Exponent Calculator

What is an Exponent?

Exponentiation is a mathematical operation, written as $a^n$, involving the base a and an exponent n. In the case where n is a positive integer, exponentiation corresponds to repeated multiplication of the base, n times.

Let me break that down with something concrete.

The calculator above accepts negative bases, but does not compute imaginary numbers. It also does not accept fractions, but can be used to compute fractional exponents, as long as the exponents are input in their decimal form. I've found this limitation rarely matters in everyday calculations.

Picture this. You need to multiply 3 by itself four times. 3 × 3 × 3 × 3. That's a lot to write, and it gets messy fast. In math, we use exponents so we can say the same thing with less effort: $3^4$. Here, 3 is the base. 4 is the exponent. The exponent tells you how many times to use the base as a factor.

Think of folding a piece of paper in half, then again, and again. After four folds, you have $2^4 = 16$ layers. Each fold doubles what came before. The exponent quietly keeps track of this progression without you having to count manually.

Anytime you see $a^n$, it means a multiplied by itself n times. Exponents appear when something grows or shrinks at the same rate like doubling bacteria or interest on savings. After t steps, you have $2^t$ as many as you started with. This isn't just theory, it's how compound interest actually works in your bank account.

If you're ever unsure, pause and ask: What's my base? What is the exponent asking me to do? That's all it takes to see what exponents really mean. I tell my students to verbalize the question out loud. It helps more than you'd expect.

Exponent Rules and Properties

When exponents that share the same base are multiplied, the exponents are added. This is probably the most useful rule you'll learn.

EX: $2^2 \times 2^4 = 4 \times 16 = 64$

Here's the same thing another way: $2^2 \times 2^4 = 2^{(2 + 4)} = 2^6 = 64$. See how much cleaner that second approach is?

When an exponent is negative, the negative sign is removed by reciprocating the base and raising it to the positive exponent. This trips people up constantly, but once you get it, it's simple.

EX: $2^{(-3)} = \frac{1}{2^3} = \frac{1}{8}$

When exponents that share the same base are divided, the exponents are subtracted. Think of it as the reverse of multiplication.

EX: $\frac{2^2}{2^4} = \frac{4}{16} = \frac{1}{4}$

Breaking it down: $\frac{2^2}{2^4} = 2^{(2-4)} = 2^{-2} = \frac{1}{2^2} = \frac{1}{4}$

When exponents are raised to another exponent, the exponents are multiplied. This is where things get interesting.

EX: $(2^2)^4 = 4^4 = 256$

Same calculation, cleaner: $(2^2)^4 = 2^{(2 \times 4)} = 2^8 = 256$

When multiplied bases are raised to an exponent, the exponent is distributed to both bases.

EX: $(2 \times 4)^2 = 8^2 = 64$

Alternatively: $(2 \times 4)^2 = 2^2 \times 4^2 = 4 \times 16 = 64$

Similarly, when divided bases are raised to an exponent, the exponent is distributed to both bases.

EX: $(2/5)^2 = (2/5) \times (2/5) = 4/25$

Or: $(2/5)^2 = 2^2/5^2 = 4/25$

When an exponent is 1, the base remains the same. This seems obvious, but it's worth stating explicitly.

When an exponent is 0, the result of the exponentiation of any base will always be 1, although some debate surrounds 00 being 1 or undefined. For many applications, defining 00 as 1 is convenient and standard practice.

Shown below is an example of an argument for $a^0=1$ using one of the previously mentioned exponent laws.

If $a^n \times a^m = a^{(n+m)}$

Then $a^n \times a^0 = a^{(n+0)} = a^n$

Thus, the only way for $a^n$ to remain unchanged by multiplication, and this exponent law to remain true, is for $a^0$ to be 1. The logic is airtight once you work through it.

When an exponent is a fraction where the numerator is 1, the nth root of the base is taken. Shown below is an example with a fractional exponent where the numerator is not 1. It uses both the rule displayed, as well as the rule for multiplying exponents with like bases discussed above. Note that the calculator can calculate fractional exponents, but they must be entered into the calculator in decimal form.

It is also possible to compute exponents with negative bases. They follow much the same rules as exponents with positive bases. Exponents with negative bases raised to positive integers are equal to their positive counterparts in magnitude, but vary based on sign. If the exponent is an even, positive integer, the values will be equal regardless of a positive or negative base. If the exponent is an odd, positive integer, the result will again have the same magnitude, but will be negative.

While the rules for fractional exponents with negative bases are the same, they involve the use of imaginary numbers since it is not possible to take any root of a negative number. An example is provided below for reference, but please note that the calculator provided cannot compute imaginary numbers, and any inputs that result in an imaginary number will return the result "NAN," signifying "not a number." The numerical solution is essentially the same as the case with a positive base, except that the number must be denoted as imaginary.

Let me show you the compact notation that mathematicians use:

The exponent formula is straightforward: $a^n = a \times a \times \dots \times a\ (n\ \text{times})$. The base a is raised to the power of n, is equal to n times multiplication of a.

For example: $2^5 = 2 \times 2 \times 2 \times 2 \times 2 = 32$. Simple, right?

Multiplying exponents:

Example: $2^3 \cdot 2^4 = 2^{(3+4)} = 2^7 = 128$

Example: $3^2 \cdot 4^2 = (3 \cdot 4)^2 = 12^2 = 144$

Dividing exponents:

Example: $2^5 / 2^3 = 2^{(5-3)} = 2^2 = 4$

Example: $8^2 / 2^2 = (8/2)^2 = 4^2 = 16$

Power of exponent:

Example: $(2^3)^4 = 2^{(3 \cdot 4)} = 2^{12} = 4096$ that's a big number from small inputs.

