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Math & School
Exponent Calculator — Powers, Negative & Fractional Exponents
Compute base to the power of exponent, with the expanded form and the rules shown
An exponent (also called a power or index) is shorthand for repeated multiplication. In the expression b^n, the base b is the number being multiplied and the exponent n tells you how many times to use it as a factor. So 2^5 = 2 × 2 × 2 × 2 × 2 = 32 and 10^3 = 10 × 10 × 10 = 1,000. Exponents are everywhere in real math — area and volume (squares and cubes), scientific notation, compound interest, computer storage (powers of 2), and every growth-or-decay formula you'll meet in algebra and science.
The definition stretches naturally beyond whole-number powers, and that's where most of the confusion lives:
Zero exponent: any nonzero base to the power 0 equals 1. So 7^0 = 1 and (−3)^0 = 1. The reason: b^n ÷ b^n = b^(n−n) = b^0, and anything divided by itself is 1.
Negative exponent: a negative power means take the reciprocal — b^(−n) = 1 ÷ b^n. So 2^(−3) = 1 ÷ 2^3 = 1 ÷ 8 = 0.125. A negative exponent never makes the answer negative; it makes it a fraction.
Fractional exponent: a fraction in the exponent means a root. b^(1/2) is the square root, b^(1/3) is the cube root, and in general b^(m/n) = (the n-th root of b), raised to m. So 8^(1/3) = 2 (the cube root of 8) and 16^(0.75) = 16^(3/4) = (16^(1/4))^3 = 2^3 = 8.
Decimal exponent: 5^2.5 just means 5^(5/2) = √(5^5) ≈ 55.90. The calculator handles these directly.
A few important edge cases. A negative base with a fractional exponent often has no real value — (−4)^0.5 would be the square root of a negative number, which is imaginary, so this tool reports that instead of guessing. 0^0 is famously ambiguous: most calculators and the JavaScript Math.pow convention return 1, and we follow that, but we flag it because in some contexts it's left undefined. And the order of operations matters: −3^2 means −(3^2) = −9, while (−3)^2 = 9 — the parentheses change everything.
This calculator raises any base to any exponent — positive, negative, zero, decimal or fractional — shows the expanded multiplication for small whole-number powers, applies the right rule (reciprocal for negative, root for fractional), and explains each step so you can check your homework or sanity-check a spreadsheet.
Calculator
Fill in the fields and click "Calculate" for instant results.
📰 Formula
• Definition: b^n = b × b × … × b (n factors), for a whole number n • Zero power: b^0 = 1 for any b ≠ 0 • Negative power: b^(−n) = 1 / b^n • Product of powers: b^m × b^n = b^(m+n) • Quotient of powers: b^m / b^n = b^(m−n) • Power of a power: (b^m)^n = b^(m×n) • Fractional power (root): b^(m/n) = (ⁿ√b)^m → b^(1/2) = √b, b^(1/3) = ∛b
📰 Formula
• Definition: b^n = b × b × … × b (n factors), for a whole number n • Zero power: b^0 = 1 for any b ≠ 0 • Negative power: b^(−n) = 1 / b^n • Product of powers: b^m × b^n = b^(m+n) • Quotient of powers: b^m / b^n = b^(m−n) • Power of a power: (b^m)^n = b^(m×n) • Fractional power (root): b^(m/n) = (ⁿ√b)^m → b^(1/2) = √b, b^(1/3) = ∛b
🧪 Worked examples
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Example 2
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Example 3
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Example 4
⚠️ Common mistakes
- Reading −3^2 as 9 — it's −(3^2) = −9; only (−3)^2 equals 9.
- Thinking a negative exponent makes the answer negative (it gives a reciprocal, not a negative number).
- Multiplying the base by the exponent (2^5 is 32, not 2 × 5 = 10).
- Forgetting that b^(1/n) is a root, so 27^(1/3) = 3, not 9.
- Trying to take a fractional power of a negative base, which has no real value.
💡 Tips
- A negative exponent means 'flip and make positive': b^(−n) = 1 ÷ b^n.
- A fractional exponent is a root: the denominator is the root, the numerator is the power.
- Wrap a negative base in parentheses, e.g. (−2)^4, so the sign is included before raising to the power.
- Powers of 2 (2, 4, 8, 16, 32, 64, 128, 256, 1024…) are worth memorizing — they drive computer storage and binary.
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❓ Frequently asked questions
How do I calculate an exponent or power?
Multiply the base by itself as many times as the exponent says. For 3^4, multiply 3 × 3 × 3 × 3 = 81. The exponent is the count of factors, not a multiplier of the base.
Why does any number to the power of 0 equal 1?
Because b^n ÷ b^n = b^(n−n) = b^0, and any nonzero quantity divided by itself is 1. So 5^0 = 1, 100^0 = 1, and (−7)^0 = 1. The base just can't be zero.
What does a negative exponent mean?
It means take the reciprocal of the positive power: b^(−n) = 1 ÷ b^n. For example 2^(−3) = 1 ÷ 8 = 0.125. The result is a fraction, never a negative number (unless the base itself is negative).
What is a fractional exponent?
A fractional exponent is a root. The denominator is the root and the numerator is the power, so b^(m/n) = (n-th root of b)^m. For instance 8^(2/3) = (∛8)^2 = 2^2 = 4, and b^(1/2) is just the square root of b.
Is −3^2 equal to 9 or −9?
By the order of operations, −3^2 means −(3^2) = −9, because exponents bind tighter than the negative sign. To get 9 you must write (−3)^2. This calculator treats a base you type as negative as already inside the parentheses.
What is 0 to the power of 0?
0^0 is debated. Many calculators, spreadsheets and the JavaScript convention return 1, which is what this tool does, but some math contexts leave it undefined. We return 1 and add a note so you know it's a special case.
Can you raise a negative number to a fractional power?
Not within the real numbers in most cases. (−4)^0.5 is the square root of a negative number, which is imaginary. Odd roots like (−8)^(1/3) = −2 do have real values, but to keep results unambiguous this calculator flags negative-base fractional powers instead of guessing.
How do I square or cube a number?
Squaring is raising to the power 2 (multiply the number by itself once): 6^2 = 36. Cubing is raising to the power 3: 4^3 = 64. Enter the exponent as 2 or 3 and the calculator shows the expanded multiplication.
What's the difference between an exponent and a multiple?
A multiple adds the number repeatedly (5 × 3 = 5 + 5 + 5 = 15); an exponent multiplies it repeatedly (5^3 = 5 × 5 × 5 = 125). They grow at very different speeds, which is why exponential growth outpaces linear growth so fast.