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11.4 Addition and Subtraction of Radicals

Similar Radicals • Adding and Subtracting Radicals

When adding or subtracting algebraic expressions, including those with radicals, we combine similar terms. Thus, radicals must be similar, differing only in numerical coefficients, to be added. This means they must have the same order and same radicand.

In order to add radicals, we first express each radical in its simplest form, rationalize any denominators, and then combine those that are similar. For those that are not similar, we can only indicate the addition.

EXAMPLE 1 Adding similar radicals

$\begin{array}{l} 2 \sqrt{7} - 5 \sqrt{7} + \sqrt{7} = - 2 \sqrt{7} & all similar radicals \end{array}$ $\begin{array}{l} 2 \sqrt{7} - 5 \sqrt{7} + \sqrt{7} = - 2 \sqrt{7} & all similar radicals \end{array}$

This result follows the distributive law, as it should. We can write

$2 \sqrt{7} - 5 \sqrt{7} + \sqrt{7} = (2 - 5 + 1) \sqrt{7} = - 2 \sqrt{7}$ $2 \sqrt{7} - 5 \sqrt{7} + \sqrt{7} = (2 - 5 + 1) \sqrt{7} = - 2 \sqrt{7}$

We can also see that the terms combine just as

$2 x - 5 x + x = - 2 x$ $2 x - 5 x + x = - 2 x$
$\begin{array}{l} \sqrt[5]{6} + 4 \sqrt[5]{6} - 2 \sqrt[5]{6} = 3 \sqrt[5]{6} & all similar radicals \end{array}$ $\begin{array}{l} \sqrt[5]{6} + 4 \sqrt[5]{6} - 2 \sqrt[5]{6} = 3 \sqrt[5]{6} & all similar radicals \end{array}$

We note in (c) that we are able only to indicate the final subtraction because the radicals are not similar.

EXAMPLE 2 Adding radicals

$\begin{array}{rcl} \sqrt{2} + \sqrt{8} & = & \sqrt{2} + \sqrt{4 \times 2} = \sqrt{2} + \sqrt{4} \sqrt{2} \\ = & \sqrt{2} + 2 \sqrt{2} = 3 \sqrt{2} \end{array}$ $\begin{array}{rcl} \sqrt{2} + \sqrt{8} & = & \sqrt{2} + \sqrt{4 \times 2} = \sqrt{2} + \sqrt{4} \sqrt{2} \\ = & \sqrt{2} + 2 \sqrt{2} = 3 \sqrt{2} \end{array}$
$\begin{array}{rcl} \sqrt[3]{24} + \sqrt[3]{81} & = & \sqrt[3]{8 \times 3} + \sqrt[3]{27 \times 3} = \sqrt[3]{8} \sqrt[3]{3} + \sqrt[3]{27} \sqrt[3]{3} \\ = & 2 \sqrt[3]{3} + 3 \sqrt[3]{3} = 5 \sqrt[3]{3} \end{array}$ $\begin{array}{rcl} \sqrt[3]{24} + \sqrt[3]{81} & = & \sqrt[3]{8 \times 3} + \sqrt[3]{27 \times 3} = \sqrt[3]{8} \sqrt[3]{3} + \sqrt[3]{27} \sqrt[3]{3} \\ = & 2 \sqrt[3]{3} + 3 \sqrt[3]{3} = 5 \sqrt[3]{3} \end{array}$

CAUTION

Notice that $\sqrt{8}, \sqrt[3]{24},$ $\sqrt{8}, \sqrt[3]{24},$ and $\sqrt[3]{81}$ $\sqrt[3]{81}$ were simplified before performing the additions. We also note that $\sqrt{2} + \sqrt{8}$ $\sqrt{2} + \sqrt{8}$ is not equal to $\sqrt{2 + 8} .$ $\sqrt{2 + 8} .$

We note in the illustrations of Example 2 that the radicals do not initially appear to be similar. However, after each is simplified, we are able to recognize the similar radicals.

