Introduction

Suppose we need to evaluate an expression like 1+\sqrt{4}, where we have a square root sign. What would be the order of operations? It turns out that the root signs ( \sqrt{\phantom{x}} or \sqrt[3]{\phantom{x}} ) can be thought of as parentheses. In other words, we find the value under the square root first: \begin{align*} 1 + {\color{blue}\sqrt{4}} &= \\[5pt] 1 + {\color{blue}2} &= \\[5pt] 3 \end{align*}

Similarly, let's evaluate \big(1-\sqrt[3]{-27}\big) \div 2 .

To evaluate expressions with radicals, we use PEMDAS as before, but with the following additional rules:

  • expressions underneath radicals are treated the same as parentheses

  • radicals themselves are treated as exponents

Following the order of operations, we obtain:

\begin{align*} \require{cancel} \big(1-{\color{blue}\sqrt[3]{-27}}\big) \div 2 &= \\[5pt] \big(1-\sqrt[3]{(-3)^3}\big) \div 2 &= \\[5pt] (1-({\color{blue}-3})) \div 2 &= \\[5pt] 4 \div 2 &= \\[5pt] 2 \end{align*}

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Example: Evaluating Expressions Containing Radicals and One or Two Additional Operations

What is the value of 7\times \sqrt[3]{-8}?

EXPLANATION

To evaluate the given expression, we use PEMDAS. In terms of the order of operations:

  • expressions underneath radicals are treated the same as parentheses

  • radicals themselves are treated as exponents

Therefore, we first evaluate the value of the radical, and we obtain

\begin{align*} \require{cancel} 7\times{\color{blue}\sqrt[3]{-8}} &=\\[3pt] 7\times\sqrt[3]{(-2)\cdot (-2)\cdot (-2)} &=\\[3pt] 7\times\sqrt[3]{(-2)^3} &=\\[3pt] 7\times\left({\color{blue}-2}\right) &=\\[3pt] -14. \end{align*}

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Practice: Evaluating Expressions Containing Radicals and One or Two Additional Operations

What is the value of $2\times\sqrt{64}?$

a
 $18$
b
 $12$
c
 $14$
d
 $16$
e
 $8$
Practice: Evaluating Expressions Containing Radicals and One or Two Additional Operations

$\sqrt[3]{-27} - 15 =$

a
 $-42$
b
 $-15$
c
 $-12$
d
 $-24$
e
 $-18$
Example: Evaluating Expressions Containing Radicals and Multiple Operations

Find the value of the expression \sqrt{ (3^4+7^2-2) \div 2}.

EXPLANATION

To evaluate the given expression, we use PEMDAS. In terms of the order of operations:

  • expressions underneath radicals are treated the same as parentheses

  • radicals themselves are treated as exponents

Therefore, following the order of operations, we obtain

\begin{align*} \sqrt{ (3^4+7^2-2)\div 2}&=\\[3pt] \sqrt{ (81+49-2)\div 2}&=\\[3pt] \sqrt{ (130-2)\div 2}&=\\[3pt] \sqrt{ 128\div 2}&=\\[3pt] \sqrt{ 64}&=\\[3pt] \sqrt{8 \cdot 8}&=\\[3pt] \sqrt{8^2}&=\\[3pt] 8 &. \end{align*}

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Practice: Evaluating Expressions Containing Radicals and Multiple Operations

Find the value of $(\sqrt{81} +7 )\div (2-6).$

a
 $\dfrac{43}{4}$
b
 $\dfrac{29}{4}$
c
 $4$
d
 $2$
e
 $-4$
Practice: Evaluating Expressions Containing Radicals and Multiple Operations

Find the value of the expression $\sqrt{ 2^3\times(3^2-5) \div 2}.$

a
 $2$
b
 $4$
c
 $9$
d
 $16$
e
 $3$
Expressions Resulting in Radicals

Suppose we want to simplify the following expression:

-3\sqrt{\left(2^4-1\right) \div 3}\,.

