How to Solve Decimal Exponents

Co-authored by David Jia

Last Updated: February 22, 2024 Fact Checked

Calculating exponents is a basic skill students learn in pre-algebra. Usually you see exponents as whole numbers, and sometimes you see them as fractions. Rarely do you see them as decimals. When you do see an exponent that is a decimal, you need to convert the decimal to a fraction. Then, there are a number of rules and laws regarding exponents you can use to calculate the expression.

Part 1

Part 1 of 3:

Calculating a Decimal Exponent

Download Article

1
Convert the decimal to a fraction. To convert a decimal to a fraction , consider place value. The denominator of the fraction will be the place value. The digits of the decimal will equal the numerator. ^{[1]
X
Expert Source

David Jia
Academic Tutor

Expert Interview. 14 January 2021.}
- For example, for the exponential expression $81^{{0.75}}$ , you need to convert $0.75$ to a fraction. Since the decimal goes to the hundredths place, the corresponding fraction is ${\frac {75}{100}}$ .
2
Simplify the fraction , if possible. Since you will be taking a root corresponding to the denominator of the exponent’s fraction, you want the denominator to be as small as possible. Do this by simplifying the fraction. If your fraction is a mixed number (that is, if your exponent was a decimal greater than 1), rewrite it as an improper fraction.
- For example, the fraction ${\frac {75}{100}}$ reduces to ${\frac {3}{4}}$ , So, $81^{{0.75}}=81^{{{\frac {3}{4}}}}$
Advertisement
3
Rewrite the exponent as a multiplication expression. To do this, turn the numerator into a whole number, and multiply it by the unit fraction. The unit fraction is the fraction with the same denominator, but with 1 as the numerator.
- For example, since ${\frac {3}{4}}={\frac {1}{4}}\times 3$ , you can rewrite the exponential expression as $81^{{{\frac {1}{4}}\times 3}}$ .
4
Rewrite the exponent as a power of a power. Remember that multiplying two exponents is like taking the power of a power. So $x^{{{\frac {1}{b}}}}\times a$ becomes $(x^{{{\frac {1}{b}}}})^{{a}}$ . ^{[2]
X
Research source}
- For example, $81^{{{\frac {1}{4}}\times 3}}=(81^{{{\frac {1}{4}}}})^{{3}}$ .
5
Rewrite the base as a radical expression. Taking a number by a rational exponent is equal to taking the appropriate root of the number. So, rewrite the base and its first exponent as a radical expression.
- For example, since $81^{{{\frac {1}{4}}}}={\sqrt[ {4}]{81}}$ , you can rewrite the expression as $({\sqrt[ {4}]{81}})^{{3}}$ . ^{[3]
  X
  Research source}
6
Calculate the radical expression. Remember that the index (the small number outside the radical sign) tells you which root you are looking for. If the numbers are cumbersome, the best way to do this is using the ${\sqrt[ {x}]{y}}$ feature on a scientific calculator.
- For example, to calculate ${\sqrt[ {4}]{81}}$ , you need to determine which number multiplied 4 times is equal to 81. Since $3\times 3\times 3\times 3=81$ , you know that ${\sqrt[ {4}]{81}}=3$ . So, the exponential expression now becomes $3^{{3}}$ .
7
Calculate the remaining exponent. You should now have a whole number as an exponent, so calculating should be straightforward. You can always use a calculator if the numbers are too large.
- For example, $3^{{3}}=3\times 3\times 3=27$ . So, $81^{{0.75}}=27$ .

Part 2

Part 2 of 3:

Solving a Sample Problem

Download Article

$256^{{2.25}}$ .
2
Convert the decimal to a fraction. Since $2.25$ is greater than 1, the fraction will be a mixed number.
- The decimal $0.25$ is equal to ${\frac {25}{100}}$ , so $2.25=2{\frac {25}{100}}$ .
3
Simplify the fraction, if possible. You should also convert any mixed numbers to improper fractions .
- Since ${\frac {25}{100}}$ reduces to ${\frac {1}{4}}$ , $2{\frac {25}{100}}=2{\frac {1}{4}}$ .
- Converting to an improper fraction, you have ${\frac {9}{4}}$ . So, $256^{{2.25}}=256^{{{\frac {9}{4}}}}$ .
Since ${\frac {9}{4}}={\frac {1}{4}}\times 9$ , you can rewrite the expression as $256^{{{\frac {1}{4}}\times 9}}$ .
So, $256^{{{\frac {1}{4}}\times 9}}=(256^{{{\frac {1}{4}}}})^{{9}}$ .
$256^{{{\frac {1}{4}}}}={\sqrt[ {4}]{256}}$ , so you can rewrite the expression as $({\sqrt[ {4}]{256}})^{{9}}$ .
${\sqrt[ {4}]{256}}=4$ . So, the expression is now $(4)^{{9}}$ .
$(4)^{{9}}=4\times 4\times 4\times 4\times 4\times 4\times 4\times 4\times 4=262,144$ . So, $256^{{2.25}}=262,144$

Part 3

Part 3 of 3:

