Laws of Exponents and Multiplying Monomials
Objective Learn the algebra of polynomials by
using the laws of exponents to multiply and divide monomials.
The main ideas in this lesson are the laws for multiplying and
dividing powers. In this lesson, we will deal with monomials that
are powers of a single variable.
Powers
Let's begin by reviewing powers and exponents. If a is a
variable, then a^{ 2} represents a Â· a , a^{ 3}
represents a Â· a Â· a , and more generally, a^{ b}
represents
Remember that a is called the base, b is
called the exponent, and a^{ b} is
called the power. Also, the power a^{ 4}
is read "a raised to the fourth power". In general,
when we write a^{ b}, we say "a raised to the b
power".
When multiplying monomials, we must analyze the product of the
two powers of the same base. Consider x^{ 2} Â· x^{ 3}.
Let' s analyze this by multiplying various powers of 2 together.
2^{ 2} Â· 2^{ 3} = 4 Â· 8 = 32 = 2^{ 5}
2^{ 4} Â· 2^{ 5} = 16 Â· 32 = 512 = 2^{ 9}
In both cases, the exponent of the resulting power is the sum
of the exponents in the two factors. For 2^{ 2} Â· 2^{
3} , 5 = 2 + 3, and for 2^{ 4} Â· 2^{ 5} ,9
= 4 + 5. The table shows what happens when a power of 2 is
multiplied by "2 to the first power," which is 2.
Recall that any number raised to the first power is the number
itself.
Notice each power that results. Do you see a pattern? Each
resulting power can be found by adding 1 to the exponent of the
original power. For example, 2^{ 3 }Â· 2^{ 1} = 2^{
3 + 1} or 2^{ 4 }. Using symbols, we write 2^{ n }Â·
2^{ 1} = 2^{ n + 1}. The following table shows
what happens when a power of 2 is multiplied by "2 to the
second power" or 4.
Again, notice each power that results. In this case, each
power can be found by adding 2 to the exponent of the original
power. For example, 2^{ 3 }Â· 2^{ 2} = 2^{ 3 +
2} or 2^{ 5}. Using symbols, we write 2^{ n }Â·
2^{ 2} = 2^{ n + 2}.
This confirms our earlier observation that when we multiply
two powers that have the same base, the exponent of the resulting
power is the sum of the exponents in the two factors.
Now is a good time to explore this idea for yourself. Choose
your own bases and exponents. Then evaluate both a^{ b}
Â· a^{ c} and a^{ b + c} to verify that they are
equal. Why is this true? Remember th at exponents area shorthand
that represents a repeated product of the same number or
variable. So,
In general, when we multiply a^{ b }and a^{ c},
we'll get
Key Idea
When we multiply a power of a times another power of a, the
result is a power of a , where the exponent is the sum of the
exponents of the two factors. In symbols,
a^{ b} Â· a^{ c} = a^{ b + c}
This holds true for any number a and positive integers b and c
.