The first thing to realize is that there is an invisible multiplication symbol here. The above expression is short for

-3\,{\color{red}{\times}}\,\sqrt{\left(2^4-1\right) \div 3}\,.

When multiplying a radical by a number, we usually omit the additional multiplication symbol.

Now, to simplify the given expression, we use PEMDAS. In terms of the order of operations:

  • Expressions underneath radicals are treated as if they are contained within parentheses (the P in PEMDAS), so we always deal with this first.

  • Radicals themselves are treated as exponents (the E in PEMDAS).

Therefore, following PEMDAS, we obtain

\begin{align*} -3 \sqrt{\left(2^4-1\right) \div 3} &=\\[5pt] -3\sqrt{\left(16-1\right) \div 3} &=\\[5pt] -3 \sqrt{15 \div 3} &=\\[5pt] -3 \sqrt{5} &. \end{align*}

And that's our answer!

To get a numerical approximation, we need to use a calculator. In this case, we get

-3 \sqrt{5} = -3\times \sqrt{5} \approx -6.708\,204

rounded to six decimal places.

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Example: Simplifying a Radical Expression

Simplify the expression \sqrt{((5^2-4) \div 3)-2}.

EXPLANATION

To simplify the given expression, we use PEMDAS. In terms of the order of operations,

  • expressions underneath radicals are treated the same as parentheses, and

  • radicals themselves are treated as exponents.

Therefore, following the order of operations, we obtain

\begin{align*} \sqrt{((5^2-4) \div 3)-2} &=\\[3pt] \sqrt{((25-4) \div 3)-2} &=\\[3pt] \sqrt{(21 \div 3)-2} &=\\[3pt] \sqrt{7-2} &=\\[3pt] \sqrt{5} &. \end{align*}

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Practice: Simplifying a Radical Expression

$-3\sqrt{(3^3-5) \div 2}=$

a
 $-9$
b
 $-3\sqrt{13}$
c
 $-3\sqrt{11}$
d
 $-12$
e
 $-3\sqrt{3}$
Practice: Simplifying a Radical Expression

$\sqrt{20} \times (2-6) =$

a
 $4\sqrt{20}$
b
 $-6\sqrt{20}$
c
 $-4\sqrt{20}$
d
 $-16$
e
 $20$
Example: Approximating Radical Expressions

What is the value of 2\sqrt{13\times 2 - 5} + 1 rounded to 3 decimal places?

EXPLANATION

To simplify the given expression, we use PEMDAS. In terms of the order of operations,

  • expressions underneath radicals are treated the same as parentheses, and

  • radicals themselves are treated as exponents.

Therefore, following the order of operations, we obtain

\begin{align*} 2\sqrt{13 \times 2-5} + 1 &=\\[3pt] 2\sqrt{26 - 5} + 1 &=\\[3pt] 2\sqrt{21} + 1 . \end{align*}

Now, we approximate the value of the radical using a calculator: \sqrt{21} \approx 4.582\,576 rounded to 6 decimal places.

Finally, we approximate the value of the expression:

\begin{align*} 2\sqrt{21} + 1 &\approx\\[3pt] ‭‭‭2(4.582\,576) + 1 &= \\[3pt] 9.165\,152 + 1 & = \\[3pt] 10.165\,152 & \approx \\[3pt] 10.165 \end{align*}

rounded to 3 decimal places.

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Practice: Approximating Radical Expressions

What is the value of $\sqrt{2 \times 4-6}$ rounded to $3$ decimal places?

a
 $1.548$
b
 $1.618$
c
 $1.392$
d
 $1.414$
e
 $1.212$
Practice: Approximating Radical Expressions

What is the value of $\sqrt{(24 \div 2) +12} - 3$ rounded to $3$ decimal places?

a
 $1.901$
b
 $1.899$
c
 $3$
d
 $-0.145$
e
 $2$
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