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# Exponential Terms Raised to an Exponent

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Power Properties of Exponents

There are 1,000 bacteria present in a culture. When the culture is treated with an antibiotic, the bacteria count is halved every 4 hours. How many bacteria remain 24 hours later?

### Watch This

Watch the second part of this video, starting around 3:30.

### Guidance

The last set of properties to explore are the power properties. Let’s investigate what happens when a power is raised to another power.

#### Investigation: Power of a Power Property

1. Rewrite $(2^3)^5$ as $2^3$ five times.

$(2^3)^5 = 2^3 \cdot 2^3 \cdot 2^3 \cdot 2^3 \cdot 2^3$

2. Expand each $2^3$ . How many 2’s are there?

$(2^3)^5 = \underbrace{2 \cdot 2 \cdot 2}_{2^3} \cdot \underbrace{2 \cdot 2 \cdot 2}_{2^3} \cdot \underbrace{2 \cdot 2 \cdot 2}_{2^3} \cdot \underbrace{2 \cdot 2 \cdot 2}_{2^3} \cdot \underbrace{2 \cdot 2 \cdot 2}_{2^3} = 2^{15}$

3. What is the product of the powers?

$3 \cdot 5 = 15$

4. Fill in the blank. $(a^m)^n = a^{-^\cdot-}$

$(a^m)^n = a^{mn}$

The other two exponent properties are a form of the distributive property.

Power of a Product Property: $(ab)^m = a^m b^m$

Power of a Quotient Property: $\left( \frac{a}{b} \right)^m = \frac{a^m}{b^m}$

#### Example A

Simplify the following.

(a) $(3^4)^2$

(b) $(x^2 y)^5$

Solution: Use the new properties from above.

(a) $(3^4)^2 = 3^{4 \cdot 2} = 3^8 = 6561$

(b) $(x^2 y)^5 = x^{2 \cdot 5} y^5 = x^{10} y^5$

#### Example B

Simplify $\left( \frac{3a^{-6}}{2^2 a^2} \right)^4$ without negative exponents.

Solution: This example uses the Negative Exponent Property from the previous concept. Distribute the $4^{th}$ power first and then move the negative power of $a$ from the numerator to the denominator.

$\left( \frac{3a^{-6}}{2^2 a^2} \right)^4 = \frac{3^4 a^{-6 \cdot 4}}{2^{2 \cdot 4} a^{2 \cdot 4}} = \frac{81a^{-24}}{2^8 a^8} = \frac{81}{256a^{8+24}} = \frac{81}{256a^{32}}$

#### Example C

Simplify $\frac{4x^{-3} y^4 z^6}{12x^2 y} \div \left( \frac{5xy^{-1}}{15x^3 z^{-2}} \right)^2$ without negative exponents.

Solution: This example is definitely as complicated as these types of problems get. Here, all the properties of exponents will be used. Remember that dividing by a fraction is the same as multiplying by its reciprocal.

$\frac{4x^{-3} y^4 z^6}{12x^2 y} \div \left( \frac{5xy^{-1}}{15x^3 z^{-2}} \right)^2 &= \frac{4x^{-3} y^4 z^6}{12x^2 y} \cdot \frac{225x^6 z^{-4}}{25x^2 y^{-2}}\\&= \frac{y^3 z^6}{3x^5} \cdot \frac{9x^4 y^2}{z^4}\\&= \frac{3x^4 y^5 z^6}{x^5 z^4}\\&= \frac{3y^5 z^2}{x}$

Intro Problem Revisit To find the number of bacteria remaining, we use the exponential expression $1000 (\frac{1}{2})^n$ where n is the number of four-hour periods.

There are 6 four-hour periods in 24 hours, so we set n equal to 6 and solve.

$1000 (\frac{1}{2})^6$

Applying the Power of a Quotient Property, we get:

$1000 (\frac{1^6}{2^6}) = \frac {1000 \cdot 1}{2^6} = \frac {1000}{64} = 15.625$

Therefore, there are 15.625 bacteria remaining after 24 hours.

### Guided Practice

Simplify the following expressions without negative exponents.

1. $\left( \frac{5a^3}{b^4} \right)^7$

2. $(2x^5)^{-3} (3x^9)^2$

3. $\frac{(5x^2 y^{-1})^3}{10y^6} \cdot \left( \frac{16x^8 y^5}{4x^7} \right)^{-1}$

1. Distribute the 7 to every power within the parenthesis.

$\left( \frac{5a^3}{b^4} \right)^7 = \frac{5^7 a^{21}}{b^{28}} = \frac{78,125a^{21}}{b^{28}}$

2. Distribute the -3 and 2 to their respective parenthesis and then use the properties of negative exponents, quotient and product properties to simplify.

$(2x^5)^{-3} (3x^9)^2 = 2^{-3} x^{-15} 3^2 x^{18} = \frac{9x^3}{8}$

3. Distribute the exponents that are outside the parenthesis and use the other properties of exponents to simplify. Anytime a fraction is raised to the -1 power, it is equal to the reciprocal of that fraction to the first power.

$\frac{\left(5x^2 y^{-1}\right)^3}{10y^6} \cdot \left( \frac{16x^8 y^5}{4x^7} \right)^{-1} &= \frac{5^3 x^{-6} y^{-3}}{10y^6} \cdot \frac{4x^7}{16x^8 y^5}\\&= \frac{500xy^{-3}}{160x^8 y^{11}}\\&= \frac{25}{8x^7 y^{14}}$

### Vocabulary

Power of Power Property
$(a^m)^n = a^{mn}$
Power of a Product Property
$(ab)^m = a^m b^m$
Power of a Quotient Property
$\left( \frac{a}{b} \right)^m = \frac{a^m}{b^m}$

### Practice

Simplify the following expressions without negative exponents.

1. $(2^5)^3$
2. $(3x)^4$
3. $\left( \frac{4}{5} \right)^2$
4. $(6x^3)^3$
5. $\left( \frac{2a^3}{b^5} \right)^7$
6. $(4x^8)^{-2}$
7. $\left( \frac{1}{7^2 h^9} \right)^{-1}$
8. $\left( \frac{2x^4 y^2}{5x^{-3} y^5} \right)^3$
9. $\left( \frac{9m^5 n^{-7}}{27 m^6 n^5} \right)^{-4}$
10. $\frac{(4x)^2 (5y)^{-3}}{(2x^3 y^5)^2}$
11. $(5r^6)^4 \left( \frac{1}{3} r^{-2} \right)^5$
12. $(4t^{-1} s)^3 (2^{-1} ts^{-2})^{-3}$
13. $\frac{6a^2 b^4}{18a^{-3} b^4} \cdot \left( \frac{8b^{12}}{40a^{-8} b^5} \right)^2$
14. $\frac{2(x^4 y^4)^0}{2^4 x^3 y^5 z} \div \frac{8z^{10}}{32x^{-2} y^5}$
15. $\frac{5g^6}{15g^0 h^{-1}} \cdot \left( \frac{h}{9g^{15} j^7} \right)^{-3}$
16. Challenge $\frac{a^7 b^{10}}{4a^{-5} b^{-2}} \cdot \left[ \frac{(6ab^{12})^2}{12a^9 b^{-3}} \right]^2 \div (3a^5 b^{-4})^3$
17. Rewrite $4^3$ as a power of 2.
18. Rewrite $9^2$ as a power of 3.
19. Solve the equation for $x$ . $3^2 \cdot 3^x = 3^8$
20. Solve the equation for $x$ . $(2^x)^4 = 4^8$