How do we interpret exponential terms where the exponent is not an integer but a non-integer rational number?

For example, consider the exponential term \(3^{\frac{1}{2}} \). What does this mean? This is a term which when multiplied with itself should give us 3. In simpler words, this is the square root of 3.

Now, what about \(3^{\frac{1}{5}} \)? This is a term which when multiplied with itself 5 times, should give us 3. In other words, this is the fifth root of 3.

Let's learn about these non-integer rational exponents today!

In this mini-lesson, we shall explore the topic of non-integer rational exponents,* *by finding answers to questions like what are non integer rational exponents, and how to simplify non integer rational exponents?

**Lesson Plan**

**What are Non-Integer Rational Exponents? **

Exponents with fractions are referred to as non-integer 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.

\(2^{0.5}, 5^{\frac{2}{3}}, 11^{\frac{1}{2}}\)

**How to Simplify Non-Integer Rational Exponents?**

The non-integer rational exponents can be solved in the same way in which exponents with integers are solved.

The following laws are used to solve the non-integer rational exponents.

- \( a^m \times a^n = a^{m + n}\)
- \( \dfrac{a^m}{ a^n} = a^{m - n}\)
- \( (a^m)^n= a^{m \times n}\)
- \( a^{-m} = \dfrac{1}{a^m}\)
- \( \sqrt [n] a^m= (a^m)^{\frac{1}{n}} =a^{\frac{m}{n}}\)

Here, 'a' is the base and 'm' and 'n' are the exponents, which are non zero integers.

**Example**

Observe the following examples which show how the fractional exponents are solved using the above laws.

- \(\begin{align}7^{\frac{2}{3}} \times 7^{\frac{3}{2}} &= 7^{\frac{2}{3} + \frac{3}{2}}\\&=7^ {\frac{2 \times 2 + 3 \times 3}{3 \times 3} } \\&= 7^{\frac{4 + 9}{6}}\\& = 7^{\frac{13}{6}}\end{align}\)
- \(\begin{align}(4^{\frac{-3}{5}})^{\frac{2}{3}} &= 4^{\frac{-3}{5} \times \frac{2}{3} } \\&= 4^{\frac{-2}{5}}\end{align}\)

- 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. \[ \begin{align}5^{\frac{1}{2}} &= 5^{0.2} \\11^{\frac{1}{3}} &= 5^{0.33} \\ 3^{\frac{1}{4}} &= 5^{0.25}\end{align}\]

**Solved Examples**

Example 1 |

Express \(11^{\frac{2}{3}} \) in its radical form.

**Solution**

Exponents and radicals can be transformed as follows:

\[\begin{align}11^{\frac{2}{3}}&=(11^2)^{\frac{1}{3}}\\&=121^{\frac{1}{3}}\\&=\sqrt[3] {121}\end{align} \]

\(\therefore \) The radical form is \( \sqrt[3] {121}\) |

Example 2 |

If \(10^{\frac{2}{3}} \times 10^{\frac{1}{2}} = 10^x \), find the value of x.

**Solution**

\[\begin{align}10^x&=10^{\frac{2}{3}} \times 10^{\frac{1}{2}}\\&= 10^{\frac{2}{3} + \frac{1}{2}}\\&=10^{\frac{2 \times 2 + 1 \times 3}{3 \times 2}}\\&=10^{\frac{4 + 3}{6}}\\&=10^{\frac{7}{6}}\end{align} \]

\(\therefore x = \frac{7}{6} \) |

Example 3 |

Solve \( \left(\dfrac{216}{125}\right)^{-\frac{2}{3}}\)

**Solution**

\[ \begin{align}\left(\dfrac{216}{125}\right)^{-\frac{2}{3}}&=\left(\dfrac{6^3}{5^3}\right)^{-\frac{2}{3}}\\&=\left(\dfrac{6}{5}\right)^{3 \times-\frac{2}{3}}\\&=\left(\dfrac{6}{5}\right)^{-2}\\&=\left(\dfrac{5}{6}\right)^{2}\\&=\dfrac{25}{36}\end{align}\]

\(\therefore\) The answer is \(\dfrac{25}{36} \) |

Example 4 |

Simplify \(\dfrac{36^{\frac{3}{4}} \times 16^{\frac{5}{8}}}{27 ^{\frac{5}{6}}} \)

