Hi, and welcome to this video on the properties of exponents!
Working with polynomial, radical, and rational functions often times requires us to perform
algebraic operations with powers.
Recall that a power is nothing more than a base that is raised to an exponent.
So let’s take a look at the properties of exponents that are used to simplify algebraic expressions with powers.
The product of powers property applies to powers with the same base.
When asked to multiply powers with the same base, simply add the exponents.
For example:
b^2 times b^3 equals b^5.
This rule makes intuitive sense if you expand each power like this:
(b times b), times, (b times b, times b).
Counting up the bases of b that are being multiplied results in (1, 2, 3, 4, 5)
b^5.
Therefore, the general form of the product of powers rule is:
b^m times b^n equals b^(m plus n)
The quotient of powers property also applies to powers with the same base;
however, the rule requires the subtraction of exponents.
Let’s look at an example:
b^5 over b^3 in expanded form would look like this:
b times b, times b, times b, times b, all over, b times b, times b.
Canceling out common factors of b from the numerator and the denominator would simplify to be b^2.
The quotient of powers property allows a quicker, more efficient result.
Simply subtract the exponent of the denominator from the exponent of the numerator:
5 minus 3 equals 2.
The general form for quotient of powers property is:
b^m over b^n equals b^(m minus n).
The power of a power property allows us to raise a power to another exponent.
Once again, expanding an expression makes understanding the rule a bit easier.
Suppose the power b^3 is squared.
In expanded form it would look like this:
(b times b, times b), times, (b times b, times b).
Using the product of powers property, or simply counting up the bases of b, results in b^6th power.
The general form of this property is: (b^m)^n equals b^(m) times ^(n).
Expanding on the power of a power property results in additional tools that are used to simplify
expressions with powers.
For example, suppose the base of a power is a monomial with a coefficient and a variable.
(3b^4)^2 expands to (3b^4)(3b^4).
Rearranging the factors to group the coefficients and the powers with the same base, b,
and simplifying gives us 3 times 3, times b^4, times b^4 equals 9b^8.
This is called the power of a product property, and illustrates that the exponent of each factor
in the base must be multiplied by the power outside the parentheses.
The general form for the power of product property is:
(b times c)^x equals b^x times c^x.
The power of quotient property is similar in that the exponent of a rational base is multiplied
by the exponents in both the numerator and denominator.
Let’s look at some examples:
(b^3 over c^2)^2 expands to (b^3 over c^2) times (b^3 over c^2).
The product of powers property results in the simplified expression:
b^6 over c^4.
However, applying the power of quotient property directly is often more efficient:
(b^3 over c^2)^2 equals b^(3 times 2) over c^(2 times 2), which equals b^6 over c^4.
Here’s another example:
(2b over c^2d)^3 expands to (2b over c^2d) times (2b over c^2d), times (2b over c^2d).
Rearranging factors and applying the product of powers property results in:
2 times 2, times 2, times b, times b, times b, all over, c^2 times c^2, times c^2, times d, times d, times d,
this equals 8b^3 over c^6 times d^3.
If we were to apply the power of quotient property, it would look like this:
2^(1 times 3) times b^(1times 3), all over c^(2 times 3), times d^(1 times 3),
which equals 2^3 times b^3, all over, c^6 times d^3,
which equals 8b^3, all over, c^6 times d^3.
The general rule for the power of quotient property is:
(b over c)^x equals b^x over c^x.
As you can see, these rules are essential in simplifying expressions within our work
with various algebraic functions.
Thanks for watching this review of exponents!
Happy studying!