Irrational Exponents

Before we begin this lesson on irrational exponents, let's have a quick recap about the concept of exponents. Did you know that the first modern use of the word exponent was noticed in "Arithemetica Integra," written by an English author and mathematician Micheal Stifel in 1544. This concept has been in use for years and is still considered very important today.

                Irrational Exponents:

We will begin this lesson with an overview of irrational exponents and then proceed to solving them. Let's begin!

Lesson Plan 

What do you Mean by Irrational Exponents?

Before we learn about irrational exponents, let's go through the topic of exponents one more time.

Exponents show the number of times a number is replicated in multiplication.

For example, \( 4^2 = 4 \times 4 = 16 \)

Here, the exponent 2 is a whole number.

Irrational exponent is given as the exponent which is an irrational number and it cannot be expressed in \(\frac{p}{q}\) form.

Look at the figure shown at the beginning of this page to understand how irrational exponents are represented.

For the irrational exponent \(2^\sqrt{2}\) as \(\sqrt{2} \approx 1.414\), then \(2^\sqrt{2} \approx 2^1.414\)

The decimal number 1.414 can be written as, \(\frac{1414}{1000}\), hence \(2^\sqrt{2} = 2^{\frac{1414}{1000}}\)

This shows, we can raise 2 to the power 1414 and take the 1000th root of the resulting value.

This is just an approximation, and not the actual value of our exponential term. To that, one could reply: alright, then lets calculate it more accurately, by taking a better approximation of \(\sqrt{2} \approx 1.41421\)

So, \(2^\sqrt{2} = 2^{\frac{141421}{100000}}\)

Once again, we know how to interpret and calculate the right hand side term. If you were to now say: this is still an approximation, then one could in turn say: we can take an even better approximation of square root of 2 and find the value of this term even more accurately. And we could keep doing this and come closer and closer to the actual value of this term.

How do you Simplify Irrational Exponents?

We can simplify irrational exponents using the law of exponents.

Hence the irrational exponents rules are same as the law of exponents.

They are:

  • Law of Product 
  • Law of Quotient 
  • Law of Zero Exponent
  • Law of Negative Exponent
  • Law of Power of a Power 
  • Law of Power of a Product 
  • Law of Power of a Quotient


Example 1: Solve \( 3^\sqrt{2} \times 3^\sqrt{2} \)

Solution: As per the law of product we get, 

\( 3^{\sqrt{2}+\sqrt{2}} = 3^{2\sqrt{2}} = (3^2)^\sqrt{2} = (9)^\sqrt{2}\) 

\(\therefore\) \( 3^\sqrt{2} \times 3^\sqrt{2} = (9)^\sqrt{2} \)

Example 2: Solve \( (4^\sqrt{2})^3 \)
Solution: As per the law of power of a power we get,

\( (4^\sqrt{2})^3 = (4^3)^\sqrt{2} = (64)^\sqrt{2}\)

\(\therefore\) \( (4^\sqrt{2})^3 = 64^\sqrt{2}\)

Example 3: Solve \( \dfrac{7^{\sqrt{3}+2}}{49} \)
Solution: 49 can be written as \(7^2\)

Hence, on substitution of the value we get, 

\( \dfrac{7^{\sqrt{3}+2}}{7^2} \)

The numerator can be written as, 

\(7^{\sqrt{3}+2} = 7^{\sqrt{3}} \times 7^2\)

The irrational exponent will hence be written as, 

\( \dfrac{7^{\sqrt{3}} \times 7^2}{7^2} = 7^{\sqrt{3}}\)

\(\therefore\) \( \dfrac{7^{\sqrt{3}+2}}{49} = 7^{\sqrt{3}} \)

tips and tricks
Tips and Tricks


  • The exponential term x raised to the power \(\sqrt{y}\) is mathematically well-defined. To calculate its value, we could take better and better rational approximations of \(\sqrt{y}\) for the exponent, and come closer and closer to the actual value of this exponential term.
  • An exponent can be an arbitrary real number hence, no matter whether exponent is an integer, a non-integer rational number, or an irrational number it is possible to interpret and calculate that term.

What Is the Difference Between Rational and Irrational Exponents?

