Rational Exponents

Rational exponents (or fractional exponents) are exponents written as fractions, such as \frac{1}{2}, \frac{3}{4}, \frac{2}{3}​, and . They provide a way to express powers and roots in a single form.

In general a rational exponent is written as a^{\frac{p}{q}}.

where a is the base and p, q, are integers with q ≠ 0.

It is defined as: a^{\frac{p}{q}} = \left(\sqrt[q]{a}\right)^p = \sqrt[q]{a^p}. Here, the denominator q represents the root, and the numerator p represents the power.

Rational exponents expressed in terms of roots and powers-

For example,16^{\frac{1}{2}} = \sqrt{16} = 4,27^{\frac{1}{3}} = \sqrt[3]{27} =3, \text{and} \ 8^{\frac{2}{3}} = (\sqrt[3]{8})^2 =4.

Rational exponents follow all the usual laws of exponents and are widely used in mathematics and its applications.

Formulas

The general formula for rational exponents is

a^{m/n} = ⁿ√{a^m} = {a^m}^1/n

where

• a is the Base
• m is the Numerator (the Power)
• n is the Denominator (the Root)

The rational exponent formula relates a rational exponent a^{m/n} to its equivalent radical expression, allowing for easy conversion between the two notations.

Various formulas used in exponents also hold true for rational exponents that includes:

• a^m/n × a^p/q = a^{(m/n + p/q)}
• a^m/n ÷ a^p/q = a^{(m/n - p/q)}
• a^m/n × b^m/n = (ab) ^m/n
• a^m/n ÷ b^m/n = (a÷b)^m/n
• a^-m/n = (1/a)^m/n
• a^0/n = a⁰ = 1
• (a^m/n) ^p/q = a^{m/n × p/q}
• x^m/n = y ⇔ x = y^n/m

where, a^m/n, a^p/q are exponent with same base, b^m/n is exponent with different base

Rational Exponents and Radicals

Rational exponents can easily be written as radicals.

Take the rational exponent ap/q; this can be changed to radical form as

Step 1: Observe the given rational exponent, ap/q, and now the numerator of the rational exponent is the power. In a^p/q, p is the power.

Step 2: Again observe the given rational exponent, ap/q, and now the denominator of the rational exponent is the root. In a^p/q, q is the root.

Step 3: Write the base as the radicand, the power raised to the radicand, and the root as the index. i.e., a^p/q = ^p√a^q

This is explained by the example: (3)^2/3 = ³√(3)²

Rational Exponents vs Radical Notation

Rational exponents and radical notation are interchangeable forms of expressing the same mathematical concept, where a rational exponent is equivalent to a radical expression.

Simplifying Rational Exponents

We can easily simplify rational exponents by simplifying them into their simplest form using radicals. This is explained by the example added below:

Example: Simplify (27)^4/3

27^4/3 = (³√{27})⁴...(i)

Or

27^4/3 = ³√(27)⁴...(ii)

Form eq. (i)

27^4/3 = (³√{27})⁴

27^4/3 = (3)⁴

27^4/3 = 81

Rational Exponents with Negative Bases: Rational exponents with negative bases follow the same rules as those with positive bases, with considerations for even roots resulting in complex solutions.

Examples of rational exponents with negative bases are

• (-12)^8/9
• (-3/5)^11/7

Non-Integer Rational Exponents

Non-integer rational exponents represent fractional powers or roots of numbers extending beyond whole numbers and integers.

General format of a rational exponent is: a^p/q

where

• a is Base
• p/q is Exponent

Various examples of non-integer rational exponents are

(15)^0.3, (6)^2.5, (5)^2/3, (11)^1/2, (5/6)^3/4, etc.

Simplifying Non-Integer Rational Exponents

Non-integer rational exponents are solved in the same way as exponents with integers are solved. The following exponent rules are used to solve the exponents.

• a^m × aⁿ = a^m+n
• a^m / aⁿ = a^m-n
• (a^m)ⁿ = a^{m × n}
• a^{- m} = 1/a^m
• ⁿ√a^m = (am) = a^m/n

Example: Simplify (32)^3/5

32^{3/5} = ( \sqrt[5]{32} )^3 ...(i) \ Or\\[4pts] 32^{3/5} = \sqrt[5]{(32)^3} ...(ii)\\[4pts] \text{From eq. (i)}\\[4pts]32^{3/5} = ( \sqrt[5]{32} )^3\\[4pts] 32^{3/5} = (2)^3\\[4pts]32^{3/5} = 8

Applications of Rational Exponents

Rational exponents find applications in various fields such as engineering, physics, and finance in calculations involving fractional powers and roots.

• Solving various mathematical problems.
• In the field of physics and engineering.
• In economics and investment purposes, etc.

Related Articles

• Negative Exponents
• Laws of Exponents
• How to multiply and divide exponents
• Adding and Subtracting of  Exponents

Solved Examples

Example 1: Simplify 8^{2/3}

To simplify 8^{2/3}

We rewrite 8 as 2³

so we have (2³)^{2/3}

Applying the power of a power rule,

we get 2^{{3 × (2/3)}}

= 2² = 4

Example 2: Evaluate 27^{-2/3}

To evaluate 27^{-2/3}

We rewrite 27 as (3³)

So, we have (3³)^{-2/3}

Using the power of a power rule,

we get 3^{{3 × (-2/3)}}

= 3^{-2} = 1/9

Example 3: Simplify 16^3/2

To evaluate 16^3/2

We rewrite 16 as {2⁴}

So, we have {2⁴}^3/2

Using the power of a power rule,

We get 2^{{4 × (3/2)}}

= 2^{{2 × 3}}

= 2^{6} = 64

Example 4: Calculate the values of 25^1/2

To evaluate 25^1/2

 We rewrite 25 as {5²}

So, we have {5²}^1/2

Using Power of a power rule,

= 5^{{2 × (1/2)}} = 5

Example 5: Simplify the expression: 81^3/4

To evaluate 81^3/4

We rewrite 81 as {3⁴}

So, we have {3⁴}^3/4

Using the power of a power rule,

we get 3^{{4 × (3/4)}}

= 3^{3} = 27

Practice Questions

Q1. Simplify the expression: 27^5/3

Q2. Calculate the value of expression: 8^1/3

Q3. Evaluate the expression: 16^3/4

Q4. Simplify the expression: 4^3/2

Q5. Find the value of the expression: 81^5/4