   # Multiplying with Exponents

 Our Algebra 1 tutorial software program contains over 60 topic areas. One of them is Multiplying with Exponents, and this page summarizes the main ideas of this topic. This page is intended for review, and is not a substitute for the interactive, self-paced tutorials of the MathTutor algebra 1 software program.

General Explanation

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 = b5-4 = 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 = j7*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 = p8*9*k2*9 = p72k18.

Sample Problem

Simplify the following:
```(h8n2)6
```

Solution

First we distribute the exponent over each factor in the monomial:
```h8*6n2*6
```
Then we simplify to get:
```h48n12
```