Power of a Power Calculator

Q: How do I calculate the power of a power?

To calculate the power of a power , follow these steps: Determine the first and second powers. Multiply the powers together. Raise the base to the resulting power. That's it! This is how to raise an exponent to an exponent.

Q: What is 5 to the power of 5 to the power of 5?

298,023,223,876,953,125. Here's how to calculate 5 to the power of 5 to the power of 5: Identify the powers: 5 and 5 . Multiply them together: 25 . Raise the base (5) to the power of 25: (5 5 ) 5 = 5 25 = 298,023,223,876,953,125 .

Q: What is the power of a power property?

The power of a power property states that when we raise one power to another power, we multiply the exponents . In mathematical notation, this is expressed as (b m ) n = b (m × n) , where b is the base, m is the first power, and n is the second power.

Q: Does the power of a power law work with negative exponents?

Yes , the rule also applies to negative exponents. When dealing with negative exponents, multiply as usual, bearing in mind that the signs will affect the final result. For example, (b -3 ) 7 = b (-3 × 7) = b -21 .

Creators

Claudia Herambourg

Claudia Herambourg is an aspiring computational linguist with a passion for both words and numbers. She holds a Bachelor’s degree in English Literature and Mathematical Studies from University College Cork, Ireland. She is currently pursuing her Master’s degree in Linguistics at Sorbonne Nouvelle University in Paris, specializing in computational linguistics. In her spare time, you'll probably find her reading a book of Japanese literature that nobody has ever heard of, on a train going nowhere. See full profile

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Reviewers

Joanna Śmietańska, PhD

Joanna ŚmietańskaPhD, AGH University of Krakow, Cracow, Poland

Website

Research Gate

Joanna Śmietanska-Nowak is an academic teacher at AGH University of Science and Technology in Kraków. Her research interests focus on protein crystallography and modulated systems, though she has previously worked on photocatalysis and energy storage systems, including supercapacitors and Li-ion batteries. She has participated in and led several research projects funded by the Polish National Science Centre. While working at Omni, she primarily developed calculators related to physics, chemistry, and mathematics. In her free time, she enjoys playing board games, cooking, hiking, and brewing beer. See full profile

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and Steven Wooding

Steven Wooding

Steven Wooding is a physicist by training with a degree from the University of Surrey specializing in nuclear physics. He loves data analysis and computer programming. He has worked on exciting projects such as environmentally aware radar, using genetic algorithms to tune radar, and building the UK vaccine queue calculator. Steve is now the Editorial Quality Assurance Coordinator here at Omni Calculator, making sure every calculator meets the standards our users expect. In his spare time, he enjoys cycling, photography, wildlife watching, and long walks. See full profile

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Welcome to our power of a power calculator, an easy-to-use tool for calculating an exponent raised to an exponent. In this article, you'll discover:

What is the power of a power law;
An example of how to calculate the power of a power by hand; and
How to use our raising a power to a power calculator.

Ready to dive into this powerful mathematical concept?

Definition and formula: What is the power of a power law?

The power of a power property (or law) is a fundamental concept in the exponent rules. This law states that if the base (b) raised to one power (m) is raised to another power (n), then both powers are multiplied, and the base remains the same.

The formula to calculate a power raised to a power is as follows:

\rm({b^m)^n} = \rm{b^{m \times n}}

where:

$\rm b$ — The base;
$\rm m$ — The first power;
$\rm n$ — The second power.

The power of a power calculator is far from being our one and only exponentiation-related calculator. If you're curious, don't hesitate to have a look at our multiplying exponents calculator and learn how to solve equations such as $\rm{y^a} \times \rm{x^b}$ . Be careful not to confuse these two calculators!

How to use our power of a power calculator

Our exponent raised to an exponent calculator is very easy to use. Follow these steps:

First, determine what part of the expression you want to find:
- The solution a such as $\text{a} = \rm({b^m)^n}$ ;
- The base b;
- The first power m; or
- The second power n.
Once you know what you want to calculate, fill in the three fields for which you have the data.
The raising a power to a power calculator will show the result in the target field.
But that's not all! A new section containing a step-by-step solution appears at the bottom of the calculator. For example, if you're looking to find the first power m, open this section and you'll have a quick tutorial explaining how to use the logarithmic function to find it.

Example: How to raise an exponent to an exponent by hand

Suppose you want to calculate $\text{a} = \rm({2^3)^4}$ .

First, identify the data that you have. You know:
- The base: 2;
- The first power: 3; and
- The second power: 4.
You have the necessary data to find the solution a to the expression.
Then, rewrite the expression using the exponent raised to an exponent formula:

$\text{a} = (2^3)^4 = 2^{3 \times 4} = 2^{12}$
Finally, solve the simplified expression. You can use an online calculator, a scientific calculator, or paper and pencil if you have time. Remember that $2^{12}$ is 2 multiplied by itself 12 times.

2^{12} = \underbrace{2 \times 2 \times \cdots \times 2 \times 2}_\textrm{12 times} = 4,\!096

Congratulations! You've found the result of this power of two by yourself.

FAQs

How do I calculate the power of a power?

To calculate the power of a power, follow these steps:

Determine the first and second powers.
Multiply the powers together.
Raise the base to the resulting power.
That's it! This is how to raise an exponent to an exponent.

What is 5 to the power of 5 to the power of 5?

298,023,223,876,953,125. Here's how to calculate 5 to the power of 5 to the power of 5:

Identify the powers: 5 and 5.
Multiply them together: 25.
Raise the base (5) to the power of 25: (5⁵)⁵ = 5²⁵= 298,023,223,876,953,125.

What is the power of a power property?

The power of a power property states that when we raise one power to another power, we multiply the exponents. In mathematical notation, this is expressed as (b^m)ⁿ = b^{(m × n)}, where b is the base, m is the first power, and n is the second power.

Does the power of a power law work with negative exponents?

Yes, the rule also applies to negative exponents. When dealing with negative exponents, multiply as usual, bearing in mind that the signs will affect the final result. For example, (b^-3)⁷ = b^{(-3 × 7)} = b^-21.