Exponent Calculator: Solve Powers, Negative & Fractional Exponents

Dealing with exponents, or “powers,” is a fundamental concept in mathematics. Whether you’re solving a simple power like $2^{3}$ or tackling more complex negative and fractional exponents, our calculator is designed to give you an accurate answer instantly.

Modern Exponent Calculator

Calculates exponents and roots. Enter any two values (base, exponent, or result) to solve for the third.

How to Use Our Exponent Calculator

Our calculator is simple to use. You only need two numbers to find your solution. The problem is in the format of $X^{Y}$ .

Base (X): This is the number that will be multiplied by itself. Enter this value in the first field.
Exponent (Y): Also known as the “power” or “index,” this number tells you how many times to multiply the base by itself. Enter this value in the second field.

Once you’ve entered both the base and the exponent, the calculator will provide the result.

Understanding Your Results

The result of an exponent calculation is the total value after multiplying the base by itself the number of times indicated by the exponent. For example, $5^{3}$ means you multiply 5 by itself 3 times: $5 t im es 5 t im es 5 = 125$ .

However, exponents can be more than just positive whole numbers. Understanding the rules for different types of exponents is key to mastering them.

Rules of Exponents

Positive Exponents: This is the most straightforward rule. The exponent indicates the number of times the base is multiplied.
- Example: $4^{3} = 4 t im es 4 t im es 4 = 64$ .
Negative Exponents: A negative exponent means you need to calculate the reciprocal of the base raised to the positive exponent. In simple terms, you move the power to the bottom of a fraction.
- Rule: $X^{- Y} = 1/ X^{Y}$
- Example: $5^{-2} = 1/ 5^{2} = 1/25 = 0.04$ .
Fractional Exponents: A fractional exponent involves a root. The denominator of the fraction tells you the root of the base, and the numerator is the power.
- Rule: $X^{(A / B)} = s q r t [B] X^{A}$
- Example: $2 7^{(2/3)}$ means you first find the cube root of 27 (which is 3), and then you square that result (3^2). The answer is 9.
The Zero Exponent: Any non-zero number raised to the power of zero is always 1.
- Rule: $X^{0} = 1$ (for any $X n e q 0$ )
- Example: $1, 234, 56 7^{0} = 1$ .
The One Exponent: Any number raised to the power of one is simply itself.
- Rule: $X^{1} = X$
- Example: $5 9^{1} = 59$ .

Frequently Asked Questions

How do you multiply exponents?

You can only multiply exponents directly if they have the same base. In that case, you keep the base and add the exponents.

Rule: $X^{A} t im es X^{B} = X^{(A + B)}$
Example: $3^{2} t im es 3^{4} = 3^{(2 + 4)} = 3^{6} = 729$ . You cannot simplify the multiplication of exponents with different bases, like $2^{2} t im es 3^{3}$ .

How do you divide exponents?

Similar to multiplication, you can only divide exponents if they have the same base. You keep the base and subtract the exponents (top minus bottom).

Rule: $X^{A} / X^{B} = X^{(A - B)}$
Example: $4^{5} / 4^{3} = 4^{(5 - 3)} = 4^{2} = 16$ .

How do you add or subtract exponents?

There is no simple rule for adding or subtracting exponents, even if they have the same base. You must calculate each exponent term separately and then add or subtract the results.

Example: To solve $2^{3} + 2^{4}$ :
1. Calculate $2^{3} = 8$ .
2. Calculate $2^{4} = 16$ .
3. Add the results: $8 + 16 = 24$ . You cannot simply add the exponents. $2^{(3 + 4)} = 2^{7} = 128$ , which is incorrect.

What is a power of a power?

This is when you have an exponent expression that is raised to another power. In this case, you keep the base and multiply the exponents.

Rule: $(X^{A})^{B} = X^{(A t im es B)}$
Example: $(5^{2})^{3} = 5^{(2 t im es 3)} = 5^{6} = 15, 625$ .

What is the difference between $X^{Y}$ and $Y^{X}$ ?

The order matters greatly. The base is the number being multiplied, and the exponent is how many times it’s multiplied. Switching them produces a completely different result.

Example:
- $2^{5} = 2 t im es 2 t im es 2 t im es 2 t im es 2 = 32$ .
- $5^{2} = 5 t im es 5 = 25$ .

How do you handle decimal exponents?

Our calculator can handle decimal exponents. This is just another way of writing a fractional exponent. For example, an exponent of 0.5 is the same as an exponent of 1/2, which means you are finding the square root.

Example: $3 6^{0.5} = 3 6^{(1/2)} = s q r t 36 = 6$ .

What about a negative base?

If a negative base is raised to an even exponent, the result will be positive.
- Example: $(- 2)^{4} = (- 2) t im es (- 2) t im es (- 2) t im es (- 2) = 16$ .
If a negative base is raised to an odd exponent, the result will be negative.
- Example: $(- 2)^{3} = (- 2) t im es (- 2) t im es (- 2) = - 8$ . Pay close attention to parentheses. $(- 2)^{4}$ is not the same as $- 2^{4}$ . In the second case, you would calculate $2^{4}$ first and then apply the negative sign, resulting in -16.

What is scientific notation?

Scientific notation is a way of writing very large or very small numbers using exponents with a base of 10.

Example: The speed of light is about 300,000,000 meters per second. In scientific notation, this is written as $3 t im es 1 0^{8}$ .
A human hair’s width is about 0.00007 meters. In scientific notation, this is $7 t im es 1 0^{-5}$ .

Concrete Example: Calculating Compound Interest

Exponents are essential for calculating compound interest, a core concept in finance. The formula is: $A = P (1 + r / n)^{(n t)}$ , where:

A = the future value of the investment/loan, including interest
P = the principal amount (the initial amount of money)
r = the annual interest rate (in decimal form)
n = the number of times that interest is compounded per year
t = the number of years the money is invested or borrowed for

Let’s say you invest $1,000 (P) at an annual interest rate of 5% (r = 0.05), compounded monthly (n = 12), for 10 years (t). The exponent part of the formula is (nt), which is 12 * 10 = 120.

You would calculate (1 + 0.05/12), which is approximately 1.004167. Then you’d use our exponent calculator to find 1.004167^120, which is roughly 1.647. Finally, multiply by the principal: `$1,000 \times 1.647 = $1, 647$ . After 10 years, your investment would be worth $1,647.

What is an exponent of zero? Why is it 1?

Think of the division rule. We know that $X^{A} / X^{A} = 1$ (since any number divided by itself is 1). Using the exponent rule for division, we get $X^{(A - A)} = X^{0}$ . Therefore, $X^{0}$ must be equal to 1. This holds true for any non-zero base X.

Now that you’ve mastered exponents, you might be interested in their inverse operation. Use our Logarithm Calculator to find the exponent a base needs to be raised to in order to produce a certain number. For basic roots, you can also use our Square Root Calculator.

Ismael Vargas

An experienced software developer specializing in React, JavaScript, Django and Python, with more than six years’ expertise building full‑stack applications, data visualizations and cloud‑hosted solutions. He has a strong background in API integration, testing, and AWS services, delivering polished web products.

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