Understanding Exponents

Download Article

1
Recognize an exponential expression. An exponential expression has a base and an exponent. The base is the large number in the expression. The exponent is the smaller number. ^{[4]
X
Research source}
- For example, in the expression $3^{{4}}$ , $3$ is the base and $4$ is the exponent.
2
Identify the parts of an exponential expression. The base is the number that is being multiplied. The exponent tells you how many times the base is being used as a factor in the expression. ^{[5]
X
Research source}
- For example, $3^{{4}}=3\times 3\times 3\times 3=81$ .
3
Identify a rational exponent. A rational exponent is also called a fractional exponent. It is an exponent that takes the form of a fraction. ^{[6]
X
Research source}
- For example, $4^{{{\frac {1}{2}}}}$ .
4
Understand the relationship between radicals and rational exponents. Taking a number to the ${\frac {1}{2}}$ power is like taking the square root of the number. So, $x^{{{\frac {1}{2}}}}={\sqrt {x}}$ . The same is true for other roots and exponents. The denominator of the exponent will tell you which root to take: ^{[7]
X
Research source}
- $x^{{{\frac {1}{3}}}}={\sqrt[ {3}]{x}}$
- $x^{{{\frac {1}{4}}}}={\sqrt[ {4}]{x}}$
- $x^{{{\frac {1}{5}}}}={\sqrt[ {5}]{x}}$
- For example, $81^{{{\frac {1}{4}}}}={\sqrt[ {4}]{81}}=3$ . You know that 3 is the fourth root of 81 since $3\times 3\times 3\times 3=81$
5
Understand the exponential law of powers of powers. This law says that $(x^{{a}})^{{b}}=x^{{ab}}$ . In other words taking an exponent to another power is the same as multiplying the two exponents. ^{[8]
X
Research source}
- When working with rational exponents, this law looks like $x^{{{\frac {a}{b}}}}=(x^{{{\frac {1}{b}}}})^{{a}}$ , since ${\frac {1}{b}}\times a={\frac {a}{b}}$ . ^{[9]
  X
  Research source}
- It doesn't actually matter whether you do the root or the exponent part of the problem first. However, taking the root first will give you a smaller number to work with, which usually makes the problem easier to solve. ^{[10]
  X
  Expert Source
  
  David Jia
  Academic Tutor
  
  Expert Interview. 14 January 2021.}

Expert Q&A

Search

Add New Question

Question

Why do you calculate the root before the exponent?

David Jia
Math Tutor
David Jia is an Academic Tutor and the Founder of LA Math Tutoring, a private tutoring company based in Los Angeles, California. With over 10 years of teaching experience, David works with students of all ages and grades in various subjects, as well as college admissions counseling and test preparation for the SAT, ACT, ISEE, and more. After attaining a perfect 800 math score and a 690 English score on the SAT, David was awarded the Dickinson Scholarship from the University of Miami, where he graduated with a Bachelor’s degree in Business Administration. Additionally, David has worked as an instructor for online videos for textbook companies such as Larson Texts, Big Ideas Learning, and Big Ideas Math.

David Jia

Math Tutor

Expert Answer

You can do either one first, but it's easier to take the root first because it will give you a smaller number, which will be easier to work with in your head.

Thanks! We're glad this was helpful.
Thank you for your feedback.
If wikiHow has helped you, please consider a small contribution to support us in helping more readers like you. We’re committed to providing the world with free how-to resources, and even $1 helps us in our mission. Support wikiHow
Yes No

Not Helpful 1 Helpful 6
Question

How can you easily calculate a decimal exponent?

David Jia
Math Tutor
David Jia is an Academic Tutor and the Founder of LA Math Tutoring, a private tutoring company based in Los Angeles, California. With over 10 years of teaching experience, David works with students of all ages and grades in various subjects, as well as college admissions counseling and test preparation for the SAT, ACT, ISEE, and more. After attaining a perfect 800 math score and a 690 English score on the SAT, David was awarded the Dickinson Scholarship from the University of Miami, where he graduated with a Bachelor’s degree in Business Administration. Additionally, David has worked as an instructor for online videos for textbook companies such as Larson Texts, Big Ideas Learning, and Big Ideas Math.

David Jia

Math Tutor

Expert Answer

First, take the decimal and convert it to a fraction. The numerator of the fraction is an exponent and the denominator is going to be a root. For example, if you have a number like 8^2/3, you would take the cube root of 8 first, which is 2, then square that to get 4.

Thanks! We're glad this was helpful.
Thank you for your feedback.
If wikiHow has helped you, please consider a small contribution to support us in helping more readers like you. We’re committed to providing the world with free how-to resources, and even $1 helps us in our mission. Support wikiHow
Yes No

Not Helpful 1 Helpful 6
Question

How do I find 15^1.4?

Donagan

Top Answerer

First convert 1.4 to 14/10, and reduce to 7/5. An exponent of 7/5 means first raising to the seventh power and then finding the fifth root (or first finding the fifth root and then raising to the seventh power). Thus, in this case you would raise 15 to the seventh power and then find that number's fifth root.

Thanks! We're glad this was helpful.
Thank you for your feedback.
If wikiHow has helped you, please consider a small contribution to support us in helping more readers like you. We’re committed to providing the world with free how-to resources, and even $1 helps us in our mission. Support wikiHow
Yes No

Not Helpful 14 Helpful 33

See more answers

Ask a Question

200 characters left

Include your email address to get a message when this question is answered.

Submit

Video

Tips

Submit a Tip

All tip submissions are carefully reviewed before being published

Name

Please provide your name and last initial

Submit

Thanks for submitting a tip for review!

[1]

Expert Source

David Jia
Academic Tutor

[2]

[3]

[4]

[5]

[6]

[7]

[8]

[9]

[10]

Log in

How to Solve Decimal Exponents

Steps

Calculating a Decimal Exponent

Solving a Sample Problem

Understanding Exponents

Expert Q&A

Video

Tips

You Might Also Like

References

About This Article

Reader Success Stories

Did this article help you?

Quizzes

You Might Also Like

Featured Articles

Trending Articles

Featured Articles

Featured Articles

Watch Articles

Trending Articles

Quizzes

Explore Categories