Exponents and Roots

Exponents and roots have the similar meaning in algebra as they do in arithmetic. Thus, if x represents any number then x2 = x · x, x3n means that x is to be taken as a factor n times. That is, xn is equal to x · x · x....with x appearing n times. In roots have three main parts such as index, radical and radicand.                                                    exponentsandroots

Examples on Exponents and Roots


Below are the examples on exponents and roots:

Example 1:

Write the following root value in exponent form.

`root(3)(27)`


Solution:

We can write the given root value in exponent form by using the following method.

If x is a positive integer that is greater than 1 and y is a real number then,

  `root(x)(y)` `=` `y^((1)/(x))`

Therefore we are using the above method to solve the given root problem.

That is, `root(3)(27)`` =` `27^((1)/(3))`

This is the form of exponent.

Answer: `27^((1)/(3))`


Example 2:

 Write the following root in exponent form. 

 `root(8)(5x)`


Solution: 

 We can write the given root value in exponent form by using the following method.

If x is a positive integers that is greater than 1 and y is a real number then,

  `root(x)(y)``=` `y^((1)/(x))`

Therefore we are using the above method to solve the given root problem.

That is, `root(8)(5x)` = `5x^((1)/(8))`

This is the form of exponent.

Answer: `5x^((1)/(8))`


Example 3:

Evaluate: `root(4)(81)`  

Solution:

If evaluate the above root, we can first convert root to exponent form and then evaluate.

In this problem, we can evaluate by using the following method.

First we are converting the given term in exponent form.

Then we evaluate it.

That is, `root(4)(81)`` =` `81^((1)/(4))`

Here `81=` `3^(4)`

Substitute this into the above term. Then we get

`3^4^(1/4)`

Here we can use the `a^m^(1/m)` formula:

`(a^m)^n` `=` `a^(mn)`

Therefore, `3^4^(1/4)`

Then we get 3

Answer: 3


Practice problems


1. Write the following root in exponent form.  Answer: `20^((1)/(3))`  

  `root(3)(20)`


2. Evaluate `root(4)(1296)` Answer: 6