Exponents Calculator
Type 2^10 and this exponents calculator gives 1024. Type x³ × x⁵ and it gives x⁸, with the reason: same base, so the powers add.
Both are the same question asked at different levels, and both are here. What an exponent calculator will not do — or should not — is join x³ × y⁵ into anything, because that rule does not exist, and saying so is more useful than inventing an answer.
The Rules of Exponents
Five laws cover the whole topic. These are the exponent rules every question is built from, and the laws of exponents you will see written on the board.
Product rule. xᵃ × xᵇ = xᵃ⁺ᵇ. Multiplying exponents with the same base means adding the powers, so x³ × x⁵ = x⁸. It is not multiplying the powers, which is the commonest slip — x³ × x⁵ is not x¹⁵.
Why it works is worth seeing once. x³ is three x's multiplied, x⁵ is five, so together there are eight. Counting is all the rule is.
Quotient rule. xᵃ ÷ xᵇ = xᵃ⁻ᵇ. Dividing exponents subtracts the powers, so x⁸ ÷ x³ = x⁵. Same counting: eight x's on top, three cancel against the bottom, five left.
This is also where negative powers come from. x³ ÷ x⁵ is x⁻² by the rule, and 1/x² by cancelling. Both are right, which is exactly why x⁻² means 1/x².
Power of a power. (xᵃ)ᵇ = xᵃᵇ. Here you multiply, and this is where the two rules get mixed up: (x³)⁵ is x¹⁵, not x⁸. Adding is for multiplying the terms; multiplying is for a power on top of a power.
A bracket also sends its power to everything inside. (2x)³ = 8x³, because the 2 gets cubed as well — forgetting the coefficient is the second commonest slip in the topic. Same with (x²y³)⁴ = x⁸y¹²: the 4 reaches both letters.
Negative power. x⁻ᵃ = 1/xᵃ. This one causes the most trouble, so it has a section of its own below.
Zero exponent rule. x⁰ = 1, whatever x is. It looks arbitrary until you use the quotient rule: x³ ÷ x³ is x⁰ by the rule and 1 by cancelling. The zero exponent rule is not a special case — it is what the other rules force.
What a Negative Exponent Means
x⁻³ is not a negative number. That is the whole difficulty in one line.
A negative exponent is a reciprocal: x⁻³ = 1/x³. Put a number in and it is obvious. 2⁻³ is 1/8, which is 0.125 — small, and positive. Nothing about it is negative except the sign in the exponent.
So the rule for handling negative exponents is simply: move the factor to the other side of the fraction line and make the power positive. Something on top with a negative power goes underneath. Something underneath with a negative power comes up.
That second half is worth saying out loud, because it is the part people forget. 1/x⁻³ is x³, not 1/x³. The minus was already doing its job downstairs, and moving up cancels it.
It works with numbers in front too. (2x)⁻² sends the whole bracket down, and the 2 gets squared on the way: 1/(4x²). And a fraction simply turns over — (2/3)⁻² is (3/2)², which is 9/4.
Why Different Bases Never Combine
This is where the marks go, so it is worth being blunt about it.
Every law above has the same letter on both sides. xᵃ × xᵇ works because it is x in both places. x³ × y⁵ has no such rule, and there is nothing to reach for: it is not (xy)⁸, and it does not simplify at all.
Check it with numbers. Put x = 2 and y = 3. The left side is 8 × 243 = 1944. (xy)⁸ would be 6⁸, which is 1,679,616. Nowhere close.
The calculator gathers each base separately and says how many it found, so an expression with two bases comes back with both intact rather than mashed together. That is the one thing an exponents calculator has to get right.
How to Use This Exponents Calculator
Type the expression and press =. Powers go in with ^, so x^3 means x cubed, and brackets work as you would expect. Negative and fractional powers are both fine: x^-3 and x^(1/2) are read correctly.
It works out plain powers as well, so 2^10 comes back as 1024 rather than being left alone. Simplifying expressions with exponents and working out a single power are the same machinery, and both are here.
The answer comes back with each base appearing once, negatives moved underneath, and coefficients worked out — 6x⁵/(2x²) becomes 3x³, with the 6 and the 2 divided as well as the powers subtracted.
One tap opens the laws that were used and why, and under that the working step by step, ending with a check you can do yourself: put numbers in for the letters and both forms must come to the same value. That check catches an added power where a subtracted one belonged, which is the error that survives a re-read.
One detail worth knowing, because most tools get it wrong: −x² is read as −(x²), not (−x)². The power binds tighter than the sign, so −x² is negative and (−x)² is positive. With a negative base the sign survives an odd power and cancels on an even one — (−x)³ is −x³, but (−x)⁴ is x⁴.
Exponents are also the other half of logarithms — log₂8 = 3 says exactly what 2³ = 8 says — so the log calculator is next door if you are working the other way. It is free to use online, with no account.
Exponents FAQ
What are the rules of exponents?
Why can't you combine x cubed and y to the fifth?
What does a negative exponent mean?
Is minus x squared the same as (minus x) squared?
What is anything to the power of zero?
Same base and you can add, subtract or multiply the powers. Different bases and you leave them alone. Every one of the laws of exponents follows from those two sentences.