The differences between rational and irrational exponents are, 

  • Rational exponents can be expressed in \(\dfrac{p}{q}\) form while irrational exponents cannot be expressed in \(\dfrac{p}{q}\) form.
  • Irrational exponents are non repeating or infinite decimals while rational exponents are rational numbers.
  • The value of an irrational exponent when calculated is approximate in nature while the value of rational exponent is exact.   
important notes to remember
Important Notes 
  • Irrational exponents follow all the laws of exponents which are given as:
  1. \(a^m \times a^n = a^{m+n}\)
  2. \(a^m \div a^n = a^{m-n}\)
  3. \(a^0 = 1\)
  4. \(a^(-m) = \frac{1}{a^{m}}\)
  5. \((a^m)^n = a^{mn}\)
  6. \((ab)^m = a^{m}b^{m}\)
  7. \((\frac{a}{b})^m = \frac{a^m}{b^m}\)

Solved Examples

Example 1



Solve the below given question.

\( (3^\sqrt{2})^\sqrt{2} \)


As per the Law of Power of a Product, we get \((a^m)^n = a^{mn}\).

Applying the Law of Power of a Product we get,

\((3^\sqrt{2})^\sqrt{2} = 3^{\sqrt{2} \times \sqrt{2}}\)

As \((\sqrt{2})^2 = 2\)

Hence, the question is written as, 

\(3^{\sqrt{2} \times \sqrt{2}} = 3^2 = 9\)

\(\therefore\) Answer is 9
Example 2



Help Nicolas in simplifying the irrational exponents given below.

\( \dfrac{3^{\sqrt{2}+2}}{3^{\sqrt{2}-2}} \)


Nicolas will apply the Law of Quotient for the simplification of exponents.

He will write the exponents as,

\( \dfrac{3^{\sqrt{2}+2}}{3^{\sqrt{2}-2}} = (3)^{(\sqrt{2}+2)-(\sqrt{2}-2)}\)

As, the negative sign reverses all the signs within the brackets,

Hence, the exponents will now become,

\( (3)^{\sqrt{2}+2-\sqrt{2}+2} = (3^4) = 81\)

\(\therefore\) The answer is 81

Interactive Questions

Here are a few activities for you to practice.

Select/Type your answer and click the "Check Answer" button to see the result.


Let's Summarize

We hope you enjoyed learning about irrational exponents and irrational exponent definition with the simulations and practice questions. Now, you will be able to easily solve problems on simplifying irrational exponents, multiplying irrational exponents, rational and irrational exponents by using irrational exponents calculator and irrational exponent rules.

The mini-lesson targeted the fascinating concept of irrational exponents. The math journey around polynomial expressions starts with what a student already knows, and goes on to creatively crafting a fresh concept in the young minds. Done in a way that not only it is relatable and easy to grasp, but also will stay with them forever. Here lies the magic with Cuemath.

About Cuemath

At Cuemath, our team of math experts are dedicated to make learning fun for our favorite readers, the students! Through an interactive and engaging learning-teaching-learning approach, the teachers explore all angles of a topic. Be it problems, online classes, videos, or any other form of relation, it’s the logical thinking and smart learning approach that we at Cuemath believe in.

FAQs on Irrational Exponents

1. What is the exponent of a power?

The number which raised to the base of a power is known as exponent of a power. 

For example, for in the power \(5^8\), 8 is the exponent. 

2. What are 7 rules of exponents?

There are 7 rules of exponents,

  • Law of Product, \(a^m \times a^n = a^{m+n}\)
  • Law of Quotient, \(a^m \div a^n = a^{m-n}\) 
  • Law of Zero Exponent, \(a^0 = 1\)
  • Law of Negative Exponent, \(a^(-m) = \frac{1}{a^{m}}\)
  • Law of Power of a Power, \((a^x)^y = a^{xy}\) 
  • Law of Power of a Product, \((ab)^m = a^{m}b^{m}\) 
  • Law of Power of a Quotient, \((\frac{a}{b})^m = \frac{a^m}{b^m}\)

3. Are exponents rational numbers?

Exponents can be rational and irrational numbers.

4. Does the power rule work for irrational exponents?

Yes, the power rule work for irrational exponents.

5. Can an irrational number raised to an irrational power be rational?

Yes, an irrational number raised to an irrational power can be rational.

6. How do you solve irrational exponents?

The irrational exponents can be solved using the laws of exponents.

7. Is root 2 rational or irrational?

Root 2 is irrational.

8. Is "e" a rational or irrational number?

"e" is an irrational number.

9. How do you prove a root is irrational?

A root cannot be expressed in the \(\dfrac{p}{q}\) form, hence they are irrational.

10. Are logarithms exponents?

Yes logarithms are exponents.

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