Q. 54 - Intermediate Algebra

Question

Explain the difference of finding the $n^{th}$root of a number when the index is even compared to when the index is odd.

Step-by-Step Solution

Answer

When $n$is odd, the $n^{th}$root is positive or negative depending on the sign of the radical.

When n is even, the $n^{th}$root is positive or not real depending on the sign of the radical. 

Step 1. Explanation

We know that $n^{th}$ root of a number $x$, is a number $r$, which when raised to power $n$, gives $x$.

If $a^n=x$, then $a$ is the $n^{th}$root of $x$.

The principal $n^{th}$root of $x$ can be written as $\sqrt[n]{x}$.

Here, $n$ is the index of radical.

For any integer, when $n\ge 2$

1. When $n$ is odd, then $\sqrt[n]{x^n}=x$.
2. When $n$ is even, then $\sqrt[n]{x^n}=\left|x\right|$.

Step 2. Example

For example.

If we assume a number $3$.

1. $\sqrt[3]{3^3}=3$, since index is odd number.
2. $\sqrt[4]{3^4}=\left|3\right|$, since index is even number.