What does an exponent mean?
An exponent is shorthand for repeated multiplication. The small raised number counts how many copies of the base you multiply.
So 3⁴ = 3 × 3 × 3 × 3 = 81, and 2¹⁰ = 1024. Powers grow fast: 2²⁰ is already 1048576. That growth is why a calculator with exponents is handy for anything past the first few powers, where mental math gets slow.
To enter a power here, use ^. Type 3^4 for 3⁴, and put brackets around any exponent that is more than a single number, like 8^(2/3).
What are the rules of exponents?
The exponent rules tell you how to combine powers with the same base. You add exponents when you multiply, subtract when you divide and multiply when you raise a power to a power.
These rules only work when the bases match. You can't combine 2³ × 3⁴ into one power, so you just work out each part.
The zero and negative rules make sense if you follow a pattern. 2³ = 8, 2² = 4 and 2¹ = 2, so each time the exponent drops by 1, the value is cut in half. Keep going and you get 2⁰ = 1, then 2⁻¹ = 1/2 and 2⁻² = 1/4. Nothing about the rules changes when the exponent crosses zero.
How do you calculate negative exponents?
Flip the base to its reciprocal and make the exponent positive. So 2⁻³ = 1/2³ = 1/8, which is 0.125 as a decimal.
A negative exponent never makes the answer negative. It makes the number smaller (for a base above 1) by moving it under a fraction bar. As a negative exponent calculator, this page shows that flip as its own step.
With a fraction as the base, the flip turns it upside down. (1/2)⁻² = 2² = 4, and (3/4)⁻¹ = 4/3. Powers of ten work the same way: 10⁻² = 1/100 = 0.01.
You can check a negative power with the product rule. 2⁻³ × 2³ = 2⁰ = 1, and sure enough 1/8 × 8 = 1.
How do you solve fractional exponents?
The bottom of the fraction is the root and the top is the power. For 8^(2/3), take the cube root of 8 to get 2, then square it to get 4.
Taking the root first keeps the numbers small. A fractional exponent calculator follows the same order, and you can check each line by hand.
- Read the exponent 2/3: the 3 means cube root and the 2 means square.
- Cube root first: ∛8 = 2.
- Then square: 2² = 4.
- So 8^(2/3) = 4.
Decimal exponents are fractions in disguise. 4^1.5 is 4^(3/2): √4 = 2, then 2³ = 8. And 2^0.5 is the square root of 2, so the calculator gives the exact answer √2 ≈ 1.41421356237.
Is −3² the same as (−3)²?
No. −3² = −9, because the exponent applies only to the 3. (−3)² = 9, because the brackets make the whole negative number the base.
The same rule shows up with every even power: −2⁴ = −16 but (−2)⁴ = 16. With an odd power the sign survives either way, so (−2)³ = −8.
Negative bases with fractional exponents need care. (−8)^(1/3) = −2, since a cube root of a negative number is fine. But (−4)^(1/2) has no real answer, because no real number squared gives −4, and the calculator will tell you so.
How do powers of ten and scientific notation work?
A power of ten moves the decimal point. A positive exponent moves it right and a negative exponent moves it left, so 4 × 10³ = 4000 and 5 × 10⁻⁴ = 0.0005.
Scientific notation writes very large or very small numbers as a number between 1 and 10 times a power of ten. That keeps the zeros under control. 320000 becomes 3.2 × 10⁵, and 0.0015 becomes 1.5 × 10⁻³.
The product rule makes multiplying these numbers quick. Multiply the front parts and add the exponents: (1.2 × 10⁴) × (3 × 10²) = 3.6 × 10⁶, which is 3600000. You can type scientific notation here with a capital E, so 3.2E5 means 3.2 × 10⁵.
What are common mistakes with exponents?
The first one is multiplying the base by the exponent. 2³ is 2 × 2 × 2 = 8, not 2 × 3 = 6.
The second is spreading a power over a sum. (2 + 3)² = 25, but 2² + 3² = 13. A power distributes over multiplication, never over addition.
Stacked powers trip people up too. In 2^3^2 the top pair goes first: 3² = 9, then 2⁹ = 512. If you meant (2³)² = 64, type the brackets.
How do you solve for an unknown exponent?
When the exponent is the unknown, as in 2ˣ = 8, write both sides with the same base or use a logarithm. Since 8 = 2³, the answer is x = 3.
This page evaluates powers, so for an equation, use the algebra solver. It handles 2^x = 8 and 3^(x+1) = 27 (x = 2) with steps. When the answer isn't a whole number, like 2^x = 10, the solver gives x = log₂(10) ≈ 3.32192809489. The log calculator explains how that log is computed.