Exponents Calculator

Calculator Use

This is an online calculator for exponents. Calculate the power of large base integers and real numbers. You can also calculate numbers to the power of large exponents less than 2000, negative exponents, and real numbers or decimals for exponents.

For larger exponents try the Large Exponents Calculator

For instructional purposes the solution is expanded when the base x and exponent n are small enough to fit on the screen. Generally, this feature is available when base x is a positive or negative single digit integer raised to the power of a positive or negative single digit integer. Also, when base x is a positive or negative two digit integer raised to the power of a positive or negative single digit integer less than 7 and greater than -7.

For example, 3 to the power of 4:

\( x^n = \; 3^{4} \)

\( = \;3 \cdot 3 \cdot 3 \cdot 3 \)

\( = 81 \)

For example, 3 to the power of -4:

\( x^n = \;3^{-4} \)

\( = \dfrac{1}{3^{4}} \)

\( = \; \dfrac{1}{3 \cdot 3 \cdot 3 \cdot 3} \)

\( = \; \dfrac{1}{81} \)

\( = 0.012346 \)

Exponent Notation:

Note that -42 and (-4)2 result in different answers: -42 = -1 * 4 * 4 = -16, while (-4)2 = (-4) * (-4) = 16. If you enter a negative value for x, such as -4, this calculator assumes (-4)n.

"When a minus sign occurs with exponential notation, a certain caution is in order. For example, (-4)2 means that -4 is to be raised to the second power. Hence (-4)2 = (-4) * (-4) = 16. On the other hand, -42 represents the additive inverse of 42. Thus -42 = -16. It may help to think of -x2 as -1 * x2 ..."[1]

Examples:

• 3 raised to the power of 4 is written 34 = 81.
• -4 raised to the power of 2 is written (-4)2 = 16.
• -3 raised to the power of 3 is written (-3)3 = -27. Note that in this case the answer is the same for both -33 and (-3)3 however they are still calculated differently. -33 = -1 * 3 * 3 * 3 = (-3)3 = -3 * -3 * -3 = -27.
• For 0 raised to the 0 power the answer is 1 however this is considered a definition and not an actual calculation.

Exponent Rules:

\( x^m \cdot x^n = x^{m+n} \) \( \dfrac{x^m}{x^n} = x^{m-n} \) \( (x^m)^n = x^{m \cdot n} \) \( (x \cdot y)^m = x^m \cdot y^m \) \( \left(\dfrac{x}{y}\right)^m = \dfrac{x^m}{y^m} \) \( x^{-m} = \dfrac{1}{x^m} \) \( \left(\dfrac{x}{y}\right)^{-m} = \dfrac{y^m}{x^m} \) \( x^1 = x \) \( x^0 = 1 \) \( 0^0 = 1 \; (definition) \) \( if \; x^m = y \; then \; y = \sqrt[m]{x} = y^{\frac{1}{m}} \) \( x^{\frac{m}{n}} = \sqrt[n]{x^m} \)

References

[1] Algebra and Trigonometry: A Functions Approach; M. L. Keedy and Marvin L. Bittinger; Addison Wesley Publishing Company; 1982, page 11.

For more detail on Exponent Theory see Exponent Laws.

To calculate fractional exponents use our Fractional Exponents Calculator.

To calculate root or radicals use our Roots Calculator.