Math : Powers

Power

The power of a number says how many times  to use the number in multiplication.

Powers are also called Exponents or Indices ( index ).

Example 1

Find 5 to the power 2 or 5 to the second power or 5 squared

5^2 = 5 * 5 = 25

Example 2

Find 5 to the power 3 or 5 to the third power or 5 cube

5^3 = 5 * 5 * 5 = 125

Example 3

Find 5 to the power 4

5^4=5*5*5*5 = 625

Negative Exponent

Negative exponent which means how many times to divide one by that number

Example

2^(-3) = 1/(2*2*2) = 1/8

3^(-2) = 1/(3*3) = 1/9

We can say that 2^(-3) = 1/(2^3)

Note 1: If the exponent is 1, that is any number have the exponent 1 then the solution be the number itself.

5^1 = 5 , 8^1 = 8 , (-3)^1 = -3

Note 2 : if the exponent is 0, that is any number which has the exponent as 0 then the solution be 1.

5^0 = 1

That is,
  5^3=5*5*5=125
  5^2=5*5=25
  5^1=5
5^0= ?
5^-1=1/5
5^-2=1/5*5=1/25

In the above example, the exponent keeps on reducing by 1 and to the right side the number gets divisible by 5

When 5/5 we gets 1

Hence 5^0=1

Note 3 : If whole number 0 has exponent 0 that is 0^0 which gives 1

Let's consider the closest value of 0,
 0.1^0.1 = 0.7943282
0.01^0.01=0.954992586
0.001^0.001=0.993116

So as we approach to 0 we gets 1
That is, limit x->0, x^x=1

Laws of Exponent

 1. x^0 = 1

 2. x^1 = x

 3. x^-1 = 1/x  [ x^-m = 1/x^m]

 4. x^m * x^n = x^(m+n)

 5. x^m/x^n = x^(m-n)

 6. (x^m)^n = x^(m*n)

 7. (x*y)^m = x^m * y^n

 8. (x/y)^m = x^m/y^m