Introduction

Suppose that we wish to evaluate the expression \sqrt x + 3 for the value x=9. This type of expression is called a radical expression because it contains the radical symbol \sqrt{\phantom 0}. Remember that \sqrt x means "the positive square root of the number x. "

To evaluate this expression at the given value, we replace x with 9 in the given expression and simplify.

\begin{align*} \sqrt x + 3 &=\\[3pt] \sqrt 9 + 3 &=\\[3pt] \sqrt{3^2} + 3 &=\\[3pt] 3 + 3 &=\\[3pt] 6 \end{align*}

Important: You may have heard somewhere that \sqrt{9} = \pm 3, because (-3)^2=9 and (3)^2 = 9. However, when we write down \sqrt{9}, we always mean "the positive square root of 9 ".

If we want the negative square root of 9 , we write -\sqrt{9} = -3. Sometimes, we want both, in which case we would write \pm\sqrt{9}=\pm 3.

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Example: Evaluating a Radical Expression that Simplifies to a Whole Number

Evaluate \sqrt{x} + 2x -1 when x=16.

EXPLANATION

We replace x with 16 in the given expression, and simplify:

\begin{align*} \sqrt{x} + 2x -1 &=\\[3pt] \sqrt{16} + 2\cdot 16 -1 &=\\[3pt] \sqrt{4^2} + 32 -1 &=\\[3pt] 4 + 32 -1 &=\\[3pt] 35 \end{align*}

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Practice: Evaluating a Radical Expression that Simplifies to a Whole Number

Evaluate $x(\sqrt{x}-1)$ when $x=9.$

a
 $36$
b
 $9$
c
 $27$
d
 $18$
e
 $-18$
Practice: Evaluating a Radical Expression that Simplifies to a Whole Number

Evaluate $2\sqrt{2x} + 8$ when $x=18.$

a
 $30$
b
 $32$
c
 $16$
d
 $20$
e
 $4$
Example: Evaluating a Radical Expression that Does Not Simplify to a Whole Number

Evaluate \sqrt{3x-1} for x=3.

EXPLANATION

We replace x with 3 in the given expression, and simplify:

\begin{align*} \sqrt{3x-1} &=\\[3pt] \sqrt{3\cdot 3 -1}&=\\[3pt] \sqrt{9 -1}&=\\[3pt] \sqrt{8}& \end{align*}

We can simplify even further using the product rule:

\begin{align*} \sqrt{8}&=\\[3pt] \sqrt{4\cdot 2}&=\\[3pt] \sqrt{2^2\cdot 2}&=\\[3pt] \sqrt{2^2}\cdot \sqrt 2&=\\[3pt] 2\sqrt 2 \end{align*}

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Practice: Evaluating a Radical Expression that Does Not Simplify to a Whole Number

Evaluate $\sqrt{5x+4}$ for $x=8.$

a
 $4\sqrt {11}$
b
 $2\sqrt {11}$
c
 $2\sqrt {22}$
d
 $8\sqrt {11}$
e
 $4\sqrt {22}$
Practice: Evaluating a Radical Expression that Does Not Simplify to a Whole Number

Evaluate $\sqrt{7x-4}$ for $x=7.$

a
 $3\sqrt {15}$
b
 $3\sqrt 3$
c
 $9\sqrt 5$
d
 $3\sqrt 5$
e
 $5\sqrt {3}$
Example: Evaluating a Radical Expression Containing Fractions

Given that f(x) = \dfrac{\sqrt{9x+3}}{\sqrt{x-2}}, find f(5).

EXPLANATION

We replace x with 5 in the given expression, and simplify:

\begin{align*} f(5) & = \dfrac{\sqrt{9(5)+3}}{\sqrt{(5)-2}} \\[5pt] &= \dfrac{\sqrt{48}}{\sqrt{3}} \\[5pt] &= \sqrt{\dfrac{48}{3}}\\[5pt] &= \sqrt{16}\\[5pt] &= \sqrt{4^2}\\[5pt] &= 4 \end{align*}

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Practice: Evaluating a Radical Expression Containing Fractions

Given $f(x) = \dfrac{\sqrt{x}}{\sqrt{8-x}},$ find $f(6).$

a
 $\sqrt{2}$
b
 $1$
c
 $2$
d
 $\dfrac{1}{2}$
e
 $\sqrt{3}$
Practice: Evaluating a Radical Expression Containing Fractions

Given $h(x) = \dfrac{\sqrt{x+6}}{\sqrt{x}},$ find $h(2).$

a
 $6$
b
 $2\sqrt{2}$
c
 $2$
d
 $4\sqrt{2}$
e
 $8$
Example: Evaluating a Radical Expression Containing a Cube Root

Evaluate \sqrt[3]{x+6} at x=2.

EXPLANATION

We replace x with 2 in the given expression, and simplify as much as possible: \begin{align*} \sqrt[3]{x+6} &=\\[3pt] \sqrt[3]{2+6} &=\\[3pt] \sqrt[3]{8} &=\\[3pt] \sqrt[3]{2^3} &=\\[3pt] 2 \end{align*}

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Practice: Evaluating a Radical Expression Containing a Cube Root

Evaluate $\sqrt[3]{x+3}$ at $x=5.$

a
 $4$
b
 $\sqrt{2}$
c
 $2$
d
 $1$
e
 $3$
Practice: Evaluating a Radical Expression Containing a Cube Root

Evaluate $\sqrt[3]{7x+5}$ at $x=7.$

a
 $9\sqrt[3]{2}$
b
 $3\sqrt[3]{2}$
c
 $18$
d
 $3\sqrt[3]{6}$
e
 $27\sqrt[3]{2}$
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