EXAMPLE 3 Simplify each—combine similar radicals

.4-2 Full Alternative Text
$\begin{array}{rcl} \sqrt{24} + \sqrt{\frac{3}{2}} & = & \sqrt{4 \times 6} + \sqrt{\frac{3 \times 2}{2 \times 2}} = \sqrt{4} \sqrt{6} + \sqrt{\frac{6}{4}} & rationalize denominator \\ = & 2 \sqrt{6} + \frac{\sqrt{6}}{2} = \frac{4 \sqrt{6} + \sqrt{6}}{2} = \frac{5}{2} \sqrt{6} & combine similar radicals \end{array}$ $\begin{array}{rcl} \sqrt{24} + \sqrt{\frac{3}{2}} & = & \sqrt{4 \times 6} + \sqrt{\frac{3 \times 2}{2 \times 2}} = \sqrt{4} \sqrt{6} + \sqrt{\frac{6}{4}} & rationalize denominator \\ = & 2 \sqrt{6} + \frac{\sqrt{6}}{2} = \frac{4 \sqrt{6} + \sqrt{6}}{2} = \frac{5}{2} \sqrt{6} & combine similar radicals \end{array}$

EXAMPLE 4 Radicals with literal numbers

\begin{array}{rcl} \sqrt{\frac{2}{3 a}} - 2 \sqrt{\frac{3}{2 a}} & = & \sqrt{\frac{2 (3 a)}{3 a (3 a)}} - 2 \sqrt{\frac{3 (2 a)}{2 a (2 a)}} = \sqrt{\frac{6 a}{9 a^{2}}} - 2 \sqrt{\frac{6 a}{4 a^{2}}} \\ = & \frac{1}{3 a} \sqrt{6 a} - \frac{2}{2 a} \sqrt{6 a} = \frac{1}{3 a} \sqrt{6 a} - \frac{1}{a} \sqrt{6 a} \\ = & \frac{\sqrt{6 a} - 3 \sqrt{6 a}}{3 a} = \frac{- 2 \sqrt{6 a}}{3 a} = - \frac{2}{3 a} \sqrt{6 a} \end{array}

$\begin{array}{rcl} \sqrt{\frac{2}{3 a}} - 2 \sqrt{\frac{3}{2 a}} & = & \sqrt{\frac{2 (3 a)}{3 a (3 a)}} - 2 \sqrt{\frac{3 (2 a)}{2 a (2 a)}} = \sqrt{\frac{6 a}{9 a^{2}}} - 2 \sqrt{\frac{6 a}{4 a^{2}}} \\ = & \frac{1}{3 a} \sqrt{6 a} - \frac{2}{2 a} \sqrt{6 a} = \frac{1}{3 a} \sqrt{6 a} - \frac{1}{a} \sqrt{6 a} \\ = & \frac{\sqrt{6 a} - 3 \sqrt{6 a}}{3 a} = \frac{- 2 \sqrt{6 a}}{3 a} = - \frac{2}{3 a} \sqrt{6 a} \end{array}$

EXAMPLE 5 Adding radicals—roof truss

A roof truss for a house has been designed as shown in Fig. 11.5. Find an expression for the exact number of linear feet of wood needed to construct the truss.

Figure 11.5 Full Alternative Text

The left side of the truss is shown in Fig. 11.6. By the Pythagorean theorem,

Figure 11.6 Full Alternative Text

\begin{array}{rcl} x^{2} & = & 16^{2} + 8^{2} \\ x & = & \sqrt{320} = \sqrt{64 (5)} = 8 \sqrt{5} \end{array}

$\begin{array}{rcl} x^{2} & = & 16^{2} + 8^{2} \\ x & = & \sqrt{320} = \sqrt{64 (5)} = 8 \sqrt{5} \end{array}$

Therefore, $a = \frac{8 \sqrt{5}}{2} = 4 \sqrt{5} .$ $a = \frac{8 \sqrt{5}}{2} = 4 \sqrt{5} .$

Since triangles ADE and ACB are similar triangles, $\frac{b}{8} = \frac{4 \sqrt{5}}{16},$ $\frac{b}{8} = \frac{4 \sqrt{5}}{16},$ or $b = 2 \sqrt{5} .$ $b = 2 \sqrt{5} .$ Side c can then be found using the Pythagorean theorem:

c^{2} = a^{2} + b^{2} = {(4 \sqrt{5})}^{2} + {(2 \sqrt{5})}^{2} = 4^{2} {(\sqrt{5})}^{2} + 2^{2} {(\sqrt{5})}^{2} = 16 (5) + 4 (5) = 100

