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Exponential Terms Raised to an Exponent

Multiply to raise exponents to other exponents

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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 remained 24 hours later?

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 \begin{align*}(2^3)^5\end{align*} as \begin{align*}2^3\end{align*} five times.

\begin{align*}(2^3)^5 = 2^3 \cdot 2^3 \cdot 2^3 \cdot 2^3 \cdot 2^3\end{align*}

2. Expand each \begin{align*}2^3\end{align*}. How many 2’s are there?

\begin{align*}(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}\end{align*}

3. What is the product of the powers?

\begin{align*}3 \cdot 5 = 15\end{align*}

4. Fill in the blank. \begin{align*}(a^m)^n = a^{-^\cdot-}\end{align*}

\begin{align*}(a^m)^n = a^{mn}\end{align*}

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

Power of a Product Property: \begin{align*}(ab)^m = a^m b^m\end{align*}

Power of a Quotient Property: \begin{align*}\left( \frac{a}{b} \right)^m = \frac{a^m}{b^m}\end{align*}

Solve the following problems

Simplify the following.

\begin{align*}(3^4)^2\end{align*}

\begin{align*}(3^4)^2 = 3^{4 \cdot 2} = 3^8 = 6561\end{align*}

\begin{align*}(x^2 y)^5\end{align*}

\begin{align*}(x^2 y)^5 = x^{2 \cdot 5} y^5 = x^{10} y^5\end{align*}

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

This example uses the Negative Exponent Property from the previous concept. Distribute the \begin{align*}4^{th}\end{align*} power first and then move the negative power of \begin{align*}a\end{align*} from the numerator to the denominator.

\begin{align*}\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}}\end{align*}

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

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.

\begin{align*}\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}\end{align*}

Examples

Example 1

Earlier, you were asked how many bacteria remained 24 hours later. 

To find the number of bacteria remaining, we use the exponential expression \begin{align*}1000 (\frac{1}{2})^n\end{align*} 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.

\begin{align*}1000 (\frac{1}{2})^6\end{align*}

Applying the Power of a Quotient Property, we get:

\begin{align*}1000 (\frac{1^6}{2^6}) = \frac {1000 \cdot 1}{2^6} = \frac {1000}{64} = 15.625\end{align*}

Therefore, there are 15.625 bacteria remaining after 24 hours.

Simplify the following expressions without negative exponents.

Example 2

\begin{align*}\left( \frac{5a^3}{b^4} \right)^7\end{align*}

Distribute the 7 to every power within the parenthesis.

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

Example 3

\begin{align*}(2x^5)^{-3} (3x^9)^2\end{align*}

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

\begin{align*}(2x^5)^{-3} (3x^9)^2 = 2^{-3} x^{-15} 3^2 x^{18} = \frac{9x^3}{8}\end{align*}

Example 4

\begin{align*}\frac{(5x^2 y^{-1})^3}{10y^6} \cdot \left( \frac{16x^8 y^5}{4x^7} \right)^{-1}\end{align*}

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.

\begin{align*}\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}}\end{align*}

Review

Simplify the following expressions without negative exponents.

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

Answers for Review Problems

To see the Review answers, open this PDF file and look for section 6.3. 

Vocabulary

Power of a Power Property

The power of a power property states that (a^m)^n = a^{mn}.

Power of a Quotient Property

The power of a quotient property states that \left( \frac{a}{b} \right)^m = \frac{a^m}{b^m}.

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