How exponents work
An exponent tells you how many times to multiply the base by itself. Writing base^power as 2^10 means ten 2s multiplied together, which equals 1024. The base is the number being repeated and the power (or exponent) is the count of repetitions.
The same rule extends smoothly past whole numbers. A power of 1 returns the base unchanged, a power of 0 always returns 1, and the result grows or shrinks much faster than the inputs suggest because each step multiplies rather than adds.
Reading fractional and negative powers
Fractional powers are roots. Raising a number to the 0.5 power gives its square root, to 1/3 gives its cube root, and to 2/3 gives the cube root squared. So 27^(1/3) is 3, and 8^(2/3) is 4.
Negative powers are reciprocals. base^(-n) equals 1 divided by base^n, so 2^(-3) is 1/8, or 0.125. Combining the two, a negative fractional power both inverts and takes a root.
Practical tips
A few quick habits make exponents easier to estimate and check.
- Any nonzero base to the power 0 is 1, which is a fast sanity check.
- For powers of 10, the exponent equals the number of zeros: 10^6 is 1 followed by six zeros.
- To combine powers of the same base, add the exponents: base^a × base^b = base^(a+b).
- To undo an exponent and solve for the power itself, use a logarithm.
Common mistakes and limits
A negative base with a fractional power has no real answer, because it would require an even root of a negative number. For example (-4)^0.5 is undefined in real numbers, and this calculator flags such cases.
Watch the order of operations: -2^2 is read as -(2^2) = -4, while (-2)^2 = 4. Parentheses around a negative base change the result. Very large powers can also overflow to infinity once they exceed what a computer can represent.
Formula
result = base^power