Exponents and Radicals

Section 2.5 Exponents and Radicals

Goals:

E: Be able to apply exponents and radicals to simplify expressions.
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F: Be able to identify functions and operations that could be combined to produce a given function.
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F: Be able to determine a composition of functions given in any form (graph, table or equation).
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F: Be able to perform arithmetic operations on functions given in any form.
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Below is a list of exponent rules where \(a\) and \(b\) are positive real numbers, and \(m\) and \(n\) are real numbers.

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\(\displaystyle a^n \cdot a^m=a^{m+n}\)
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\(\displaystyle \frac{a^m}{a^n}=a^{m-n}\)
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\(\displaystyle \left(a^m \right)^n=a^{mn}\)
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\(\displaystyle \left(ab \right)^m=a^m \cdot b^m\)
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\(\displaystyle \left(\frac{a}{b} \right)^m=\frac{a^m}{b^m}\)
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\(\displaystyle a^{-n}=\frac{1}{a^n}\)
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\(\displaystyle a^0=1\)
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Problem 2.5.1.

Use the exponent rules from above to simplify the following expressions. Write your answer without negative exponents. State which rule you used in each step of your solution.

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\(\displaystyle \frac{x^2y^3}{xy^5}\)
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\(\displaystyle \frac{1}{t^{-5}}\)
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\(\displaystyle 5x^2y(2x^4y^-3)^2\)
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\(\displaystyle \left(\frac{-3a^2bc^{-2}}{-a^3b^2c^2}\right)^3\)
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\(\displaystyle \displaystyle \left(\frac{8x^2y^{-2}}{3z^{-4}}\right) \left(\frac{-9z^{-2}}{4x^3y^{-1}} \right)\)
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\(\displaystyle (3x^ay^bz^c)(-y^fz^g)\)
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We can expand our use of exponents to include radicals by representing \(\sqrt[n]{a}\) as \(a^{\frac{1}{n}}\) and \(\sqrt[n]{a^m}\) as \(a^{\frac{m}{n}}\text{,}\) where \(n>0\) and \(m\) is a real number. We can then apply the rules listed above.

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Example 2.5.2.

Simplify. Assume all variables represent positive real numbers.

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\(\displaystyle \sqrt[3]{27x^6y^9}\)
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\(\displaystyle (x^2 \sqrt{4x})^2\)
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\(\displaystyle c(b^3+1)^{\frac{1}{2}}\)
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Problem 2.5.3.

Simplify. Assume all variables represent positive real numbers.

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\(\displaystyle \sqrt[4]{x^{12}y^8z^4}\)
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\(\displaystyle (\sqrt[3]{m^6n^4})(\sqrt[3]{m^4n^3})\)
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\(\displaystyle \frac{\sqrt{r^8t^4}}{\sqrt{r^4t^2}}\)
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Problem 2.5.4.

Perform the indicated operation and simplify.

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\(\displaystyle \frac{\sqrt{x}y-x^2y^2+x^3}{x}\)
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\(\displaystyle \frac{p^3-3p+\sqrt{p}}{\sqrt{p}}\)
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\(\displaystyle \sqrt{x}(x^3-2x-1)\)
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Example 2.5.5.

Let \(a(x)=10^x\text{,}\) \(b(x)=10^{2x^2}\text{,}\) \(c(x)=x^3\text{.}\) Write a formula for each function below, using the exponent rules to simplify the formula and write without negative exponents.

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\(\displaystyle k(x)=(c \circ b)(x)\)
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\(\displaystyle f(x)=(c(x) \cdot a(x))^4\)
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\(\displaystyle g(x)=(b(x))^{-2}\)
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\(\displaystyle h(x)=2 \left( \frac{a(x)}{b(x)} \right)\)
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\(\displaystyle j(x)=a(x)\cdot b(x)\)
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Problem 2.5.6.

Let \(w(x)=3e^x\text{,}\) \(y(x)=e^{-x}\text{,}\) \(z(x)=\frac{1}{x^2}\text{.}\) First, write the given function as a combination of \(w\text{,}\) \(y\text{,}\) and \(z\text{,}\) using any arithmetic operations and the operation of composition, as appropriate. Then, use the exponent rules to simplify the formula if possible. For example,

\begin{equation*} f(x)=3e^{e^{-x}} \end{equation*}

is the composition of \(w(x)\) and \(y(x)\text{,}\) so

\begin{equation*} f(x)=(w \circ y)(x)=w(y(x)) \end{equation*}

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\(\displaystyle m(x)=e^{-x} \cdot 3e^x\)
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\(\displaystyle h(x)=\frac{3e^x}{e^{-x}}\)
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\(\displaystyle g(x)=\frac{1}{(e^{-x})^2}\)
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\(\displaystyle k(x)=\left(\frac{3e^x}{x^2} \right)^2\)
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\(\displaystyle j(x)=(3e^x)^{-4}\)
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