EXPONENTS:
See: Square Roots
See: Scientific Notation

Just as Multiplication is "rapid addition":
3 x 4 = 12:  3 + 3 + 3 + 3  or 4 + 4 + 4
5 x 8 = 40: 5 + 5 + 5 + 5 + 5 + 5 + 5 + 5 or 8 + 8 + 8 + 8 + 8

And Division is "rapid subtraction":
8/ 2 = 8 - 2 = 6:  6 - 2 = 4:  4 - 2 = 2:  2 - 2 = 0
2 can be subtracted from 8, four  times.

EXPONENTS or POWERS are "Rapid Multiplication".

The Most common Exponent is '2' or the 2nd Power or Squared

BASE ^ Power
N^0 = 1: Any Number (Except 0) to the ZERO Power is: ONE (1).

N^1 = N: Any Number  (Except 0)  to the First Power is that NUMBER. Every number has an implied 1st Power.  This is important when you numbers are replaced with variables:

N^2 = N x N: This is the second power or N Squared. The answer will always be a POSITIVE. (Signed numbers: Like Signs)

N^3 = N x N x N: This is the Third power or N Cubed. (be careful with negative Values)
-3^2  is not the same as (-3)^2.  Order of Operations:

A Negative Power makes a reciprocal: 
N^-1 = 1/N
N^-2 = 1/N^2

A Negative Exponent should never be left in the Denominator of a fraction.
1/N^-2 -> N^2

Many calculators have a Square or Cube Button, but you can also use the Exponent (^ Power)  Button: 

4^2 = 16.    16^(1/2) power = 4

2^3 = 2 x 2 x 2 = 8.  8^(1/3) power is 2

.5 can also be used for the Square Root or 1/2 Power, BUT since 1/3 is a never ending repeating value, (.3333333...) you cannot use that to get an equivalent answer.

A PERFECT SQUARE is when N is an Integer:
1^2 = 1 x 1 = 1
2^2 = 2 x 2 = 4
3^2 = 3 x 3 = 9
4^2 = 4 x 4 = 16
5^2 = 5 x 5 = 25
6^2 = 6 x 6 = 36
7^2 = 7 x 7 = 49
8^2 = 8 x 8 = 64
9^2 = 9 x 9 = 81
10^2 = 10 x 10 = 100
11^2 = 11 x 11 = 121
12^2 = 12 x 12 = 144
13^2 = 13 x 13 = 169
15^2 = 15 x 15 = 255
20^2 = 20 x 20 = 400

Revised: April 24, 2024