Non Integer Rational Exponents
When a number is raised to the power of rational numbers or decimals, i.e., when an exponent is a rational number or a decimal then the exponent is a noninteger rational exponent. For example, consider the exponential term 3^{½}. This implies a number when multiplied with itself would give us 3 ⇒ (y × y = 3). In simpler words, this is the square root of 3 which in its radical form is √3. Similarly, the fifth root of 3 is \(3^{\frac{1}{5}}\) or \(3^{0.2}\). In such exponentiations, the powers are nonintegers and are called the noninteger rational exponents. Let's learn to simplify the noninteger rational exponents.
1.  What Are Non Integer Rational Exponents? 
2.  How to Simplify Non Integer Rational Exponents? 
3.  Solved Examples 
4.  Practice Questions 
5.  FAQs on Non Integer Rational Exponents 
What Are Non Integer Rational Exponents?
Exponents with fractions and decimals are referred to as noninteger rational exponents. The general format of a rational exponent is: \(a^{\frac{p}{q}}\). Here 'a' is the base and the rational number, \(\frac{p}{q} \) is the exponent. Observe the following examples of non integer rational exponents. \(2^{0.5}, 5^{\frac{2}{3}}, 11^{\frac{1}{2}}\). Consider \(27^{\frac{2}{3}}\), where 2/3 is the non integer rational exponent. It can also be written in the radical form as: ^{3}√(27)^{2 }= (^{3}√27)^{2}
How to Simplify Non Integer Rational Exponents?
The noninteger rational exponents can be solved in the same way by which the exponents with integers are solved. Let us recall that if 'a' is the base and 'm' and 'n' are the exponents, which are non zero integers, the following exponent rules are used to solve the exponents.
 a^{m} × a^{n} = a^{m+n}
 a^{m} / a^{n} = a^{mn}
 (a^{m})^{n} = a^{m} ^{×}^{ n}
 a^{ m} = 1/a^{m}
 \( \sqrt [n] a^m= (a^m)^{\frac{1}{n}} =a^{\frac{m}{n}}\)
Observe the following examples which show how the same exponential laws are used to solve the fractional exponents as well.
 \(\begin{align}7^{\frac{2}{3}} \times 7^{\frac{3}{2}} &= 7^{\frac{2}{3} + \frac{3}{2}}\\&=7^ {\frac{2 \times 2 + 3 \times 3}{6} } \\&= 7^{\frac{4 + 9}{6}}\\& = 7^{\frac{13}{6}}\end{align}\)
 \(\begin{align}(4^{\frac{3}{5}})^{\frac{2}{3}} &= 4^{\frac{3}{5} \times \frac{2}{3} } \\&= 4^{\frac{2}{5}}\end{align}\)
Related topics on Non Integer Rational Exponents
 Operations on Exponential Terms
 Negative Exponents
 Exponents Formula
 Exponential Equations
 Fractional Exponents
Tips And Tricks
 Exponents and radicals can be converted from one form to another. \[ a^{\frac{p}{q}} =\sqrt[q] {a^p} \]
 Rational exponents can also be written as decimal exponents. \[ \begin{align}5^{\frac{1}{2}} &= 5^{0.5} \\11^{\frac{1}{3}} &= 5^{0.33} \\ 3^{\frac{1}{4}} &= 5^{0.25}\end{align}\]
Solved Examples on Non Integer Rational Exponents

Example 1:Express \(11^{\frac{2}{3}} \) in its radical form.
Solution:Exponents and radicals can be transformed as follows:
\[\begin{align}11^{\frac{2}{3}}&=(11^2)^{\frac{1}{3}}\\&=121^{\frac{1}{3}}\\&=\sqrt[3] {121}\end{align} \] Therefore, \(11^{\frac{2}{3}} \) in its radical form = \(\sqrt[3]{121}\)

Example 2:If \(10^{\frac{2}{3}} \times 10^{\frac{1}{2}} = 10^x \), find the value of x.
Solution:\[\begin{align}10^x&=10^{\frac{2}{3}} \times 10^{\frac{1}{2}}\\&= 10^{\frac{2}{3} + \frac{1}{2}}\\&=10^{\frac{2 \times 2 + 1 \times 3}{3 \times 2}}\\&=10^{\frac{4 + 3}{6}}\\&=10^{\frac{7}{6}}\end{align} \] Therefore, x = 7/6. 
Example 3:Solve \( \left(\dfrac{216}{125}\right)^{\frac{2}{3}}\)
Solution:\[ \begin{align}\left(\dfrac{216}{125}\right)^{\frac{2}{3}}&=\left(\dfrac{6^3}{5^3}\right)^{\frac{2}{3}}\\&=\left(\dfrac{6}{5}\right)^{3 \times\frac{2}{3}}\\&=\left(\dfrac{6}{5}\right)^{2}\\&=\left(\dfrac{5}{6}\right)^{2}\\&=\dfrac{25}{36}\end{align}\] Therefore, \(\left(\frac{216}{125}\right)^{\frac{2}{3}}\) = 25/36
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Practice Questions on Non Integer Rational Exponents
Frequently Asked Questions(FAQs)
What Is a Non Integer Rational Exponent?
The Non Integer rational exponent is an exponent that can be represented in the form of a fraction p/q. \(a^{\frac{p}{q}}= \sqrt[q] {a^p}\). p/q can be a fraction or a decimal.
Does an Exponent have to be an Integer?
An exponent need not be an integer. It can also be a decimal, fraction, a root number, or a negative number. The exponents can also be non integer rational exponents as a^{¾ }or m^{¼ }or n^{0.25}
Do Non Integer Rational Exponents Add or Multiply?
The exponents are added if the bases are multiplied. a^{m} × a^{n} = a^{m+n} whereas, the exponents are multiplied if one exponent is placed over another exponent. (a^{m})^{n} = a^{m} ^{×}^{ n }, where m and n are non integer rational exponents.
How Do you Solve Non Integer Rational Exponents?
Know the laws of exponents that are used to solve for the integers as exponents.
 a^{m} × a^{n} = a^{m+n}
 a^{m} / a^{n} = a^{mn}
 (a^{m})^{n} = a^{m} ^{×}^{ n}
 a^{ m} = 1/a^{m}
 \( \sqrt [n] a^m= (a^m)^{\frac{1}{n}} =a^{\frac{m}{n}}\)
Apply the same rules for solving non integer rational exponents also.
Are Non Integer Rational Exponents negative?
Non Integer Rational Exponents can be non positive integers,in the form of \(a^{\frac{p}{q} }\). For example: a^{½}
Where Are Negative Non Integer Rational Exponents Used in Real Life?
In real life, negative non integer rational exponents are used as to show how small a thing is. For example, Zoologists use negative exponents to measure the different parts of bats which are extremely small to measure.