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Exponents and Roots

Exponents and roots (radicals) have the similar meaning in algebra as they do in arithmetic. Exponent of a number shows you how many times the number is to be used in a multiplication. It is shown as a small number to the right and above the base number. 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. Roots are nothing but the square roots. It is defined as, nth root of power value. The value of n can be different for different applications. The form y1/q is exponential form, if q is positive, then the $\sqrt[q]{y}$ is called the square roots form. In roots have three main parts such as index, radical and radicand.

Exponents and Roots

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What are Exponents?

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In mathematics, the number of multiplication of a same number is represented as an exponent. The base number is a number which is to be multiplied and the number of times of the base is in the power of the base.
For example, 895. This represents the base value 89 is to be multiplied 5 times.

Examples on Exponents and Roots

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Below are the examples on exponents and roots:

Solved Examples

Question 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.


 

Correct answer is `27^((1)/(3))`
Question 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 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(8)(5x)` = `5x^((1)/(8))`

This is the form of exponent.


 

Correct answer is `5x^((1)/(8))`
Question 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
 

Correct answer is 3

Exponents and Radicals Practice

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Practice Problems

Question 1: Write the following root in exponent form.

`root(3)(20)`


Question 2: Evaluate `root(4)(1296)`
More topics in Exponents and Roots
Powers and Exponents Positive Exponents
Negative Exponents Zero Exponent
Multiplying Exponents Dividing Exponents
Exponent rules Fractional Exponents
Rational Exponents Perfect Cubes
Equivalent Ratios Multiplying and Dividing Exponents
Exponentials and Logarithms
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