**Solution**

\[ \begin{align}\dfrac{36^{\frac{3}{4}} \times 16^{\frac{5}{8}}}{27 ^{\frac{5}{6}}}&=\dfrac{(6^2)^{\frac{3}{4}} \times (2^4)^{\frac{5}{8}}}{(3^3) ^{\frac{5}{6}}}\\&=\dfrac{(2^2 \times 3^2)^{\frac{3}{4}} \times (2^4)^{\frac{5}{8}}}{(3^3) ^{\frac{5}{6}}} \\&=\dfrac{2^{2 \times\frac{3}{4}} \times 3^{2 \times\frac{3}{4}} \times 2^{4 \times \frac{5}{8}}}{3^ {3 \times \frac{5}{6}}}\\&=\dfrac{2^{\frac{3}{2}} \times 3^{\frac{3}{2}} \times 2^{\frac{5}{2}}}{3^ {\frac{5}{2}}}\\&=2^{\frac{3}{2}+\frac{5}{2}} \times 3^{\frac{3}{2} - \frac{5}{2}}\\&=2^{\frac{3+5}{2}} \times 3^{\frac{3-5}{2}}\\&=2^{\frac{8}{2}} \times 3^{\frac{-2}{2}}\\&=2^4 \times 3^{-1}\\&=\dfrac{16}{3}\end{align}\]

\(\therefore \) The answer is \(\dfrac{16}{3} \) |

Simplify the following expression.

\[\left(\dfrac{p^{\frac{2}{3}} \cdot q^{-\frac{1}{3}}}{p^{\frac{1}{4}} \cdot q^{\frac{3}{5}}}\right)^{\frac{4}{5}}\]

**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**

The mini-lesson targeted the fascinating concept of non integer rational exponents. The math journey around non integer rational exponents 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 is dedicated to making 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 worksheets, online classes, doubt sessions, or any other form of relation, it’s the logical thinking and smart learning approach that we, at Cuemath, believe in.

**FAQs on Non-Integer Rational Exponents**

### 1. Does an exponent have to be an integer?

The exponent need not be an integer. It can also be a decimal, fraction, a root number, or a negative number.

### 2. Do exponents add or multiply?

The exponents add or multiply only if the bases are equal.

The exponents are added if the bases are multiplied. \( a^m \times a^n = a^{m + n}\)

The exponents are multiplied if one exponent is placed over another exponent. \((a^m)^n = a^{m \cdot n} \)

### 3. What are the rules for rational exponents?

The rules for rational exponents are the same as for any other types of exponents.

- \( a^m \times a^n = a^{m + n}\)
- \( \dfrac{a^m}{ a^n} = a^{m - n}\)
- \( (a^m)^n= a^{m \times n}\)

### 4. What is the rational exponent property?

The rational exponent property states that a rational number exponent can be represented in the form of a fraction \(\dfrac{p}{q} \). \[ a^{\frac{p}{q}}= \sqrt[q] {a^p}\]

The exponent can also be written in the form of a surd.

### 5. How do you solve exponents step by step?

The following three steps are to be followed to simplify the exponents.

- Split the bases into their respective prime factors.
- Simplify it further to have a single base and a single exponent of the form \(a^n \).
- All the bases of the same type should be computed and written together following the above-mentioned rules of exponents.

### 6. Are negative exponents rational?

Rational exponents may be non positive integers, or non negative integers. For example, \[a^{\frac{-p}{q} }\]

### 7. How do you solve rational equations with negative exponents?

Rational equations with non positive integers as exponents are calculated in the same way as the positive exponents. The negative exponent is changed to non negative integers by moving the base to the other side of the fraction line. \[ a^{-n} = \dfrac{1}{a^n} \]

### 8. What are the 5 rules of exponents?

The five rules of exponents are as follows.

- \( a^m \times a^n = a^{m + n}\)
- \( \dfrac{a^m}{ a^n} = a^{m - n}\)
- \( (a^m)^n= a^{m \times n}\)
- \( a^{-m} = \dfrac{1}{a^m}\)
- \( \sqrt [n] a^m= (a^m)^{\frac{1}{n}} =a^{\frac{m}{n}}\)

The exponents in the above expression are non zero integers.

### 9. How do you divide rational exponents?

A rational exponent is divided by splitting the exponent. \[a^{\frac{p}{q}} = (a^{\frac{1}{q}})^p\]

### 10. Where are negative exponents used in real life?

In real life, non positive integers are used as exponents 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.