As previously discussed, in the problem 5 ³, 5 is called the base and three is the exponent. It is read as “ 5 to the 3rd power”, or “ 5 raised to the 3^rd.” The exponent says how many times to multiply the base by itself. In other words, take (5)(5)(5) which is 125. Therefore 5³=125.

This sets up some important properties of exponents.

 

x ^a x ^b = x ^{a + b} or y ² y ³ = y ^{2 + 3} = y⁵

 

Note- The bases (numbers on bottom, x and y) have got to be the same!

 

Example

 

(2 x ⁴) (3x⁵)

 

2(3) = 6, and 4 + 5 = 9

6x⁹

 

Answer!

 

 

 

And another important property

 

(x y)^c= x^c y^c or (2z)³=2³ z³ =8 z³

(x^y)^z = x^yz or (c²)⁴= c⁸

 

 

 

 

Example

(4y³)²

 

Apply the property.

4² y⁶

 

4² = 16

16 y⁶

 

Answer!

 

Here is a property dealing with dividing numbers that are the same base (x) and have exponents.

 

x^b = x ^b-a or y⁸ = y³

x^a y⁵

 

 

 

 

 

Example

 

3⁷ Apply the property. 7-5 = 2.

3⁵

3²

 

Square the 3.

 

3² = 9

 

Answer!

 

 

And the last property of exponents,

or