## Section 12.7 Radicals with Indices Greater than Two

ΒΆThe expression

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

For example,

If \(n\) is a even positive integer and \(x\) is a positive real number, then

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,

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,

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,

Also,

If \(n\) is an odd positive integer, then it is always the case that

For example,

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

when neither \(x\) nor \(y\) are negative. For example,

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

### Exercises Exercises

Simplify each radical expression.

###### 1.

\(\sqrt[4]{81}\)

\(\sqrt[4]{81}=3\)

###### 2.

\(3\sqrt[3]{-27}\)

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

###### 3.

\(\frac{\sqrt[5]{32}}{8}\)

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

###### 4.

\(\sqrt[3]{-125 \cdot 64}\)

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

###### 5.

\(-\sqrt[7]{-128}\)

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