
## Section11.7Radicals 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[3]{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 $\sqrt[4]{81}$ 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[4]{81}=4 \text{ and } -\sqrt[4]{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[3]{-27}=-3 \text{, } \sqrt[3]{0}=0 \text{, and } \sqrt[3]{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[4]{16}=2 \text{ and } -\sqrt[4]{16}=-2\text{.} \end{equation*}

Also,

\begin{equation*} \sqrt[4]{-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[5]{32 \cdot -243}\amp=\sqrt[5]{32} \cdot \sqrt[5]{-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[4]{-81 \cdot 625} \neq \sqrt[4]{-81} \sqrt[4]{625} \end{equation*}

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

\begin{align*} \sqrt[4]{81 \cdot 625} \amp= \sqrt[4]{81} \sqrt[4]{625}\\ \amp=3 \cdot 5\\ \amp=15 \end{align*}

### Subsection11.7.1Exercises

###### 1

$\sqrt[4]{81}$

Solution

$\sqrt[4]{81}=3$

###### 2

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

Solution

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

###### 3

$\frac{\sqrt[5]{32}}{8}$

Solution

\begin{aligned}[t] \frac{\sqrt[5]{32}}{8}\amp=\frac{2}{8}\\ \amp=\frac{1}{4} \end{aligned}

###### 4

$\sqrt[3]{-125 \cdot 64}$

Solution

\begin{aligned}[t] \sqrt[3]{-125 \cdot 64}\amp=\sqrt[3]{-125} \cdot \sqrt[3]{64}\\ \amp=-5 \cdot 4\\ \amp=-20 \end{aligned}

###### 5

$-\sqrt[7]{-128}$

Solution

\begin{aligned}[t] -\sqrt[7]{-128}\amp=-(-2)\\ \amp=2 \end{aligned}