$c^{2} = a^{2} + b^{2} = {(4 \sqrt{5})}^{2} + {(2 \sqrt{5})}^{2} = 4^{2} {(\sqrt{5})}^{2} + 2^{2} {(\sqrt{5})}^{2} = 16 (5) + 4 (5) = 100$

Thus, $c = \sqrt{100} = 10.$ $c = \sqrt{100} = 10.$ The total amount of linear feet needed for the truss is $32 + 1 + 1 + 10 + 10 + 8 \sqrt{5} + 8 \sqrt{5} + 2 \sqrt{5} + 2 \sqrt{5} = 54 + 20 \sqrt{5} ft$ $32 + 1 + 1 + 10 + 10 + 8 \sqrt{5} + 8 \sqrt{5} + 2 \sqrt{5} + 2 \sqrt{5} = 54 + 20 \sqrt{5} ft$ (approximately 99 ft).

Exercises11.4

In Exercises 1 and 2, simplify the resulting expressions if the given changes are made in the indicated examples of this section.

In Example 3(a), change $\sqrt{27}$ $\sqrt{27}$ to $\sqrt{45}$ $\sqrt{45}$ and then find the resulting simplified expression.
In Example 4, change $\sqrt{\frac{2}{3 a}}$ $\sqrt{\frac{2}{3 a}}$ to $\sqrt{\frac{1}{6 a}}$ $\sqrt{\frac{1}{6 a}}$ and then find the resulting simplified expression.

In Exercises 3–42, express each radical in simplest form, rationalize denominators, and perform the indicated operations.