Radical of exponent:

Example: $2\sqrt{2^6} = 2^{(6 / 2)} = 2^3 = 8$

Negative exponent:

Example: $2^{-3} = \frac{1}{2^3} = \frac{1}{8} = 0.125$

Zero exponent:

Example: $4^0 = 1$. Always.

When we multiply two expressions that have the same base, we can add the exponents. If you see $a^m \times a^n$, you don't need to multiply the bases; you keep the base the same and add the exponents: $a^m \times a^n = a^{m+n}$.

Let's see this with numbers. Suppose you have $2^3 \times 2^4$. That's 2 × 2 × 2 (three times), then 2 × 2 × 2 × 2 (four more times). All together, that's seven factors of 2, or $2^7$. When I first learned this, it felt like discovering a cheat code.

When dividing expressions with the same base, subtract the exponents. This is a natural extension of the product rule, since division undoes multiplication: $a^m / a^n = a^{m-n}$.

For example, $5^6 / 5^2 = 5^{6-2} = 5^4$. This rule works as long as the base a is not zero. Dividing by zero is undefined in mathematics that's a hard boundary we can't cross.

If you have a base raised to one exponent, and then that whole thing raised to another exponent, you multiply the exponents: $(a^m)^n = a^{m \times n}$.

If you see $(3^2)^4$, you multiply the exponents: 2 × 4 = 8, so $(3^2)^4 = 3^8$. I use this rule constantly when simplifying algebraic expressions.

If you are raising a product to an exponent, you can give the exponent to each factor in the product: $(ab)^n = a^n b^n$.

For instance, $(2 \times 5)^3 = 2^3 \times 5^3 = 8 \times 125 = 1000$. You can break apart the product, handle each piece, and multiply the results.

If you raise a fraction to an exponent, you can apply the exponent to the numerator and denominator separately: $(a/b)^n = a^n / b^n$.

Suppose you have $(3/4)^2$. That's $3^2 / 4^2 = 9/16$. Clean and simple.

As we discussed earlier, any nonzero base raised to the power zero equals 1: $a^0 = 1$ (for a ≠ 0).

This rule is a direct result of the quotient rule. If you have $a^m / a^m$, the quotient rule tells us $a^{m-m} = a^0$. But any number divided by itself is 1, so $a^0 = 1$. The math forces this conclusion.

A negative exponent means you take the reciprocal of the base and then apply the positive exponent: $a^{-n} = 1 / a^n$.

This is just the quotient rule, extended to cases where the exponent in the denominator is larger than the numerator. Each rule fits naturally with the others. If you ever feel lost, write out what the exponent is asking you to do, and use these patterns as a guide. They are the foundation for working with exponents, whether you're simplifying algebraic expressions or calculating values in science and finance.

How to Solve Exponents Manually

Let's take one example for each main type and work through them together. Manual calculation builds understanding and helps you verify calculator results.

Common Exponent Mistakes to Avoid

It's easy to accidentally multiply or divide the base when you're really supposed to combine exponents. Here are some ways to pause and check your work.

A negative exponent does not make your answer negative. It signals a reciprocal, not a sign change. I've marked this mistake on hundreds of papers over the years.

Exponent rules only work when the bases or exponents match up. Take a moment to confirm compatibility before combining terms. Rushing past this check causes more errors than anything else.

Exponents should always be handled before other operations unless you see parentheses changing the order. If in doubt, follow the order of operations. Remember PEMDAS or BODMAS depending on where you learned math.

Fractional exponents can look like division, but they're about roots. Ask yourself if the question is really about finding a root, not halving the base. This confusion is completely natural at first.

Remember to apply the exponent to every part inside the parentheses, not just one piece of a product or quotient. Distribution works differently with exponents than with regular multiplication.

If you're working with variables, it often helps to simplify first, then substitute values. This can prevent small errors and confusion, especially when the algebra gets complex.

Any nonzero number to the power of zero is always one. It's a pattern that surprises many, but it holds true every time without exception.

Exponent Theory and Additional Resources

Algebra and Trigonometry: A Functions Approach by M. L. Keedy and Marvin L. Bittinger, published by Addison Wesley Publishing Company in 1982, page 11, is an excellent reference if you want more detail on Exponent Theory and can see Exponent Laws explained thoroughly.

To calculate fractional exponents, use our Exponent Calculator. To calculate root or radicals, use our Roots Calculator. These specialized tools handle edge cases better than general-purpose calculators.

Exponents help us describe growth, patterns, and repeated multiplication in a way that's both practical and powerful. The more you practice, the more naturally exponent rules will come to you. When you need support, tools like Symbolab's Exponents Calculator can help you learn each step, not just find answers. That distinction matters more than most people realize.

How Do You Calculate Exponents?

Exponents can be calculated by multiplying a number (the base) by itself a certain number of times (the exponent). It's that fundamental operation repeated simple in concept, sometimes challenging in execution.

What Are the 4 Types of Exponents?

The four types of exponents are positive integer, negative integer, zero, and fractional. Each behaves according to its own rules, though they all connect to the same underlying principles.

What Are Exponent Rules?

Exponent rules are the properties and formulas that govern the behavior of exponents. Some common exponent rules include:

• Exponent Product rule: $a^m * a^n = a^{(m+n)}$
• Exponent Quotient rule: $a^m / a^n = a^{(m-n)}$
• Exponent Power rule: $a^m * a^n = a^{(mn)}$
• Rule for exponentiating an exponent: $a^{(m^n)} = (a^m)^n$
• Zero Exponent rule: $a^0 = 1$
• Identity Exponent rule: $a^1 = a$

These six rules handle virtually every situation you'll encounter. Master them and you've mastered exponents.