Lynn Marecek; MaryAnne Anthony-Smith; Andrea Honeycutt Mathis; Jay Abramson; Sharon North; Amy Baldwin; Alyssa Howell

5.1.4 Using Power Properties for Exponents

5.1.4 • Using Power Properties for Exponents

Activity

Work with a partner to complete the activity. Discuss any differences in the simplified results you find. Make sure you both understand each step you take to simplify.

Quotient to a Negative Power Property

If $a$ and $b$ are real numbers, $a \neq 0$ , $b \neq 0$ , and $n$ is an integer, then

$(\frac{a}{b})^{- n} = (\frac{b}{a})^{n}$

Complete problems 1 - 3 using the product property for exponents. Double click on mathematical expressions/equations to enlarge.

1. $(\frac{3}{7})^{- 5}$

$(\frac{7}{3})^{5} = (\frac{7^{5}}{3^{5}}) = \frac{16, 807}{243}$

2. $(- \frac{y}{z})^{- 7}$

Compare your answer:

$(- \frac{z}{y})^{7} = - \frac{z^{7}}{y^{7}}$

3. $(\frac{g}{h})^{- 6}$

Compare your answer:

$(\frac{h}{g})^{6} = \frac{h^{6}}{g^{6}}$

Power Property for Exponents

If $a$ is a real number and $m$ and $n$ are integers, then

$(a^{m})^{n} = a^{m \cdot n}$

Complete problems 4 - 6 using the power property for exponents. Double click on mathematical expressions/equations to enlarge.

4. $(4^{2})^{5}$

Compare your answer:

$4^{10} = 1, 048, 576$

5. $(n^{3})^{7}$

Compare your answer:

$n^{21}$

6. $(2^{3})^{2} (d^{4})^{3}$

Compare your answer:

$(2^{6}) (d^{12}) = 64 d^{12}$

Product to a Power Property for Exponents

If $a$ and $b$ are real numbers and $m$ is a whole number, then

$(a b)^{m} = a^{m} b^{m}$

Complete problems 7 - 9 using the product to a power property for exponents.

7. $(- 2 t v)^{7}$

Compare your answer:

$(- 2)^{7} t^{7} v^{7} = - 128 t^{7} v^{7}$

8. $(4 p r^{2})^{5}$

Compare your answer:

$4^{5} p^{5} r^{10} = 1, 024 p^{5} r^{10}$

9. $(3 h^{3})^{- 2}$

Compare your answer:

$(\frac{1}{3 h^{3}})^{2} = \frac{1}{3^{2} h^{6}} = \frac{1}{9 h^{6}}$

Choose one example above and explain each step to your partner. Listen as your partner does the same for you.

Self Check

Simplify the following expression.

$(\frac{3^3x^{-2}}{q^3})^3$

$\frac{19,683x^6}{q^9}$
$\frac{729}{x^6q^9}$
$\frac{19,683}{x^6q^9}$
$\frac{27}{x^2q^3}$

Additional Resources

Quotient to a Negative Power Property

Suppose now we have a fraction raised to a negative exponent. Let’s use our definition of negative exponents to lead us to a new property.

$(\frac{3}{4})^{- 2}$

Step 1 - Use the definition of a negative exponent, $a^{- n} = \frac{1}{a^{n}}$ .

$\frac{1}{(\frac{3}{4})^{2}}$

Step 2 - Simplify the denominator.

$\frac{1}{\frac{9}{16}}$

Step 3 - Simplify the complex fraction.

$\frac{16}{9}$

But we know that $\frac{16}{9}$ is $(\frac{4}{3})^{2}$ .

Step 4 - This tells us that

$(\frac{3}{4})^{- 2} = (\frac{4}{3})^{2}$

To get from the original fraction raised to a negative exponent to the final result, we took the reciprocal of the base—the fraction—and changed the sign of the exponent.

This leads us to the Quotient to a Negative Power Property.

Quotient to a Negative Power Property

If $a$ and $b$ are real numbers, $a \neq 0$ , $b \neq 0$ , and $n$ is an integer, then

$(\frac{a}{b})^{- n} = (\frac{b}{a})^{n}$

Now that we have negative exponents, we can use the Product Property we learned in Activity 5.1.2 with expressions that have negative exponents, too.

Power Property for Exponents

Now let’s look at an exponential expression that contains a power raised to a power. See if you can discover a general property.

Notice the 6 is the product of the exponents, 2 and 3. We see that $(x^{2})^{3}$ is $x^{2 \cdot 3}$ or $x^{6}$ .

We multiplied the exponents. This leads to the Power Property for Exponents.

Power Property for Exponents

If $a$ is a real number and $m$ and $n$ are integers, then

$(a^{m})^{n} = a^{m \cdot n}$

To raise a power to a power, multiply the exponents.