$3 \sqrt{7} + 5 \sqrt{7}$ $3 \sqrt{7} + 5 \sqrt{7}$
$8 \sqrt{11} - 3 \sqrt{11}$ $8 \sqrt{11} - 3 \sqrt{11}$
$\sqrt{28} + \sqrt{5} - 3 \sqrt{7}$ $\sqrt{28} + \sqrt{5} - 3 \sqrt{7}$
$8 \sqrt{6} - \sqrt{12} - 5 \sqrt{6}$ $8 \sqrt{6} - \sqrt{12} - 5 \sqrt{6}$
$2 \sqrt{18} - \sqrt{27} + \sqrt{50}$ $2 \sqrt{18} - \sqrt{27} + \sqrt{50}$
$\sqrt{32} + 5 \sqrt{24} - \sqrt{54}$ $\sqrt{32} + 5 \sqrt{24} - \sqrt{54}$
$\sqrt{5} + \sqrt{16 + 4}$ $\sqrt{5} + \sqrt{16 + 4}$
$\sqrt{7} + \sqrt{36 + 27}$ $\sqrt{7} + \sqrt{36 + 27}$
$2 \sqrt{3 t^{2}} - 3 \sqrt{12 t^{2}}$ $2 \sqrt{3 t^{2}} - 3 \sqrt{12 t^{2}}$
$4 \sqrt{2 n^{2}} - \sqrt{50 n^{2}}$ $4 \sqrt{2 n^{2}} - \sqrt{50 n^{2}}$
$\sqrt{18 y} - 3 \sqrt{8 y}$ $\sqrt{18 y} - 3 \sqrt{8 y}$
$\sqrt{27 x} + 3 \sqrt{18 x}$ $\sqrt{27 x} + 3 \sqrt{18 x}$
$2 \sqrt{28} + 3 \sqrt{175}$ $2 \sqrt{28} + 3 \sqrt{175}$
$\sqrt{100 + 25} - 7 \sqrt{80}$ $\sqrt{100 + 25} - 7 \sqrt{80}$
$3 \sqrt{200} - \sqrt{162} - \sqrt{288}$ $3 \sqrt{200} - \sqrt{162} - \sqrt{288}$
$2 \sqrt{44} - \sqrt{99} + \sqrt{2} \sqrt{88}$ $2 \sqrt{44} - \sqrt{99} + \sqrt{2} \sqrt{88}$
$3 \sqrt{75 R} + 2 \sqrt{48 R} - 2 \sqrt{18 R}$ $3 \sqrt{75 R} + 2 \sqrt{48 R} - 2 \sqrt{18 R}$
$2 \sqrt{28} - \sqrt{108} - 6 \sqrt{175}$ $2 \sqrt{28} - \sqrt{108} - 6 \sqrt{175}$
$\sqrt{40} + \sqrt{\frac{5}{2}}$ $\sqrt{40} + \sqrt{\frac{5}{2}}$
$3 \sqrt{84} - \sqrt{\frac{3}{7}}$ $3 \sqrt{84} - \sqrt{\frac{3}{7}}$
$\sqrt{\frac{1}{2}} + \sqrt{\frac{25}{2}} - 4 \sqrt{18}$ $\sqrt{\frac{1}{2}} + \sqrt{\frac{25}{2}} - 4 \sqrt{18}$
$\sqrt{6} - \sqrt{\frac{2}{3}} - \sqrt{\frac{1}{24}}$ $\sqrt{6} - \sqrt{\frac{2}{3}} - \sqrt{\frac{1}{24}}$
$\sqrt[3]{81} + \sqrt[3]{3000}$ $\sqrt[3]{81} + \sqrt[3]{3000}$
$\sqrt[3]{- 16} + \sqrt[3]{54}$ $\sqrt[3]{- 16} + \sqrt[3]{54}$
$\sqrt[4]{32} - \sqrt[8]{4}$ $\sqrt[4]{32} - \sqrt[8]{4}$
$\sqrt[6]{\sqrt{2}} - \sqrt[12]{2^{13}}$ $\sqrt[6]{\sqrt{2}} - \sqrt[12]{2^{13}}$
$5 \sqrt{a^{3} b} - \sqrt{4 a b^{5}}$ $5 \sqrt{a^{3} b} - \sqrt{4 a b^{5}}$
$\sqrt{2 R^{2} I} + \sqrt{8} \sqrt{I^{3}}$ $\sqrt{2 R^{2} I} + \sqrt{8} \sqrt{I^{3}}$
$\sqrt{6} \sqrt{5} \sqrt{3} - \sqrt{40 a^{2}}$ $\sqrt{6} \sqrt{5} \sqrt{3} - \sqrt{40 a^{2}}$
$3 \sqrt{60 b^{2} n} - b \sqrt{135 n}$ $3 \sqrt{60 b^{2} n} - b \sqrt{135 n}$
$\sqrt[3]{24 a^{2} b^{4}} - \sqrt[3]{3 a^{5} b}$ $\sqrt[3]{24 a^{2} b^{4}} - \sqrt[3]{3 a^{5} b}$
$\sqrt[5]{32 a^{6} b^{4}} + 3 a \sqrt[5]{243 a b^{9}}$ $\sqrt[5]{32 a^{6} b^{4}} + 3 a \sqrt[5]{243 a b^{9}}$
$\sqrt{\frac{a}{c^{5}}} - \sqrt{\frac{c}{a^{3}}}$ $\sqrt{\frac{a}{c^{5}}} - \sqrt{\frac{c}{a^{3}}}$
$\sqrt{\frac{14 x}{21 y}} + \sqrt{\frac{27 y}{8 x}}$ $\sqrt{\frac{14 x}{21 y}} + \sqrt{\frac{27 y}{8 x}}$
$\sqrt[3]{a b^{- 1}} - \sqrt[3]{8 a^{- 2} b^{2}}$ $\sqrt[3]{a b^{- 1}} - \sqrt[3]{8 a^{- 2} b^{2}}$
$\sqrt{\frac{2 x y^{- 1}}{3}} + \sqrt{\frac{27 x^{- 1} y}{8}}$ $\sqrt{\frac{2 x y^{- 1}}{3}} + \sqrt{\frac{27 x^{- 1} y}{8}}$
$\sqrt{\frac{T - V}{T + V}} - \sqrt{\frac{T + V}{T - V}}$ $\sqrt{\frac{T - V}{T + V}} - \sqrt{\frac{T + V}{T - V}}$
$\sqrt{\frac{16}{x} + 8 + x} - \sqrt{1 - \frac{1}{x}}$ $\sqrt{\frac{16}{x} + 8 + x} - \sqrt{1 - \frac{1}{x}}$
$\sqrt{4 x + 8} + 2 \sqrt{9 x + 18}$ $\sqrt{4 x + 8} + 2 \sqrt{9 x + 18}$
$3 \sqrt{50 y - 75} - \sqrt{8 y - 12}$ $3 \sqrt{50 y - 75} - \sqrt{8 y - 12}$

In Exercises 43–48, express each radical in simplest form, rationalize denominators, and perform the indicated operations. Then use a calculator to verify the result.

