##### Exercise $$\PageIndex{71}$$

$$\dfrac{8 - \sqrt{3}}{2 + \sqrt{18}}$$

$$\dfrac{-16 + 2\sqrt{3} + 24\sqrt{2} - 3\sqrt{6}}{14}$$

##### Exercise $$\PageIndex{72}$$

$$\dfrac{6 - \sqrt{2}}{4 + \sqrt{12}}$$

##### Exercise $$\PageIndex{73}$$

$$\dfrac{\sqrt{3} - \sqrt{2}}{\sqrt{3} + \sqrt{2}}$$

$$5 - 2\sqrt{6}$$

##### Exercise $$\PageIndex{74}$$

$$\dfrac{\sqrt{6a} - \sqrt{8a}}{\sqrt{8a} + \sqrt{6a}}$$

##### Exercise $$\PageIndex{75}$$

$$\dfrac{\sqrt{2b} - \sqrt{3b}}{\sqrt{3b} + \sqrt{2b}}$$

$$2\sqrt{6} - 5$$

## Exercises For Review

##### Exercise $$\PageIndex{76}$$

Simplify $$(\dfrac{x^5y^3}{x^2y})^5$$

##### Exercise $$\PageIndex{77}$$

Simplify $$(8x^3y)^2(x^2y^3)^4$$

$$64x^{14}y^{14}$$

##### Exercise $$\PageIndex{78}$$

Write $$(x-1)^4(x-1)^{-7}$$ so that only positive exponents appear.

##### Exercise $$\PageIndex{79}$$

Simpify $$\sqrt{27x^5y^{10}z^3}$$

$$3x^2y^5z\sqrt{3xz}$$
##### Exercise $$\PageIndex{80}$$
Simplify $$\dfrac{1}{2 + \sqrt{x}}$$ by rationalizing the denominator.