## Section13.7Radicals with Indices Greater than Two

The expression

\begin{equation*} \sqrt[n]{x} \end{equation*}

is called a radical expression with an index of $n\text{.}$ It is generally read as "the ($n^{th}$-root) of $x\text{.}$" We usually restrict the index to integers greater than or equal to two.

If $n$ is an odd positive integer, then

\begin{equation*} \sqrt[n]{x}=y \text{ if and only if } y^n=x\text{.} \end{equation*}

For example,

\begin{equation*} \sqrt{8}=2 \text{ because } 2^3=8\text{.} \end{equation*}

If $n$ is a even positive integer and $x$ is a positive real number, then

\begin{equation*} \sqrt[n]{x}=y \text{ if and only if } y \text{ is the positive number such that } y^n=x\text{.} \end{equation*}

For example, both $(-3)^4=81$ and $3^4=81\text{,}$ but $\sqrt{81}$ has the positive value of $3\text{.}$ If we want to refer to the negative fourth-root of $81\text{,}$ we need to write a negative sign in front of the radical sign. In summary,

\begin{equation*} \sqrt{81}=4 \text{ and } -\sqrt{81}=-3\text{.} \end{equation*}

When no index of a radical expression is written, we assume that the index is two and we call the radical a square root.

When the index of a radical expression is three, it's not incorrect to refer to the radical as a third root, but it's more common to refer to the radical as a cube root. No other indexed-radical has a special name.

If the index, $n\text{,}$ is odd, the $\sqrt[n]{x}$ has a real number value for all real number values of $x\text{.}$ Furthermore, $\sqrt[n]{x}$ is negative if $x$ is negative, $\sqrt[n]{x}$ is zero if $x$ is zero, and $\sqrt[n]{x}$ is positive if $x$ is positive. For example,

\begin{equation*} \sqrt{-27}=-3 \text{, } \sqrt{0}=0 \text{, and } \sqrt{27}=3\text{.} \end{equation*}

If the index, $n\text{,}$ is even, the $\sqrt[n]{x}$ has a real number value if and only if $x$ is a non-negative real number. There are always two real number even roots of a positive real number, and $\sqrt[n]{x}$ represents the positive root whereas $-\sqrt[n]{x}$ represents the negative root. For example,

\begin{equation*} \sqrt{16}=2 \text{ and } -\sqrt{16}=-2\text{.} \end{equation*}

Also,

\begin{equation*} \sqrt{-16} \text{ is not a real number.} \end{equation*}

If $n$ is an odd positive integer, then it is always the case that

\begin{equation*} \sqrt[n]{x \cdot y}=\sqrt[n]{x} \cdot \sqrt[n]{y}\text{.} \end{equation*}

For example,

\begin{align*} \sqrt{32 \cdot -243}\amp=\sqrt{32} \cdot \sqrt{-243}\\ \amp=2\cdot -3\\ \amp=-6 \end{align*}

However, If $n$ is an even positive integer, then, over the real numbers it is only the case that

\begin{equation*} \sqrt[n]{x \cdot y}=\sqrt[n]{x} \cdot \sqrt[n]{y} \end{equation*}

when neither $x$ nor $y$ are negative. For example,

\begin{equation*} \sqrt{-81 \cdot 625} \neq \sqrt{-81} \sqrt{625} \end{equation*}

because neither of the fourth-roots of negative numbers exist (as real numbers). However,

\begin{align*} \sqrt{81 \cdot 625} \amp= \sqrt{81} \sqrt{625}\\ \amp=3 \cdot 5\\ \amp=15 \end{align*}

### ExercisesExercises

###### 1.

$\sqrt{81}$

Solution

$\sqrt{81}=3$

###### 2.

$3\sqrt{-27}$

Solution

\begin{aligned}[t] 3\sqrt{-27}\amp=3 \cdot -3\\ \amp=-9 \end{aligned}

###### 3.

$\frac{\sqrt{32}}{8}$

Solution

\begin{aligned}[t] \frac{\sqrt{32}}{8}\amp=\frac{2}{8}\\ \amp=\frac{1}{4} \end{aligned}

###### 4.

$\sqrt{-125 \cdot 64}$

Solution

\begin{aligned}[t] \sqrt{-125 \cdot 64}\amp=\sqrt{-125} \cdot \sqrt{64}\\ \amp=-5 \cdot 4\\ \amp=-20 \end{aligned}

###### 5.

$-\sqrt{-128}$

Solution

\begin{aligned}[t] -\sqrt{-128}\amp=-(-2)\\ \amp=2 \end{aligned}