Simply using exponents 4^2 / 3^6

45.5625

Step-by-step explanation:4 x 4= 16 and 3 x 3 x 3 x 3 x 3 x 3= 729

16/729

SteIn 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:

3

4

34 = 3 × 3 × 3 × 3 = 81© 2015 MathsIsFun.com v0.82

In General

So in general:

an tells you to multiply a by itself,

so there are n of those a's:  exponent definition

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

2^4 = 2 × 2 × 2 × 2 = 16

Negative Exponents

Negative? What could be the opposite of multiplying?

Dividing!

A negative exponent means how many times to divide one by the number.

Example: 8-1 = 1 ÷ 8 = 0.125

You can have many divides:

Example: 5-3 = 1 ÷ 5 ÷ 5 ÷ 5 = 0.008

But that can be done an easier way:

5-3 could also be calculated like:

1 ÷ (5 × 5 × 5) = 1/53 = 1/125 = 0.008

Negative? Flip the Positive!

negative-exponent  

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

My favorite method is to start with "1" and then multiply or divide as many times as the exponent says, then you will get the right answer, for example:

Example: Powers of 5

 .. etc..  exponent 5 times larger or smaller

521 × 5 × 525

511 × 55

5011

5-11 ÷ 50.2

5-21 ÷ 5 ÷ 50.04

 .. etc..  

If you look at that table, you will see that positive, zero or negative exponents are really part of the same (fairly simple) pattern.

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

(-5,-28)

i graphed to on desmos.