How to Calculate a Number Raised to a Decimal Power

A decimal power usually means raising a number to a decimal exponent, as in $16^{0.75}$. The exponent tells you how many times to apply the base in a generalized sense; it does not mean multiply the base by the decimal. You can calculate the result exactly when the exponent corresponds to a convenient root, or use a calculator for a decimal approximation.

The method below treats a finite decimal exponent as the exact decimal you entered. If you meant a decimal number as the base instead—for example, $1.6^3$—the same exponent rules apply; you simply start with a decimal base.

Convert the decimal exponent to a fraction

A finite decimal is a rational number, so rewrite the exponent as a fraction and reduce it. For example, $0.75=75/100=3/4$, and $0.2=2/10=1/5$.

For a positive base $a$ and a reduced fraction $m/n$, use

$$a^{m/n}=\sqrt[n]{a^m}=(\sqrt[n]{a})^m.$$

In practice, choose the order that makes the arithmetic easiest. First take the root and then raise it to a power, or first raise the base and then take the root. For a positive base, both give the same real result.

Worked example: calculate $16^{0.75}$

First convert the exponent:

$$0.75=\frac{75}{100}=\frac{3}{4}.$$

Then apply the fractional-exponent rule:

$$16^{0.75}=16^{3/4}=(\sqrt[4]{16})^3=2^3=8.$$

So $16^{0.75}=8$ exactly. The fourth root is convenient here because $\sqrt[4]{16}=2$.

Another exact example is $256^{0.375}$. Since $0.375=3/8$ and $\sqrt[8]{256}=2$, the value is $2^3=8$.

Calculate a decimal approximation

If the root is not easy to work out, enter the expression using your calculator’s power key, often shown as ^, $x^y$, or $y^x$. Enter the base, the power key, and the exponent, then evaluate. Use parentheses around a negative base or a complicated exponent if your calculator’s entry format requires them.

For example, to calculate $10^{1.3}$, enter $10^1.3$. The result is approximately

$$10^{1.3}\approx 19.95262315.$$

If your calculator does not have a general power key, use the identity

$$a^y=\exp(y\ln a), \qquad a>0.$$

For this example, calculate $\exp(1.3\ln 10)$. Make sure the calculator is evaluating the entire product $1.3\ln 10$ before applying the exponential function. This method works for any positive base and any real exponent.

A calculator may display only a limited number of digits. Treat the displayed value as an approximation, and keep extra digits until the final step if you will use the result in another calculation.

What if the base is negative?

A negative base can have a real-valued decimal power in some cases, but not all. Reduce the decimal exponent to a fraction first. If the denominator is odd, the real root of the negative base exists. For example,

$$(-32)^{0.2}=(-32)^{1/5}=-2,$$

because $(-2)^5=-32$. If the reduced denominator is even, the corresponding even root of a negative number is not real. For instance, $(-16)^{0.75}=(-16)^{3/4}$ has no real value because it requires a fourth root of a negative number.

Some calculators or computer systems may return a complex number for such an expression, or report an error. That is a different domain convention from ordinary real-number arithmetic. Also, when the powers are defined as real numbers, a negative exponent means take a reciprocal: for nonzero $a$, $a^{-y}=1/a^y$. A zero base with a negative exponent is undefined.

Common mistakes

• Treating the exponent as multiplication: $16^{0.75}$ does not mean $16\times0.75$. The decimal is the exponent.
• Using the wrong root: For $a^{3/4}$, the denominator $4$ specifies a fourth root and the numerator $3$ specifies a third power.
• Forgetting to simplify the fraction: Reduce the exponent before checking whether a negative base has a real result. For example, $0.6=3/5$, whose denominator is odd.
• Assuming every decimal exponent works with a negative base: $(-a)^{1/2}$ is not real when $a>0$, even though a calculator may offer a complex-number result.
• Rounding too early: Keep the full calculator result during intermediate steps, then round to the precision the problem requests.