Explore

Simplifying d^th-degree Radical Expressions Solutions

• Writing a radical expression with a base raised to a rational exponent allows you to use the    exponent rules    to simplify.   
  

• Radical expressions follow the exponent rules because they can be written in the form: $\sqrt[d]{x^n}=x^{\frac{n}{d}}$where n and d are    whole    numbers.

• The rules for simplifying radicands are true for    Real Numbers ℝ   .

• To simplify a radical expression, simplify out exact    roots    from the radicand using the    index   .

Note

Remember to use the Formula Sheet as a reference for exponent rules. This lesson extends from what began in Algebra 1 with exponent rules and simplifying radicals. (Algebra 1: Principles of Secondary Mathematics)

• The index d determines the    number    of roots in a simplified expression.

• The most common index is    2   , which is the square root.

  • The    principal    root is the non-negative square root of a non-negative real number. 

Note

You will learn how to work with negative radicands under the set of complex numbers in a later lesson. 

• For an    ODD    index of  $\sqrt[d]{x}$:

• For an    EVEN    index of  $\sqrt[d]{x}$:

• Once terms are simplified out of the radicand with an even index, determine if    absolute value bars    are needed as part of the simplified expression.

• For simplified roots:

Simplifying Examples Chart
	odd index	even index
$\mathbb{W}$ radicand	$\sqrt[3]{8}={\left(2^3\right)}^{\frac{1}{3}}=2$	$$\begin{gathered} \sqrt{9}={\left(3^2\right)}^{\frac{1}{2}}=3 \\ –\sqrt{9}=–{\left(3^2\right)}^{\frac{1}{2}}=–3 \\ \pm \sqrt{9}=\pm {\left(3^2\right)}^{\frac{1}{2}}=\pm 3 \end{gathered}$$
$–\mathrm{\mathbb{Z}}$ radicand	$\sqrt[3]{–8}={\left(–2^3\right)}^{\frac{1}{3}}=–2$	$\sqrt{–16} \mathrm{no} \mathrm{ℝ} \mathrm{roots}$
Variable radicand	$\sqrt[5]{h^{30}{m}^{15}}=h^{\frac{30}{5}}{m}^{\frac{15}{5}}=h^6{m}^3$	$\sqrt[4]{a^{24}}=a^{\frac{24}{4}}=a^6$ $\sqrt[4]{n^{12}{p}^{32}}=n^{\frac{12}{4}}{p}^{\frac{32}{4}}=\left|n^3\right|p^8$

Note

Problems with a number and variable, as well as a radicand remainder, are in the following examples.

Example 1

Simplify. Write answers in simplified radical form. 

Note

Because the index is even, the simplified terms raised to an odd power need absolute value bars.

Example 2

Simplify. Write answers in simplified radical form.

$\sqrt[3]{54p^5{q}^9}$

$$\begin{gathered} 54=2·3^3 \\ {2}^{\frac{1}{3} }{3}^{\frac{3}{3} }{p}^{\frac{5}{3} }{q}^{\frac{9}{3}} \end{gathered}$$

$$\begin{gathered} 2^{\frac{1}{3}} 3^1 p^{1\frac{2}{3}} q^3 \\ 3p{q}^3 \sqrt[3]{2p^2} \end{gathered}$$

Note

Because the index is odd, absolute value bars are not needed because negative values are possible with odd roots under ℝ.

Example 3

Simplify. Write answers in simplified radical form.

 $\sqrt[4]{81a^{15}{b}^{21}}$

$$\begin{gathered} 81=3^4 \\ {3}^{\frac{4}{4} }{a}^{\frac{15}{4} }{b}^{\frac{21}{4}} \\ {3}^1 a^{3\frac{3}{4}} b^{5\frac{1}{4}} \\ 3\left|a^3{b}^5\right| \sqrt[4]{a^{3}b} \end{gathered}$$