$3 \sqrt{45} + 3 \sqrt{75} - 2 \sqrt{500}$ $3 \sqrt{45} + 3 \sqrt{75} - 2 \sqrt{500}$
$2 \sqrt{40} + 3 \sqrt{90} - 5 \sqrt{250}$ $2 \sqrt{40} + 3 \sqrt{90} - 5 \sqrt{250}$
$2 \sqrt{\frac{2}{3}} + \sqrt{24} - 5 \sqrt{\frac{3}{2}}$ $2 \sqrt{\frac{2}{3}} + \sqrt{24} - 5 \sqrt{\frac{3}{2}}$
$\sqrt{\frac{2}{7}} - 2 \sqrt{\frac{7}{2}} + 5 \sqrt{56}$ $\sqrt{\frac{2}{7}} - 2 \sqrt{\frac{7}{2}} + 5 \sqrt{56}$
$2 \sqrt[3]{16} - \sqrt[3]{\frac{1}{4}}$ $2 \sqrt[3]{16} - \sqrt[3]{\frac{1}{4}}$
$5 \sqrt[4]{810} - \sqrt[4]{\frac{5}{8}}$ $5 \sqrt[4]{810} - \sqrt[4]{\frac{5}{8}}$

In Exercises 49–56, solve the given problems.

Find the exact sum of the positive roots of $x^{2} - 2 x - 2 = 0$ $x^{2} - 2 x - 2 = 0$ and $x^{2} + 2 x - 11 = 0.$ $x^{2} + 2 x - 11 = 0.$
For the quadratic equation $a x^{2} + b x + c = 0,$ $a x^{2} + b x + c = 0,$ if a, b, and c are integers, the sum of the roots is a rational number. Explain.
Without calculating the actual value, determine whether $10 \sqrt{11} - \sqrt{1000}$ $10 \sqrt{11} - \sqrt{1000}$ is positive or negative. Explain.
The adjacent sides of a parallelogram are $\sqrt{12}$ $\sqrt{12}$ and $\sqrt{27}$ $\sqrt{27}$ units long. What is the perimeter of the parallelogram?
The two legs of a right triangle are $2 \sqrt{2}$ $2 \sqrt{2}$ and $2 \sqrt{6}$ $2 \sqrt{6}$ units long. What is the perimeter of the triangle?
The current I (in A) passing through a resistor R (in $Ω$ $Ω$ ) in which P watts of power are dissipated is $I = \sqrt{P / R} .$ $I = \sqrt{P / R} .$ If the power dissipated in the resistors shown in Fig. 11.7 is W watts, what is the sum of the currents in radical form?

Fig. 11.7
A rectangular piece of plywood 4 ft by 8 ft has corners cut from it, as shown in Fig. 11.8. Find the perimeter of the remaining piece in exact form and in decimal form.

Fig. 11.8
Three squares with areas of ${150 cm}^{2}, {54 cm}^{2},$ ${150 cm}^{2}, {54 cm}^{2},$ and ${24 cm}^{2}$ ${24 cm}^{2}$ are displayed on a computer monitor. What is the sum (in radical form) of the perimeters of these squares?

Answers to Practice Exercises

$2 \sqrt{7} - \sqrt{10}$ $2 \sqrt{7} - \sqrt{10}$
$7 \sqrt{11}$ $7 \sqrt{11}$
$\frac{(2 - 5 y) \sqrt{10 x y}}{5 y}$ $\frac{(2 - 5 y) \sqrt{10 x y}}{5 y}$

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Table of Contents for 11.4 Addition and Subtraction of Radicals

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Table of Contents for
11.4 Addition and Subtraction of Radicals