What are exponents?

An exponent (also called a “power”) is a compact way to describe repeated multiplication. When you write a^b, you’re saying “multiply a by itself b times.” Here, a is the base and b is the exponent. For example, 2^5 means: 2 × 2 × 2 × 2 × 2, which equals 32.

Exponents show up everywhere: compound growth (money, bacteria, population), scientific notation (very large or small numbers), physics formulas, computer science, and everyday arithmetic shortcuts. Once you really get the “base repeated b times” idea, the rules of exponents stop feeling like memorization and start feeling like common sense.

• Base (a): the repeated factor.
• Exponent (b): how many times the base is multiplied by itself (for integers).
• Power: the whole expression a^b or the result.

This calculator is designed to be both fast and educational: it gives the number immediately, but it also explains what happened, especially when the exponent is a small integer where the “repeated multiplication” story is the clearest.

The core formula (a^b)

For positive integers b, the definition is simple: a^b is a multiplied by itself b times. But exponent notation also includes a few special cases that are worth learning because they show up constantly.

• Exponent of 1: a^1 = a.
• Zero exponent: for any non-zero base, a^0 = 1.
• Negative exponent: a^(-b) = 1 / a^b (for a ≠ 0).
• Fractional exponent: a^(1/n) means the n-th root of a (real outputs require a ≥ 0 when n is even).
• General decimals: calculators compute these using logarithms/exponentials; negative bases can leave the real numbers.

The main reason exponent rules feel powerful is that they compress huge multiplication patterns into tiny symbols. Below are the exponent laws people use most often — plus the intuition behind each one.

Exponent laws (the ones that actually matter)

These rules are the cheat codes of algebra. They all come from one idea: write out what the exponent means, then count how many identical factors you have.

Rule: a^m × a^n = a^(m+n). Why? Because you are combining m copies of a with n copies of a.

• 2^3 × 2^4 = (2×2×2) × (2×2×2×2) = 2^7 = 128

Rule: a^m / a^n = a^(m-n) (for a ≠ 0). Division cancels matching factors.

• 10^5 / 10^2 = 10^3 = 1000

Rule: (a^m)^n = a^(m·n). Raising something to n means repeating that whole block n times.

• (3^2)^4 = 3^8 = 6561

Rule: (ab)^n = a^n b^n. Multiplying the same product repeatedly multiplies each part repeatedly.

• (2×5)^3 = 10^3 = 1000 and also 2^3×5^3 = 8×125 = 1000

Rule: a^(-n) = 1/a^n. Think of it as moving a factor across a fraction bar.

• 2^-3 = 1/2^3 = 1/8 = 0.125

If you can explain these in your own words, you won’t need to memorize anything. The calculator reinforces the idea by printing a “steps” line for small integer exponents.

Worked examples (with human-friendly logic)

Problem: What is 4^3?
Logic: Multiply 4 by itself three times: 4×4×4. That’s 16×4 = 64.

Problem: What is 10^-2?
Logic: Negative means reciprocal: 10^-2 = 1/10^2 = 1/100 = 0.01. This is why negative exponents are common in scientific notation for tiny numbers.

Problem: Compare -3^2 and (-3)^2.
Logic: Exponents happen before the leading negative sign unless parentheses force otherwise. So -3^2 is -(3^2) = -9, but (-3)^2 = 9.

Problem: What is 9^0.5?
Logic: 0.5 is the same as 1/2, so this is a square root: 9^(1/2) = √9 = 3.

Want the calculator to show the clearest “multiply the base repeatedly” steps? Use an integer exponent like 7 or -4. For very large exponents, it still computes instantly, but the multiplication expansion is hidden to keep the page readable.

How this Exponents Calculator works

The calculator reads your base (a) and exponent (b), validates they’re real numbers, then uses JavaScript’s power function (equivalent to a^b) to compute the result. On top of that, it adds three learning-friendly layers:

• Step expansion (small integers): if b is an integer and its absolute value is small, we print the multiplication expansion (and the reciprocal form for negative exponents).
• Scientific notation (optional): useful when the answer is huge or tiny.
• Size meter: a playful log-based meter so you can “feel” how big the number is without counting digits.

One tricky corner: if the base is negative and the exponent is not an integer, the true math result is often a complex number. Many school and everyday contexts stay in real numbers, so this calculator shows a friendly warning in that case.

Bottom line: you get speed, clarity, and share-ready output for homework screenshots, quick checks, or study notes.

Frequently Asked Questions

• What does a^b mean?

  It means “a raised to the power b.” If b is a positive integer, it’s a multiplied by itself b times. Example: 3^4 = 3×3×3×3 = 81.

• What is a negative exponent?

  A negative exponent flips the power into a fraction: a^(-n) = 1/a^n (for a ≠ 0). Example: 2^-3 = 1/8.

• Is 0^0 allowed?

  In many areas of math, 0^0 is considered “indeterminate” (it depends on context). This calculator warns you instead of pretending there’s one universal answer.

• Why does (-2)^0.5 give a warning?

  Because the square root of a negative number is not a real number. (-2)^0.5 involves complex numbers. We keep outputs real for clarity.

• Can I use decimals for the exponent?

  Yes. Decimals represent fractional powers (roots) and more general real exponents. For positive bases, you’ll get a real answer. For negative bases with non-integer exponents, you’ll get a warning about complex results.

• How do I simplify an expression like (2^3)(2^5)?

  Use the product rule: 2^3 × 2^5 = 2^(3+5) = 2^8 = 256. If you want more help with fractions, LCM/GCF, or rounding, check the related calculators below.

Make it shareable (and get clicks)

Exponents are everywhere in “wow facts”: viruses doubling, money compounding, pixels and bytes scaling, and “how many grains of rice on a chessboard?” puzzles. If you want this page to pop on social, try sharing a mini-challenge like:

• Screenshot challenge: “Guess 2^20 before I calculate it.”
• Science flex: “Bacteria doubling every 20 minutes — how many after 6 hours?”
• Money growth: “$1,000 grows 7% yearly — what’s it after 10 years?”

The copy/share buttons make it easy to drop the exact result into a post, group chat, or homework thread.