Product to a Power Property for Exponents

We will now look at an expression containing a product that is raised to a power. Can you find this pattern?

$(2 x)^{3}$

Step 1 - What does this mean?

$2 x \cdot 2 x \cdot 2 x$

Step 2 - We group the like factors together.

$2 \cdot 2 \cdot 2 \cdot x \cdot x \cdot x$

Step 3 - How many factors of 2 and of $x$ ?

$2^{3} \cdot x^{3}$

Notice that each factor was raised to the power and $(2 x)^{3}$ is $2^{3} \cdot x^{3}$ .

The exponent applies to each of the factors. This leads to the Product to a Power Property for Exponents .

Product to a Power Property for Exponents

If $a$ and $b$ are real numbers and $m$ is a whole number, then

$(a b)^{m} = a^{m} b^{m}$

To raise a product to a power, raise each factor to that power.

Quotient to a Power Property for Exponents

Now we will look at an example that will lead us to the Quotient to a Power Property.

$(\frac{x}{y})^{3}$

Step 1 - This means

$\frac{x}{y} \cdot \frac{x}{y} \cdot \frac{x}{y}$

Step 2 - Multiply the fractions.

$\frac{x \cdot x \cdot x}{y \cdot y \cdot y}$

Step 3 - Write with exponents.

$\frac{x^{3}}{y^{3}}$

Notice that the exponent applies to both the numerator and the denominator.

We see that $(\frac{x}{y})^{3}$ is $\frac{x^{3}}{y^{3}}$ .

This leads to the Quotient to a Power Property for Exponents.

Quotient to a Power Property for Exponents

If $a$ and $b$ are real numbers, $b \neq 0$ , and $m$ is an integer, then

$(\frac{a}{b})^{m} = \frac{a^{m}}{b^{m}}$

To raise a fraction to a power, raise the numerator and denominator to that power.

Try it

Using Power Properties for Exponents

Simplify each expression using the property identified.

Use the Quotient to a Negative Power Property for Exponents.

1. $(\frac{3}{7})^{- 5}$

2. $(- \frac{r}{s})^{- 2}$

Use the Power Property for Exponents.

3. $(x^{3})^{5}$

4. $(2^{2})^{3}$

5. $(h^{4})^{5} (h^{2})^{2}$

Use the Product to a Power Property for Exponents.

6. $(- 2 c d)^{6}$

7. $(4 z)^{- 3}$

Use the Quotient to a Power Property for Exponents.

8. $(\frac{2 x y^{2}}{z})^{3}$

9. $(\frac{4 p^{- 3}}{q^{2}})^{2}$

Compare your answers:

Here is how to simplify the expressions:

$(\frac{3}{7})^{- 5} = (\frac{7}{3})^{5} = \frac{7^{5}}{3^{5}} = \frac{16, 807}{243}$
$(- \frac{r}{s})^{- 2} = (- \frac{s}{r})^{2} = \frac{s^{2}}{r^{2}}$
$(x^{3})^{5} = x^{15}$
$(2^{2})^{3} = 2^{6} = 64$
$(h^{4})^{5} (h^{2})^{2} = (h^{20}) (h^{4}) = h^{24}$
$(- 2 c d)^{6} = (- 2)^{6} c^{6} d^{6} = 64 c^{6} d^{6}$
$(4 z)^{- 3} = \frac{1}{(4 z)^{3}} = \frac{1}{4^{3} z^{3}} = \frac{1}{64 z^{3}}$
$(\frac{2 x y^{2}}{z})^{3}$

Step 1 - Use Quotient to a Power Property, $(\frac{a}{b})^{m} = \frac{a^{m}}{b^{m}}$ .

$\frac{(2 x y^{2})^{3}}{z^{3}}$

Step 2 - Use the Product to a Power Property, $(a b)^{m} = a^{m} b^{m}$ .

$\frac{8 x^{3} y^{6}}{z^{3}}$

9. $(\frac{4 p^{- 3}}{q^{2}})^{2}$

Step 1 - Use Quotient to a Power Property, $(\frac{a}{b})^{m} = \frac{a^{m}}{b^{m}}$ .

$\frac{(4 p^{- 3})^{2}}{(q^{2})^{2}}$

Step 2 - Use the Product to a Power Property, $(a b)^{m} = a^{m} b^{m}$ .

$\frac{4^{2} (p^{- 3})^{2}}{(q^{2})^{2}}$

Step 3 - Simplify using the Power Property, $(a^{m})^{n} = a^{m \cdot n}$ .

$\frac{16 p^{- 6}}{q^{4}}$

Step 4 - Use the definition of negative exponent.

$\frac{16}{q^{4}} \cdot \frac{1}{p^{6}}$

Step 5 - Simplify.

$\frac{16}{p^{6} q^{4}}$