Power Rule Calculator: Calculate a Number’s Power

The power rule calculator raises a base number to an exponent: enter the base and the exponent to get the resulting power. For example, entering 2 and 3 calculates 2³, which equals 8.

Formula

The formula is P = bⁿ. In words, multiply the base by itself the number of times indicated by a positive whole-number exponent. Here, P is the power, b is the base, and n is the exponent. Both inputs and the result are dimensionless numbers unless the base represents a measured quantity; in that case, its units are raised to the exponent too.

The calculator accepts negative and decimal values as well as whole numbers. Its result is rounded to two decimal places, though values that are exact to fewer decimal places display without unnecessary trailing zeros.

Worked Example

Using the default base 2 and exponent 3:

1. Write the expression: P = 2³.
2. Multiply the base by itself three times: 2 × 2 × 2.
3. Calculate the product: P = 8.

The calculator result is 8.

What the Power Rule Means

A power is a compact way to express repeated multiplication. In the expression bⁿ, the base is the number being multiplied, and the exponent tells how many factors of that base appear. Thus, 5³ means 5 × 5 × 5, or 125. The expression is read as “five to the third power” or “five cubed.”

Despite the name, an exponent does not always mean repeated multiplication in the ordinary whole-number sense. The same notation extends to zero, negative, and fractional exponents through consistent mathematical rules. These extensions make powers useful in algebra, science, computing, and descriptions of growth or scaling.

• Square: 7² = 7 × 7 = 49.
• Cube: 3³ = 3 × 3 × 3 = 27.
• Zero exponent: 9⁰ = 1.
• Negative exponent: 2⁻³ = 1 ÷ 2³ = 1/8.
• Fractional exponent: 16^1/2 = √16 = 4.

How to Calculate Different Exponents

For a positive whole-number exponent, repeated multiplication is usually the clearest method. For larger exponents, a calculator or exponentiation algorithm is more practical. For example, 2¹⁰ is 1,024; multiplying ten factors by hand is possible, but easy to mistype.

A zero exponent gives 1 for any nonzero base. This convention follows the exponent laws: for nonzero b, dividing b^m by itself gives 1, while subtracting the exponents gives b^m−m, or b⁰. The expression 0⁰ is a special case whose treatment depends on the mathematical setting; do not assume it follows the ordinary nonzero-base rule.

A negative exponent means take the reciprocal of the corresponding positive power: b⁻ⁿ = 1/bⁿ, provided b is not zero. For instance, 10⁻² = 1/100 = 0.01. A negative exponent does not make the result negative; it makes a nonzero result a reciprocal.

A fractional exponent is related to roots. For a positive base, b^1/n is the nth root of b. For example, 27^1/3 = 3. More generally, 8^2/3 = (∛8)² = 2² = 4. The order of operations can be handled either way when the real-valued expression is defined.

Exponent Laws and Useful Shortcuts

Exponent laws simplify expressions, but each law has conditions and applies to specific operations. For nonzero bases where division is involved, useful rules include:

• Multiply powers with the same base: b^mbⁿ = b^m+n. Example: 2³ × 2⁴ = 2⁷.
• Divide powers with the same base: b^m/bⁿ = b^m−n, when b is nonzero. Example: 5⁶/5² = 5⁴.
• Raise a power to a power: (b^m)ⁿ = b^mn. Example: (3²)³ = 3⁶.
• Raise a product to a power: (ab)ⁿ = aⁿbⁿ.
• Raise a quotient to a power: (a/b)ⁿ = aⁿ/bⁿ, when b is nonzero.

These rules do not let you distribute an exponent over addition: in general, (a + b)ⁿ is not equal to aⁿ + bⁿ. For example, (2 + 3)² = 25, whereas 2² + 3² = 13.

Negative Bases, Units, and Rounding

Parentheses matter when the base is negative. The expression (−2)² means (−2) × (−2) = 4. By contrast, −2² is ordinarily interpreted as −(2²) = −4, because exponentiation is performed before the leading minus sign. Enter a negative base with parentheses when using a written expression, and check how a particular calculator handles the sign.

For a negative base raised to a fractional exponent, a real result does not always exist. For instance, (−8)^1/3 has the real value −2, but (−8)^1/2 has no real square root. More complicated fractional exponents can require care in how the fraction is represented. Many basic calculators handle fractional powers of negative numbers by returning an error or a complex result.

Units also follow the exponent. If a cube has side length 2 metres, its volume is (2 m)³ = 8 m³, not 8 m. Squaring a length produces square units, such as square metres or square feet; cubing it produces cubic units. Convert measurements to compatible units before combining them, and do not confuse a squared unit with a unit that has simply been multiplied by two.

Finally, the displayed answer may be rounded. A result shown as 3.14 can represent a value with additional digits. Keep more precision during intermediate steps when calculations continue, and round only at the end if accuracy matters. Very large powers may exceed a calculator’s display range, while very small ones may appear as zero or scientific notation.

Frequently Asked Questions

How do you calculate a power?
For a positive whole-number exponent, multiply the base by itself that many times. For example, 4³ = 4 × 4 × 4 = 64. For zero, negative, or fractional exponents, use the corresponding exponent rules.

What is the difference between a base and an exponent?
The base is the number being raised to a power; the exponent indicates the power. In 6², 6 is the base and 2 is the exponent.

What does a negative exponent mean?
It means the reciprocal of the positive power: for nonzero b, b⁻ⁿ = 1/bⁿ. Thus, 3⁻² = 1/9, not −9.

Why is any nonzero number to the power of zero equal to 1?
This rule keeps exponent laws consistent. For nonzero b, the quotient b^m/b^m equals 1 and also equals b^m−m = b⁰.

Can a negative number have a fractional power?
Sometimes. A negative number has a real nth root when the root’s denominator is odd, such as (−8)^1/3 = −2. An even root, such as the square root of −8, is not real.

Is a power the same as multiplying by the exponent?
No. 2³ means 2 × 2 × 2 = 8, not 2 × 3 = 6. The exponent specifies repeated factors, rather than a multiplier.

Related

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• Power Calculator for Math: Exponents Explained

in b | Base | - | 2 | -
in n | Exponent | - | 3 | -
out power | Power | - | b ^ n | 2