Multiplying with Exponents

This lesson will summarize the main rules we use when working with exponents and powers.
 
There are five different rules that we will review: multiplying with exponents, dividing with exponents, power of a product, power of a power, and power of a monomial.
 
BASIC DEFINITIONS
 
In the expression

x7

"x" is called the BASE and "7" is called the EXPONENT. The exponent indicates how many times the base is used as a factor. x⁷ is equivalent to x*x*x*x*x*x*x.
 
One important definition when dealing with exponents is that a quantity raised to the zero power is equal to 1.
 
When a quantity is raised to a negative power it is the same as the inverse of that quantity:

x-5 = 1/x5

We can also raise a quantity to a fractional power. For example, 4 raised to the 1/2 power is really the 2nd root of 4 which is 2. 8 raised the 1/3 power is the cube root of 8 which is 2.
 
MULTIPLYING WITH EXPONENTS
 
For all numbers a, m, and n,

am*an = amn

For example,

j7*j6 = j13

When we multiply terms having exponents and like bases, the answer will have the same base, in this case j. Then we add the exponents.
 
DIVIDING WITH EXPONENTS
 
For all numbers a, m, and n,

am/an = am-n   if a ≠ 0

For example, b⁵/b⁴ = b^5-4 = b¹ = b.
 
POWER OF A PRODUCT
 
For all numbers a, b, and m,

(ab)m = ambm

For example, (d*h)³ = d³h³
 
POWER OF A POWER
 
For all numbers a, b, and m,

(am)n = am*n

When we raise a power to a power, the answer has the same base and the two exponents are multiplied. For example, (j⁷)⁶ = j^7*6 = j⁴².
 
POWER OF A MONOMIAL
 
For all numbers a, b, m, n, and p,

(ambn)p = am*pbn*p

When we raise a monomial to a power, we distribute the exponent over each factor in the monomial. For example, (p⁸*k²)⁹ = p^8*9*k^2*9 = p⁷²k¹⁸.

Simplify the following:

(h8n2)6

First we distribute the exponent over each factor in the monomial:

h8*6n2*6

Then we simplify to get:

h48n12