4 to the power of negative 3

The base b raised to the power of minus n is equal to 1 divided by the base b raised to the power of n:

b-n = 1 / bn

Negative exponent example

The base 2 raised to the power of minus 3 is equal to 1 divided by the base 2 raised to the power of 3:

2-3 = 1/23 = 1/(2⋅2⋅2) = 1/8 = 0.125

Negative fractional exponents

The base b raised to the power of minus n/m is equal to 1 divided by the base b raised to the power of n/m:

b-n/m = 1 / bn/m = 1 / (m√b)n

The base 2 raised to the power of minus 1/2 is equal to 1 divided by the base 2 raised to the power of 1/2:

2-1/2 = 1/21/2 = 1/√2 = 0.7071

Fractions with negative exponents

The base a/b raised to the power of minus n is equal to 1 divided by the base a/b raised to the power of n:

(a/b)-n = 1 / (a/b)n = 1 / (an/bn) = bn/an

The base 2 raised to the power of minus 3 is equal to 1 divided by the base 2 raised to the power of 3:

(2/3)-2 = 1 / (2/3)2 = 1 / (22/32) = 32/22 = 9/4 = 2.25

Multiplying negative exponents

For exponents with the same base, we can add the exponents:

a -n ⋅ a -m = a -(n+m) = 1 / a n+m

Example:

2-3 ⋅ 2-4 = 2-(3+4) = 2-7 = 1 / 27 = 1 / (2⋅2⋅2⋅2⋅2⋅2⋅2) = 1 / 128 = 0.0078125

When the bases are diffenrent and the exponents of a and b are the same, we can divide a and b first:

The exponent calculator will calculate the value of any base raised to any power. This page will cover all the related topics, including the negative exponent. Let's start with the basics.

What is an exponent?

An exponent is a way to represent how many times a number, known as the base, is multiplied by itself. It is represented as a small number in the upper right hand corner of the base. For example: x² means you multiply x by itself two times, which is x * x. Likewise, 4² = 4 * 4, etc. If the exponent is 3, in the example 5³, then the result is 5 * 5 * 5.

It's easy with small numbers, but for bases that are large numbers, decimals, or when they are raised to a power that's very large or negative, use our tool. If you wish to do exponentiation by hand, do the following:

1. Determine the base and the power it's raised to, for example 3⁵.
2. Write the base the same number of times as the exponent. 3 3 3 3 3
3. Place a multiplication symbol between each base. 3 * 3 * 3 * 3 * 3.
4. Multiply! 3 * 3 * 3 * 3 * 3 = 243.

Negative exponent calculator

The concept is rather simple when the exponent is positive, but what happens when the exponent is negative? By the definition, if it is -2, we would multiply the base times itself negative two times. In actuality, what is happening here, we take the reciprocal of the base and change the negative exponent to positive and proceed as usual. If you'd like to work it out by hand, do the following:

1. Determine the base and the exponent.
2. Write the reciprocal of the base and change the sign of the exponent to positive
3. Write the reciprocal of the base the same number of times as the exponent.
4. Place a multiplication symbol between each.
5. Multiply and get the result.

Here's a quick example: 5⁻⁴ = (1/5)⁴ = (1/5) * (1/5) * (1/5) * (1/5) = 1/625 = 0.0016

Related topics

Squaring a base (raising a number to the power of 2) and taking the square root are similar concepts, many people consider one the opposite or the undoing of the other. If you want to square the number 6, you take x * x0. Now if you want to find what two identical numbers multiply to give you 36, you take the square root of 36. This square root gives the value of 6. It can also be noted that squaring a square root removes the radical.

Likewise, cubing a base (raising a number to the power of 3) will give us a perfect cube. In case you need to calculate the cube root you can use our cube root calculator which is an excellent tool that will calculate the cube root of any number.

In modular arithmetic there are dedicated methods of exponentiation - learn more with the power mod calculator.

Besides, you may check our logarithm calculator which is the inverse function of the exponent.

Any number raised to the power of 0 equals 1. The negative exponent calculator is useful when dealing with exponential decay, which has a negative exponent in its formula.

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:

\( x^m \cdot x^n = x^{m+n} \)

\( \dfrac{x^m}{x^n} = x^{m-n} \)

\( (x^m)^n = x^{m \cdot n} \)

\( (x \cdot y)^m = x^m \cdot y^m \)

\( \left(\dfrac{x}{y}\right)^m = \dfrac{x^m}{y^m} \)

\( x^{-m} = \dfrac{1}{x^m} \)

\( \left(\dfrac{x}{y}\right)^{-m} = \dfrac{y^m}{x^m} \)

\( 0^0 = 1 \; (definition) \)

\( if \; x^m = y \; then \; y = \sqrt[m]{x} = y^{\frac{1}{m}} \)

\( x^{\frac{m}{n}} = \sqrt[n]{x^m} \)

References

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

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