The product rule for exponents states that when we multiply two or more exponents that have the same base, we add the exponents:
For example,
To see why the product rule is true, let's take the above example and work it out again, the long way.
For the two factors and we have and If we now substitute these two expressions into , we get the same result:
The product rule is just a shortcut for the process above.
Simplify
We add the exponents, as follows:
Simplify $z^{17}\cdot z^{5}.$
|
a
|
$z^{9}$ |
|
b
|
$z^{22}$ |
|
c
|
$z^{19}$ |
|
d
|
$z^{12}$ |
|
e
|
$z^{25}$ |
$x^4 \cdot x^3 =$
|
a
|
$x^{12}$ |
|
b
|
$x^5$ |
|
c
|
$x^7$ |
|
d
|
$x^3$ |
|
e
|
$x$ |
Find such that is correct.
We add the exponents, as follows:
Therefore , so
Find $n$ such that $z^5\cdot z^7 \cdot z^{2}=z^{n}$ is correct.
|
a
|
$9$ |
|
b
|
$14$ |
|
c
|
$10$ |
|
d
|
$7$ |
|
e
|
$12$ |
Find $n$ such that $y^2\cdot y^3 \cdot y^{2}=y^{n}$ is correct.
|
a
|
$7$ |
|
b
|
$3$ |
|
c
|
$6$ |
|
d
|
$4$ |
|
e
|
$5$ |
Simplify
Here we have two variables, and We will start by grouping the two variables separately:
This way, we can apply the product rule to the two variables independently:
Simplify $(xz)(x^2y^2)(x^3z^3).$
|
a
|
$x^6 y^2 z^3$ |
|
b
|
$x^3 y^2 z^3$ |
|
c
|
$x^3 y^3 z^3$ |
|
d
|
$x^6 y^2 z^4$ |
|
e
|
$x^3 y z^2$ |
Use exponents to rewrite the product $ b \cdot b^2 \cdot c^3 \cdot b^4 \cdot c^5 \cdot c^6 \cdot b^7.$
|
a
|
$b^{14}\cdot c^{14}$ |
|
b
|
$b^{18}\cdot c^{11}$ |
|
c
|
$b^{6}\cdot c^{7}$ |
|
d
|
$b^{4}\cdot c^{3}$ |
|
e
|
$b^{7}\cdot c^{6}$ |