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6.3: Power Properties of Exponents

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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.

James Sousa: Properties of Exponents

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}

Answers

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

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Date Created:

Mar 12, 2013

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Sep 19, 2014
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