The “4th Power” of a number is the number multiplied by itself four times.
Write it with a raised number 4 (the exponent) next to the base number. “number4“or “54” or “84” are examples of using an exponent 4.
Saying “3 to the power of 4” or 34 is the same as saying 3 times 3 times 3 times 3 (equals 81).
Saying “20 to the exponent 4” or 204 is the same as saying 20 x 20 x 20 x 20 (equals 16,000).
To find a number to a different power, use our simple exponent calculator. To find the number needed to find the exponent 4 of a number, use the 4th root.
Formula – How to Calculate the 4th power of a number
The exponent 4 of a number is found by multiplying that number by itself 4 times.
number4 = number x number x number x number
To do the opposite, use our 4th root calculator.
Example
14 = 1 times 1 times 1 times 1 = 1
24 = 2 x 2 x 2 x 2 = 16
34 = 3 x 3 x 3 x 3 = 81
44 = 4 times 4 times 4 times 4 = 256
54 = 5 times 5 times 5 times 5 = 625
To the Power of 4 Table
Frequently Asked Questions
A number to the power of 4 is the number times itself 4 times. 3 to the power of 4 is 3 x 3 x 3 x 3 (81). 10 to the power of 4 is 10 x 10 x 10 x 10 (10,000).
Multiply the number by itself 4 times over.
To type an exponent in a word processor, look for the “superscript” command (it is usually near commands such as bold or italic).
To type an exponent for the web, use the markup. It is an HTML tag that surrounds the exponent.
They mean the same thing. The number times itself 4 times (number x number x number x number).
Sources and more resources
We know how to calculate the expression 5 x 5. This expression can be written in a shorter way using something called exponents.
$$5\cdot 5=5^{2}$$
An expression that represents repeated multiplication of the same factor is called a power.
The number 5 is called the base, and the number 2 is called the exponent. The exponent corresponds to the number of times the base is used as a factor.
Example
Write these multiplications like exponents
$$5\cdot 5\cdot 5=5^{3}$$
$$4\cdot 4\cdot 4\cdot 4\cdot 4=4^{5}$$
$$3\cdot 3\cdot 3\cdot 3=3^{4}$$
Multiplication
If two powers have the same base then we can multiply the powers. When we multiply two powers we add their exponents.
The rule:
$$x^{a}\cdot x^{b}=x^{a+b}$$
Example
$$4^{2}\cdot 4^{5}=\left ( 4\cdot 4 \right )\cdot \left ( 4\cdot 4\cdot 4\cdot 4\cdot 4 \right )=4^{7}=4^{2+5}$$
Division
If two powers have the same base then we can divide the powers. When we divide powers we subtract their exponents.
The rule:
$$\frac{x^{a}}{ x^{b}}=x^{a-b}$$
Example
$$\frac{4^{2}}{ 4^{5}}=\frac{{\color{red} {\not}{4}}\cdot {\color{red} {\not}{4}}}{{\color{red} {\not}{4}}\cdot {\color{red} {\not}{4}}\cdot 4\cdot 4\cdot 4}=\frac{1}{4^{3}}=4^{-3}=4^{2-5}$$
A negative exponent is the same as the reciprocal of the positive exponent.
$$x^{-a}=\frac{1}{x^{a}}$$
Example
$$2^{-3}=\frac{1}{2^{3}}$$
When you raise a product to a power you raise each factor with a power
$$(x\cdot y)^{a}=x^{a}\cdot y^{a}$$
Example
$$(2x)^{4}=2^{4}\cdot x^{4}=16x^{4}$$
The rule for the power of a power and the power of a product can be combined into the following rule:
$$(x^{a}\cdot y^{b})^{z}=x^{a\cdot z}\cdot y^{b\cdot z}$$
Example
$$(x^{3}\cdot y^{4})^{2}=x^{3\cdot 2}\cdot y^{4\cdot 2}=x^{6}\cdot y^{8}$$
Video lessons
Rewrite the expressions
$$2\cdot 2\cdot 2$$
$$x\cdot x\cdot x\cdot x\cdot x$$
$$3^{4}$$
$$x^{3}$$
Simplify the expression
$$\left ( x^{2}\cdot y^{3}\cdot z^{5} \right )^{3}$$
The exponent of a number says how many times to use the number in a multiplication.
In 82 the "2" says to use 8 twice in a multiplication,
so 82 = 8 × 8 = 64
In words: 82 could be called "8 to the power 2" or "8 to the second power", or simply "8 squared"
Exponents are also called Powers or Indices.
Some more examples:
Example: 53 = 5 × 5 × 5 = 125
- In words: 53 could be called "5 to the third power", "5 to the power 3" or simply "5 cubed"
Example: 24 = 2 × 2 × 2 × 2 = 16
- In words: 24 could be called "2 to the fourth power" or "2 to the power 4" or simply "2 to the 4th"
Exponents make it easier to write and use many multiplications
Example: 96 is easier to write and read than 9 × 9 × 9 × 9 × 9 × 9
You can multiply any number by itself as many times as you want using exponents.
Try here:
algebra/images/exponent-calc.js
So in general:
an tells you to multiply a by itself, so there are n of those a's: |
Another Way of Writing It
Sometimes people use the ^ symbol (above the 6 on your keyboard), as it is easy to type.
Example: 2^4 is the same as 24
Negative Exponents
Negative? What could be the opposite of multiplying? Dividing!
So we divide by the number each time, which is the same as multiplying by 1number
Example: 8-1 = 18 = 0.125
We can continue on like this:
Example: 5-3 = 15 × 15 × 15 = 0.008
But it is often easier to do it this way:
5-3 could also be calculated like:
15 × 5 × 5 = 153 = 1125 = 0.008
Negative? Flip the Positive!
That last example showed an easier way to handle negative exponents:
|
More Examples:
4-2 | = | 1 / 42 | = | 1/16 = 0.0625 |
10-3 | = | 1 / 103 | = | 1/1,000 = 0.001 |
(-2)-3 | = | 1 / (-2)3 | = | 1/(-8) = -0.125 |
What if the Exponent is 1, or 0?
1 | If the exponent is 1, then you just have the number itself (example 91 = 9) | |
0 | If the exponent is 0, then you get 1 (example 90 = 1) | |
But what about 00 ? It could be either 1 or 0, and so people say it is "indeterminate". |
It All Makes Sense
If you look at that table, you will see that positive, zero or negative exponents are really part of the same (fairly simple) pattern:
.. etc.. | |||
52 | 5 × 5 | 25 | |
51 | 5 | 5 | |
50 | 1 | 1 | |
5-1 | 15 | 0.2 | |
5-2 | 15 × 15 | 0.04 | |
.. etc.. |
Be Careful About Grouping
To avoid confusion, use parentheses () in cases like this:
With () : | (−2)2 = (−2) × (−2) = 4 |
Without () : | −22 = −(22) = −(2 × 2) = −4 |
With () : | (ab)2 = ab × ab |
Without () : | ab2 = a × (b)2 = a × b × b |
305, 1679, 306, 1680, 1077, 1681, 1078, 1079, 3863, 3864
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