Multiplying Exponents with Fractions – Rules – Examples

Rules for Multiplying Exponents with Fractions

If the exponent is in the fractional form, the fractional exponent rule is used. If the base of an expression is a fraction that is raised to an exponents, we use the same exponent rules that are used for bases that are whole numbers. The fractional exponent rule is given by:

When the fractional bases are same

Multiplying exponents with fractions can initially seem complex, but understanding the rules and breaking it down step by step makes it manageable.

Here’s a detailed explanation:

Rules for Multiplying Exponents with Fractions

Product of Powers Rule:

When multiplying like bases, we add the exponents. This rule applies to both whole numbers and fractions in the exponents.

a^m×aⁿ = a^(m+n)

Power of a Power Rule:

When raising a power to another power, we multiply the exponents. This also applies to fractional exponents.

{(a^m)}ⁿ = a^mn

Fractional Exponents:

A fractional exponent can be expressed as a root. For example:

a^m/n= n✓a^m

This is useful when dealing with fractional exponents.

Examples

Example 1: Multiplying with Fractional Exponents
Let’s multiply

a^1/2×a^1/3.

Using the product of powers rule:

a^1/2×a^1/3 =a^1/2+1/3

To add the fractions, find a common denominator:

1/2 = 3/6, 1/3 = 2/6

Now add the fractions:

3/6+2/6=5/6

So: a^1/2×a^1/3 = a^5/6

Example 2: Multiplying Different Bases with Fractional Exponents

Consider multiplying 2^1/4×3^1/4

Since the bases are different, we can’t combine the exponents directly:

2^1/4×3^1/4 = (2×3)^1/4 = 6^1/4

Example 3: Raising a Power to a Fractional Exponent

Let’s raise {(a)²}^1/3

Using the power of a power rule:

{(a)²}^1/3 = (a)^2x(1/3) =a^2/3

Example 4: Combining Rules

Let’s combine both rules in one problem:

{(a^1/2×a^1/3)}^3/2

First, apply the product of powers rule:

(a^1/2×a^1/3) = a^1/2+1/3 =a^3/6+2/6

= a^5/6

​Now, raise it to the 3/2 power:

= {(a^5/6)}^3/2

When using the power of a power rule, multiply the numerators and denominators of the fractional exponents.

= a^5/6×3/2

Simplifying: Always simplify the fractions where possible to get the final exponent in its simplest form.

= a^15/12 = a^5/4

Summary of Key Steps

• Adding Fractions: When adding exponents that are fractions, convert to a common denominator.
• Multiplying Fractions: When using the power of a power rule, multiply the numerators and denominators of the fractional exponents.
• Simplifying: Always simplify the fractions where possible to get the final exponent in its simplest form.

Understanding these steps will help we handle multiplying exponents with fractions effectively.