Rational Exponents

Rational exponents (also called fractional exponents) are expressions with exponents that are rational numbers. Rational exponents follow similar properties as integer exponents, including the product, quotient, and power rules. Rational exponents are used across various fields like physics, engineering and finance.

In this article, we will discuss the rational exponent's definition, their formula, solved examples and others in detail.

What are Rational Exponents?

Rational exponents are exponents that are expressed as fractions. They are a way of representing roots and powers simultaneously. We know that exponents are the way of representing numbers with powers, i..e. of the form a^m and rational exponents are the exponents where the exponent part(m) is a rational number, i.e. m is of the form p/q; (m = p/q).

Examples of rational exponents are: 3^4/5, (11)^-2/3, (-7/9)^1/3, etc.

Rational Exponents Definition

Rational exponents involve raising a number to a fractional power where the numerator of the fraction represents the power to which the base is raised and the denominator represents the root to be taken.

Rational Exponents Notation

General notation of rational exponents is a^p/q, where a is the base and p/q is a rational exponent. Rational exponents can also be written as 

a^p/q = ^q√(a)^p

Properties of normal exponents holds true for rational exponents also.

Rational Exponents Formulas

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.

a^{m/n} = ⁿ√{a^m}

This formula explains that raising a number a to the power of m/n is equivalent to taking the n-th root of a^m. It serves as the basis for understanding and solving expressions involving rational exponents. Various formulas used in exponets 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 exponnet with same base
• b^m/n is exponent with different base

Properties of Rational Exponents

Properties such as the power of a quotient, power of a product and power of a power apply to rational exponents, compared to integer exponents.

Rational exponents possess several properties that extend the laws of integer exponents to fractional exponents, enabling efficient manipulation of expressions involving powers and roots.

• Product Rule: For rational exponents a^{m/n} and b^{m/n} the product of two terms is (ab)^{m/n}

• Quotient Rule: When dividing two terms with rational exponents,

{a^{m/n}}/b{^m/n}} = ({a}/{b})^{m/n}

• Power Rule: To raise a term with a rational exponent to another power, the result is

(a^{m/n})^k = a^{(km/n)}

• Zero Exponent Rule: Any nonzero base raised to the power of zero is equal to one, i.e.,

a^{0} = 1

• Negative Exponent Rule: Reciprocal of a term with a rational exponent is obtained by changing the sign of the exponent, i.e.,

a^{-m/n} = 1/a^{m/n}

Rational Exponents and Radicals

Rational exponets can easily be written as radicals. This is explained using the steps added below:

Take the rational exponent a^p/q this can be changed to radical form as:

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

Step 3: Write, base as the radicand, power raising 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 and Radicals

Converting between rational exponents and radical notation involves understanding that the exponent represents both the power and the root of a number. To convert between rational exponents and radical notation, express the exponent as a fraction and identify the power and root accordingly.

Difference Between Rational Exponents and 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

Solution:

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 exponent with negative base are:

• (-12)^8/9
• (-3/5)^11/7
• (-q)^5/6
• (-a/b)^2/3

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. 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 = (a^m)^1/n = a^m/n

Applications of Rational Exponents

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

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

Examples on Rational Exponents

Example 1: Simplify 8^{2/3}

Solution:

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}

Solution:

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

Solution:

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

Solution:

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

Solution:

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 on Rational Exponents

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