## Multiplying with Exponents
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 x"x" is called the BASE and "7" is called the EXPONENT. The exponent indicates how many times the base is used as a factor. x ^{7} 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: xWe 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, aFor example, jWhen 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, aFor example, b ^{5}/b^{4} = b^{5-4} = b^{1} = b.
POWER OF A PRODUCT For all numbers a, b, and m, (ab)For example, (d*h) ^{3} = d^{3}h^{3}
POWER OF A POWER For all numbers a, b, and m, (aWhen we raise a power to a power, the answer has the same base and the two exponents are multiplied. For example, (j ^{7})^{6} = j^{7*6} = j^{42}.
POWER OF A MONOMIAL For all numbers a, b, m, n, and p, (aWhen we raise a monomial to a power, we distribute the exponent over each factor in the monomial. For example, (p ^{8}*k^{2})^{9} = p^{8*9}*k^{2*9} = p^{72}k^{18}.
Sample Problem
Simplify the following:
(h Solution
First we distribute the exponent over each factor in the monomial:
hThen we simplify to get: h |