Fractional Exponents

Generally, exponents are of the form ab, where:

• a is referred to as the base 
• b as the exponent. 

If b is expressed as a fraction, it is called a fractional exponent. 

Fractional exponents help in expressing powers and roots simultaneously, with the general form\(x^{\frac{m}{n}} \), where,

•  x is the base 
   
• \(\frac{m}{n} \)is the exponent, 

Examples for fractional exponents are \(3^{\frac{1}{2}}\) and \(6^{\frac{4}{5}}\).

The table below clearly highlights the differences between fractional and integer exponents.
 

Rules simplify multiplying/dividing numbers with fractional exponents. Familiarity with whole-number exponents doesn't prevent common errors with fractional ones, which these rules address.

Rule 1: \(a^{1/m} \times a^{1/n} = a^{(1/m + 1/n)}\)
 

Rule 2: \(a^{1/m} \div a^{1/n} = a^{(1/m - 1/n)}\)
 

Rule 3: \(a^{1/m} \times b^{1/m} = (ab)^{1/m}\)
 

Rule 4: \(a^{1/m} \div b^{1/m} = \left(\frac{a}{b}\right)^{1/m}\)
 

Rule 5: \(a^{-m/n} = \left(\frac{1}{a}\right)^{m/n}\)

Use the below formula to break down the fractional exponents into their roots and powers. 

                             \(a^{m/n} = \sqrt[n]{a^m} = \left(\sqrt[n]{a}\right)^m \)

Either take the root first and then raise it to the power, or raise to the power first and then take the root, choosing the method that simplifies the calculation.

Example: Solve \(81^{\frac{1}{2}}\)

Solution: \(81^{\frac{1}{2}}\) can also be written as 81 because \(a^\frac{1}{2}\) = square root of a.

So, \(81^{{1}{2}} = √81 = 9\).

We should follow the laws of exponents to multiply fractional exponents, especially this rule, which is as follows:

\(a^m \cdot a^n = a^{m+n} \) , and,

For multiplying fractional exponents, it becomes

\(a^{1/m} \cdot a^{1/n} = a^{1/m + 1/n} \)

For example: Multiply 323 and 312.

Solution: To multiply 323 and 312

We have to add 23 and 12 

\(23 + 12 = 76\)

Therefore, \(323 \times 312 = 376 \)

In this section, we will see how to perform division on fractional exponents. The process can be divided into two types:

Type 1: Division of exponents with the same base but different powers
Since we have the same base but different powers, we can use the exponent subtraction rule:

\(\frac{a^{1/m}}{a^{1/n}} = a^{1/m - 1/n} \)

The powers are subtracted, and the difference is written on the common base.

For example:  Divide 323 and 312.
Solution: To Divide 323 and 312
We have to subtract the given powers, 23 and 12 

\(23 - 12 = 16\)

Therefore, \(323 \times 312 = 316 \)

Type 2: Division of fractional exponents with the same power but different bases.

This is expressed as \(a^{1/m} \cdot b^{1/m} = (ab)^{1/m} \)

For example: Divide 613 and 313.

Solution: To Divide 613 and 313.
We have to divide the given bases, 

\(613 \times 313 = (63)13 = 213 \)

Students can use these tips and tricks when working on fractional exponents to maintain clarity and increase efficiency.
 

• Understand the fraction: The numerator is the power, the denominator is the root.
   
• Convert between forms: Rewrite fractional exponents as radicals to simplify calculations.
   
• Use multiplication/division rules: Apply \(a^{m/n} \cdot a^{p/q} = a^{m/n + p/q} \) and \(\frac{a^{m/n}}{a^{p/q}} = a^{m/n - p/q} \).
   
• Watch negative exponents: \(a^{-m/n} = \frac{1}{a^{m/n}} \), handle reciprocals carefully.
   
• Practice with numbers: Use fractions and decimals to get comfortable with real-life examples and avoid common mistakes. 

Fractional exponents are useful in various real-life applications, especially in science, engineering, and finance, and below are some of them

• Essentially, smooth lighting and shading in computer graphics often involve using fractional exponents in power law calculations that govern light intensity and reflection.
   
• In structural analysis for architecture and design, formulas describing how materials behave under stress or how loads are supported frequently involve roots or fractional exponents in power laws (like square or cube roots of forces).
   
• Musical scales often use roots of 2 to determine the relationships between note frequencies and pitches; for instance, the 12-tone equal temperament scale uses the 12th root of 2 as the ratio between neighboring notes.
   
• Many natural diffusion processes, like heat spread or pollutant dispersal, often follow rules involving square roots of time (which are fractional powers).
   
• In radiation and nuclear science, fractional power laws are sometimes used in dose-response models to estimate safe radiation exposure levels based on biological effects.