Back
Applying Them Together
> ... Math > Rules of Exponents > Applying Them Together

In some of the exercises you will do, there will be multiple steps to simplifying the expression, much like in the Order of Operations. Each of these rules is a tool and all the tools can be used together to simplify expressions.

Rules of Exponents - Applying them Together

Review

• Product Rule: When the bases are the same, you add the powers.
• Quotient Rule: When the bases are the same, you subtract the powers.
• Negative Exponent Rule: Move it to the other part of the fraction (numerator or denominator) and the exponent becomes positive.
• Power Rule: Multiply the powers.
• Exponents of 0 and 1: Anything raised to the 1 is itself, anything raised to the 0 is 1.
• (−1) Raised to an Exponent: If the exponent is even, the answer is positive, if the exponent is odd, the answer is negative.
• Follow the Order of Operations (PEDMAS).

### Practice Problems

Simplify and Evaluate the following expressions:
1. $${\text{a}}^{5}\,{\text{b}}^{3}\left ( {\text{a}}\,{\text{b}} \right )^{4}\,{\text{b}} =$$ (
Solution
Solution:
$${\text{a}}^{9}\,{\text{b}}^{8}$$
Details:
In this example, the first thing we need to do is look at the $$\left ({\text{a}}{\text{b}} \right )^{4}$$ part.

$$\left ({\text{a}}{\text{b}} \right )^{4}$$ is the same as $$\text{ab}$$ multiplied together 4 times.

This is the same as the variable a multiplied together 4 times and the variable b multiplied together 4 times.

Thus, the part $$\left ({\text{a}}{\text{b}} \right )^{4}$$ is the same as $${\text{a}}^{4}\,{\text{b}}^{4}$$ and we can substitute that back into our original expression.

Now we can add the exponents of factors with the same base. Remember that anything that doesn’t have an exponent actually has an invisible exponent of 1. So the variable b on the end has an exponent of 1 and we need to include that in our solution.

$${\text{a}}^{5}$$ and $${\text{a}}^{4}$$ become $${\text{a}}^{\left ( 5+4 \right )}={\text{a}}^{9}$$

$${\text{b}}^{3}$$ and $${\text{b}}^{4}$$ become $${\text{b}}^{\left ( 3+4+1 \right )}={\text{b}}^{8}$$

Our final solution is $${\text{a}}^{9}\,{\text{b}}^{8}$$.
)
2. $$\left ({\text{x}}\,{\text{y}} \right )^{3}{\text{x}}\,{\text{y}} =$$ (
Solution
Solution:
$${\text{x}}^{4}\,{\text{y}}^{4}$$
)
3. $$\dfrac{{\text{x}}^{5}\,{\text{y}}^{3}\,{\text{x}}^{2}}{{\text{x}}^{6}{\text{y}}^{2}}=$$ (
Video Solution
Solution:
$${\text{x}}\,{\text{y}}$$
Details:

(Video Source | Transcript)
)
4. $$\dfrac{{\text{m}}^{3}\,{\text{x}}^{7}}{{\text{m}}^{3}{\text{x}}^{2}}=$$ (
Solution
Solution:
$${\text{x}}^{5}$$
Details:
In this case, we are using the quotient rule on both variables m and x.

$$\dfrac{{\color{Blue} {\text{m}}}^{3}\,{\color{DarkOrange} {\text{x}}}^{7}}{{\color{Blue} {\text{m}}}^{3}{\color{DarkOrange} {\text{x}}}^{2}}$$

First, look at the quotient rule for the m variables.

$$\dfrac{{\text{m}}^{3}}{{\text{m}}^{3}} = {\text{m}}^{\left ( 3-3 \right )} = {\text{m}}^{0} = 1$$

Since $${\text{m}}^{\left ( 3-3 \right )} = {\text{m}}^{0} = 1$$ and 1 multiplied to anything is itself, we only have the x variables left.

$$\displaystyle{\color{Blue} {\text{m}}}^{\left ( 3-3 \right )}\cdot\frac{{\text{x}}^{7}}{{\text{x}}^{2}} = {\color{Blue} {\text{m}}}^{0} \cdot\frac{{\text{x}}^{7}}{{\text{x}}^{2}}={\color{Blue} 1}\cdot\frac{{\text{x}}^{7}}{{\text{x}}^{2}}$$

Now we look at the x variables and their exponents.

$$\dfrac{{\text{x}}^{7}}{{\text{x}}^{2}} = {\text{x}}^{\left ( 7-2 \right )} = {\text{x}}^{5}$$

$$\displaystyle{\color{Blue} 1}\cdot\frac{{\color{DarkOrange} {\text{x}}}^{7}}{{\color{DarkOrange} {\text{x}}}^{2}} = {\color{Blue} 1}\cdot {\color{DarkOrange} {\text{x}}}^{\left ( 7-2 \right )} = {\color{Blue} 1}\cdot {\color{DarkOrange} {\text{x}}}^{5} = {\color{DarkOrange} {\text{x}}}^{5}$$

So we have $$1 \cdot {\text{x}}^{5} = {\text{x}}^{5}$$.
)
5. $$\left ( {\text{b}}^{4}\,{\text{x}}^{3}\,{\text{y}}\,{\text{b}} \right )^{2}\,{\text{x}} =$$ (
Video Solution
Solution:
$${\text{b}}^{10}\,{\text{x}}^{7}\,{\text{y}}^{2}$$
Details:

(Video Source | Transcript)
)
6. $$\left ( {-}{\text{m}} \right )^{3}\,{\text{b}}^{2}\,{\text{mx}}^{3} =$$ (
Solution
Solution:
$$-{\text{m}}^{4}\,{\text{b}}^{2}\,{\text{x}}^{3}$$
)
7. $$\left ( -3 \right )^{3}\,{\text{a}}^{2}\,{\text{b}}^{4}\left ( -2 \right )^{2} =$$ (
Solution
Solution:
$$-108{\text{a}}^{2}\,{\text{b}}^{4}$$
)

### Need More Help?

1. Study other Math Lessons in the Resource Center.
2. Visit the Online Tutoring Resources in the Resource Center.