What is power 4 called?

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: Calculate the positive exponent (an) Then take the Reciprocal (i.e. 1/an)

More Examples:

Negative Exponent		Reciprocal of Positive Exponent		Answer
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:

Example: Powers of 5
	.. 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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