Four To The Third Power

Exponents

The exponent of a number says how many times to utilize the number in a multiplication.

In 8² the "2" says to use 8 twice in a multiplication,
and then 8ⁱⁱ = 8 × eight = 64

In words: 8^two could be called "8 to the power 2" or "8 to the second power", or simply "8 squared"

Exponents are also chosen Powers or Indices.

Some more examples:

Instance: five³ = v × five × five = 125

In words: 5³ could be chosen "5 to the third power", "5 to the ability 3" or simply "5 cubed"

Instance: 2⁴ = 2 × 2 × 2 × 2 = 16

In words: two⁴ could be called "ii to the fourth power" or "2 to the ability 4" or simply "two to the 4th"

Exponents make it easier to write and utilise many multiplications

Case: 9⁶ is easier to write and read than 9 × ix × nine × 9 × 9 × 9

You lot can multiply any number by itself every bit many times as y'all desire using exponents.

Try here:

algebra/images/exponent-calc.js

So in general:

a^northward tells you to multiply a by itself,
then in that location are due north of those a's:

Another Mode of Writing It

Sometimes people apply the ^ symbol (to a higher place the 6 on your keyboard), as it is easy to type.

Example: 2^iv is the same every bit 2^iv

2^4 = 2 × 2 × 2 × 2 = sixteen

Negative Exponents

Negative? What could be the opposite of multiplying? Dividing!

And so nosotros divide past the number each time, which is the same every bit multiplying by 1 number

Instance: 8^-1 = 1 8 = 0.125

We tin go along on similar this:

Example: 5^-three = 1 5 × 1 5 × one five = 0.008

Only it is often easier to practise it this way:

5^-3 could also be calculated like:

1 5 × five × 5 = 1 v³ = 1 125 = 0.008

Negative? Flip the Positive!

That last case showed an easier way to handle negative exponents:

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

More than Examples:

Negative Exponent		Reciprocal of Positive Exponent		Answer
4^-2	=	1 / 4²	=	one/16 = 0.0625
ten^-3	=	ane / ten^three	=	one/one,000 = 0.001
(-2)^-three	=	1 / (-2)ⁱⁱⁱ	=	1/(-eight) = -0.125

What if the Exponent is 1, or 0?

one		If the exponent is i, so you lot just have the number itself (instance 9¹ = ix)

0		If the exponent is 0, and then y'all go one (example 9⁰ = 1)

		But what about 0⁰ ? Information technology could be either one or 0, then people say information technology is "indeterminate".

Information technology All Makes Sense

If you lot wait at that table, you will see that positive, zero or negative exponents are really role of the same (adequately simple) pattern:

Case: Powers of 5
	.. etc..
5²	5 × five	25
vⁱ	5	v
5⁰	1	1
v^-1	1 5	0.2
v^-2	1 five × 1 5	0.04
	.. etc..

Be Careful About Grouping

To avert confusion, use parentheses () in cases like this:

With () :	(−2)² = (−ii) × (−two) = 4
Without () :	−2^two = −(two²) = −(2 × 2) = −four

With () :	(ab)² = ab × ab
Without () :	ab² = a × (b)ⁱⁱ = a × b × b

305, 1679, 306, 1680, 1077, 1681, 1078, 1079, 3863, 3864