Exponents & Roots Calculator

What Is an Exponent?

An exponent tells you how many times to multiply a number by itself. It is written as a small raised number to the upper right of the base. In the expression bⁿ, b is the base (the number being multiplied) and n is the exponent — also called the power. You read bⁿ as "b raised to the power of n" or "b to the nth power."

At its core, a whole-number exponent is just shorthand for repeated multiplication. Instead of writing the same factor out many times, you write the base once and record how many copies there are.

For example, 2⁵ = 2 × 2 × 2 × 2 × 2 = 32. The base 2 appears five times because the exponent is 5. Likewise, 10³ = 10 × 10 × 10 = 1,000, and 3² = 3 × 3 = 9 (read "three squared"). An exponent of 3 is often read as "cubed," so 4³ is "four cubed" = 64.

This idea extends beyond counting copies. Negative exponents represent reciprocals, and fractional exponents represent roots — the same operation that this calculator handles in Root mode. Exponentiation also has an inverse: the logarithm calculator answers the reverse question, "what power do I raise the base to in order to get this number?"

How to Use the Exponent Calculator

This calculator evaluates powers and roots — including negative and fractional exponents — and shows the full working. To compute a power:

1. Enter the base. Type the number being raised — for example 2, 10, or −4. Negative bases are allowed; the calculator tracks the sign for you.
2. Enter the exponent. Type the power. It can be a positive whole number (2⁵), a negative number (5⁻⁴), or a decimal/fraction such as 0.5 or 2/3 for roots.
3. Press Calculate. Read the answer, then open the Steps tab for a worked solution, the Visual tab to see the repeated multiplication, and Why it works for the reasoning. No login is required.

Laws of Exponents (Rules Table)

The laws of exponents are the rules that let you combine and simplify powers without expanding everything by hand. Every rule below assumes a valid base (nonzero where division or a negative exponent is involved). Use this as a quick reference table.

The first three rules — product, quotient, and power-of-a-power — are the workhorses for simplifying expressions. The product and quotient rules only apply when the bases match; you add exponents when multiplying and subtract them when dividing. The two "distributing" rules (power of a product and power of a quotient) let the exponent reach every factor inside the parentheses. Expanding a power of a sum like (a + b)ⁿ is a different problem — that is what the binomial theorem calculator is for.

The −4² vs (−4)² Trap

This is the single most common exponent error, and it comes down to order of operations. Exponents are evaluated before negation (a unary minus acts like multiplying by −1, which happens last), so the placement of parentheses changes the answer.

• −4² = −(4²) = −(16) = −16. The exponent applies only to the 4. You square first, then attach the minus sign.
• (−4)² = (−4) × (−4) = +16. The parentheses make −4 the base, so the whole negative number gets squared and the two negatives cancel.

So −4² = −16 but (−4)² = 16. Whenever you want a negative number itself raised to a power, wrap it in parentheses. Note that an odd power of a negative base stays negative — for example (−2)³ = −8 — while an even power is always positive. Squaring (exponent 2) is also the operation behind the quadratic equation calculator.

Negative Exponents

A negative exponent does not make the result negative. Instead, it tells you to take the reciprocal of the positive power. The rule is a⁻ⁿ = 1/aⁿ (with a ≠ 0).

Why? Follow the pattern of dividing by the base each time the exponent drops by one: 2³ = 8, 2² = 4, 2¹ = 2, 2⁰ = 1, 2⁻¹ = 1/2, 2⁻² = 1/4. Each step down the ladder divides by 2, so going below zero produces fractions. The quotient rule says the same thing: a⁰ ÷ aⁿ = a⁰⁻ⁿ = a⁻ⁿ, and since a⁰ = 1, that is exactly 1/aⁿ.

Worked example — 5⁻⁴:

1. Flip to a reciprocal. 5⁻⁴ = 1 / 5⁴.
2. Evaluate the positive power. 5⁴ = 5 × 5 × 5 × 5 = 625.
3. Write the result. 5⁻⁴ = 1/625 = 0.0016.

The same logic moves factors between numerator and denominator: 1/2⁻³ = 2³ = 8, because flipping a reciprocal flips the sign of the exponent. Enter a negative exponent directly into the calculator above to see this worked out.

Fractional Exponents and Roots

A fractional exponent is how powers and roots connect — and it is what makes this calculator's Root mode possible. The denominator of the fraction is the root, and the numerator is the ordinary power. In symbols:

So an exponent of 1/2 is a square root, 1/3 is a cube root, and 1/n is the nth root. You can take the root first and then the power, or the power first and then the root — both orders give the same answer, so it is usually easier to take the root first to keep the numbers small.

Worked example — 64^(1/3):

1. Read the fraction. The denominator 3 means cube root; the numerator 1 means no extra power: 64^(1/3) = ³√64.
2. Find the cube root. Ask what number cubed gives 64. Since 4 × 4 × 4 = 64, the answer is 4.
3. Result: 64^(1/3) = 4.

Worked example — 27^(2/3):

1. Split the exponent. 27^(2/3) = (³√27)² — take the cube root, then square it.
2. Cube root first. ³√27 = 3, because 3 × 3 × 3 = 27.
3. Apply the power. 3² = 9, so 27^(2/3) = 9.

Zero, One, and Special Cases

A few exponent values come up constantly and are worth memorizing:

• a⁰ = 1 for any a ≠ 0. Anything (nonzero) to the power of 0 is 1 — this follows from the quotient rule, since aⁿ ÷ aⁿ = a⁰ and a number divided by itself is 1.
• a¹ = a. A power of 1 leaves the base unchanged: 7¹ = 7.
• 1ⁿ = 1 for any n. One multiplied by itself any number of times is always 1: 1⁵ = 1, 1⁻³ = 1.
• 0ⁿ = 0 for any positive n (0³ = 0), but 0 to a negative power is undefined because it would require dividing by 0.
• 0⁰ is a special case. In algebra and combinatorics it is conventionally defined as 1, but it is context-dependent and treated as indeterminate in some limit problems.

Worked Examples

Here is one fully worked example of each exponent type. Each can be checked against the calculator above.

Example 1 — Positive integer power: 3⁵

1. Write out the factors. 3⁵ = 3 × 3 × 3 × 3 × 3.
2. Multiply step by step. 3 × 3 = 9, × 3 = 27, × 3 = 81, × 3 = 243.
3. Result: 3⁵ = 243.

Example 2 — Dividing powers: 5³ ÷ 5²

1. Apply the quotient rule. Same base, so subtract exponents: 5³⁻² = 5¹.
2. Simplify. 5¹ = 5.
3. Check by expanding. (5 × 5 × 5) ÷ (5 × 5) = 125 ÷ 25 = 5. ✓

Example 3 — Negative exponent: 10⁻²

1. Take the reciprocal. 10⁻² = 1 / 10².
2. Evaluate the power. 10² = 100.
3. Result: 10⁻² = 1/100 = 0.01.

Example 4 — Fractional exponent: 64^(1/3)

1. Convert to a root. The denominator 3 means cube root: 64^(1/3) = ³√64.
2. Solve the root. 4 × 4 × 4 = 64.
3. Result: 64^(1/3) = 4.

Example 5 — Order-of-operations trap: −2⁴

1. Apply the exponent first. The power binds only to 2: −2⁴ = −(2⁴).
2. Evaluate the power. 2⁴ = 16.
3. Attach the sign last. −2⁴ = −16. (By contrast, (−2)⁴ = +16.)

Real-World Applications of Exponents

Exponents describe anything that grows or shrinks by repeated multiplication — which is why they appear well beyond the math classroom.

Compound interest

Money that earns interest on its interest grows by a fixed factor each period, so the total is an exponent of that factor: A = P(1 + r)ⁿ, where P is the starting amount, r is the rate per period, and n is the number of periods. Invest P = $1,000 at r = 5% for n = 3 years: A = 1000 × (1.05)³ = 1000 × 1.157625 ≈ $1,157.63.

Population and bacterial doubling

When a quantity doubles each time step, the count follows N = N₀ × 2ᵗ, where N₀ is the starting amount and t is the number of doublings. Starting from N₀ = 500 bacteria that double 6 times gives N = 500 × 2⁶ = 500 × 64 = 32,000. Exponents also power scientific notation (e.g., 3 × 10⁸ m/s), a compact way to write very large or very small numbers using powers of 10.

Common Mistakes to Avoid

Studying for AP Precalculus? Exponential functions are part of Unit 2 — 25–40% of the exam. See the 2027 score calculator or what changed for 2027.

Frequently Asked Questions

• What is 6 to the power of 4?

  6 to the power of 4 equals 1,296. It means 6 multiplied by itself four times: 6 × 6 × 6 × 6 = 36 × 36 = 1,296. The base is 6 and the exponent is 4.

• How do you multiply exponents with the same base?

  Keep the base and add the exponents: aᵐ × aⁿ = aᵐ⁺ⁿ. For example, 2³ × 2⁴ = 2⁷ = 128. This product rule only applies when the bases are identical — different bases must be evaluated separately.

• How do you divide exponents?

  Keep the base and subtract the exponents: aᵐ ÷ aⁿ = aᵐ⁻ⁿ. For example, 5⁵ ÷ 5² = 5³ = 125. If the bottom exponent is larger, the result becomes a negative exponent — that is, a fraction.

• How do you calculate a fractional exponent?

  A fractional exponent is a root: a^(m/n) = ⁿ√(aᵐ). The denominator is the root and the numerator is the power. For example, 27^(2/3) = (³√27)² = 3² = 9. Use Root mode on the calculator to enter these directly.

• What does a negative exponent mean?

  A negative exponent means the reciprocal of the positive power: a⁻ⁿ = 1/aⁿ. It does not make the answer negative. For example, 5⁻⁴ = 1/5⁴ = 1/625 = 0.0016. The base just cannot be zero.

• Why is any number to the power of 0 equal to 1?

  Because of the quotient rule: aⁿ ÷ aⁿ = aⁿ⁻ⁿ = a⁰, and any number divided by itself is 1. So a⁰ = 1 for every nonzero base a. For example, 7⁰ = 1 and 100⁰ = 1.

• What is the difference between -4² and (-4)²?

  They give different answers. -4² = -(4²) = -16 because the exponent applies only to 4 and the minus sign is applied last. But (-4)² = (-4) × (-4) = 16 because the parentheses make -4 the base. Order of operations is the reason.