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Exercises 3.8 Exercises
1. Simplifying Rational Expressions.
Simplify each expression. Assume \(x\) is a non-zero real number.
\(\displaystyle \displaystyle \frac{1}{\frac{1}{x}}\)
\(\displaystyle \displaystyle \frac{x}{\frac{1}{x}}\)
\(\displaystyle \displaystyle \frac{1}{\frac{x}{x}}\)
\(\displaystyle \displaystyle \frac{1}{\frac{x}{1}}\)
\(\displaystyle \displaystyle \frac{x}{\frac{x}{1}}\)
\(\displaystyle \displaystyle \frac{x}{\frac{x}{x}}\)
2. Avoiding Common Mistakes.
Is \(\displaystyle \frac{z}{x+y}\) the same as \(\displaystyle \frac{z}{x}+\frac{z}{y}\text{?}\) Is \(\displaystyle \frac{x+y}{z}\) the same as \(\displaystyle \frac{x}{z}+\frac{y}{z}\text{?}\)
3. Expanding Products.
Multiply and simplify your final answers.
\(\displaystyle (x+y)^3\)
\(\displaystyle x(x-y)^2\)
\(\displaystyle (\sqrt{2}-3)(\sqrt{2}+3)\)
\(\displaystyle (\sqrt{4x^2+1}-2x)(\sqrt{4x^2+1}+2x)\)
Answer .
\(\displaystyle (x+y)^3 = x^3+3 x^2 y+3 x y^2+y^3\)
\(\displaystyle x(x-y)^2 = x^3-2 x^2 y+x y^2\)
\(\displaystyle (\sqrt{2}-3)(\sqrt{2}+3) = -7\)
\(\displaystyle (\sqrt{4x^2+1}-2x)(\sqrt{4x^2+1}+2x) = 1\)
4. Addition and Subtraction with Rational Expressions.
Perform the indicated addition or subtraction and simplify the expression as much as possible. Show your work carefully as a sequence of equal expressions seperated with the equal sign.
\(\displaystyle \displaystyle \frac{2x}{5} - \frac{2}{x+4}\)
\(\displaystyle \displaystyle \frac{2}{5x} + \frac{2}{x+4}\)
\(\displaystyle \displaystyle \frac{x+4}{x^2-4}-\frac{x-1}{x-2}\)
\(\displaystyle \displaystyle \frac{2}{x}+\frac{x-1}{x^2+1}\)
Answer .
\(\displaystyle \displaystyle \frac{2x}{5} - \frac{2}{x+4} = \frac{2 \left(x^2+4 x-5\right)}{5 (x+4)}\)
\(\displaystyle \displaystyle \frac{2}{5x} + \frac{2}{x+4} = \frac{4 (3 x+2)}{5 x (x+4)}\)
\(\displaystyle \displaystyle \frac{x+4}{x^2-4}-\frac{x-1}{x-2} = \frac{6-x^2}{(x-2) (x+2)}\)
\(\displaystyle \displaystyle \frac{2}{x}+\frac{x-1}{x^2+1} = \frac{3 x^2-x+2}{x \left(x^2+1\right)}\)
5. Multiplication and Division with Rational Expressions.
Perform the indicated multplication or division and simplify the expression as much as possible. Show your work carefully as a sequence of equal expressions seperated with the equal sign.
\(\displaystyle \displaystyle \frac{2x}{5} \cdot \frac{2}{x+4}\)
\(\displaystyle \displaystyle \frac{2x}{5} \div \frac{2}{x+4}\)
\(\displaystyle \displaystyle \frac{x+4}{x^2-4}\cdot\frac{x-1}{x-2}\)
\(\displaystyle \displaystyle \frac{x+4}{x^2-4}\div\frac{x-1}{x-2}\)
Answer .
\(\displaystyle \displaystyle \frac{2x}{5} \cdot \frac{2}{x+4} = \frac{4 x}{5 (x+4)} = \frac{4x}{5x+20}\)
\(\displaystyle \displaystyle \frac{2x}{5} \div \frac{2}{x+4} = \frac{1}{5} x (x+4) = \frac{x^2+4x}{5}\)
\(\displaystyle \displaystyle \frac{x+4}{x^2-4}\cdot\frac{x-1}{x-2} = \frac{(x-1) (x+4)}{(x-2)^2 (x+2)}\)
\(\displaystyle \displaystyle \frac{x+4}{x^2-4}\div\frac{x-1}{x-2} = \frac{x+4}{x^2+x-2}\)
6. Simplifying Compound Rational Expressions.
Simplify each expression as much as possible. Show your work carefully as a sequence of equal expressions seperated with the equal sign.
\(\displaystyle \displaystyle \frac{\frac{3}{2x}}{\frac{1}{3x}}\)
\(\displaystyle \displaystyle \frac{\frac{y}{x+y}+1}{x}\)
\(\displaystyle \displaystyle \frac{\frac{1}{x}-\frac{1}{y}}{\frac{1}{x^2}+\frac{1}{y^2}}\)
\(\displaystyle \displaystyle \frac{x+1-\frac{1}{x+1}}{\frac{x}{x+1}}\)
\(\displaystyle \displaystyle \frac{1}{1+(y/x)^2}\)
Answer .
\(\displaystyle \displaystyle \frac{\frac{3}{2x}}{\frac{1}{3x}} = 9/2\)
\(\displaystyle \displaystyle \frac{\frac{y}{x+y}+1}{x} = \frac{x+2y}{x^2+xy}\)
\(\displaystyle \displaystyle \frac{\frac{1}{x}-\frac{1}{y}}{\frac{1}{x^2}+\frac{1}{y^2}} = \frac{xy^2-x^2y}{x^2+y^2}\)
\(\displaystyle \displaystyle \frac{x+1-\frac{1}{x+1}}{\frac{x}{x+1}} = 2+x\)
\(\displaystyle \displaystyle \frac{1}{1+(y/x)^2} = \frac{x^2}{x^2+y^2}\)
7. Simplifying Radical Expressions.
Perform the indicated operation and simplify the expression as much as possible. Show your work carefully as a sequence of equal expressions seperated with the equal sign.
Subtract: \(\displaystyle \frac{\sqrt{4+x^2}}{x}-\frac{x}{\sqrt{4+x^2}}\)
Multiply: \(\displaystyle \frac{\sqrt{4+x^2}-2}{x} \cdot \frac{\sqrt{4+x^2}+2}{\sqrt{4+x^2}+2}\)
Answer .
\(\displaystyle \displaystyle \frac{\sqrt{4+x^2}}{x}-\frac{x}{\sqrt{4+x^2}} = \frac{4}{x \sqrt{x^2+4}} \)
\(\displaystyle \displaystyle \frac{\sqrt{4+x^2}-2}{x} \cdot \frac{\sqrt{4+x^2}+2}{\sqrt{4+x^2}+2} = \frac{x}{\sqrt{4+x^2}+2}\)
8. Avoiding Common Mistakes.
Is \(\sqrt{x^2}\) the same as \(x\text{?}\) Is \(\sqrt{4+x^2}\) the same as \(2+x\text{?}\) Is \(\sqrt{4x^2}\) the same as \(2x\text{?}\)
9. Simplifying Absolute Value Expressions.
Simplify the expression in the case that \(x \gt 2\text{,}\) \(x \lt 2\text{,}\) and \(x=2\text{,}\) if possible. Summarize your conclusions.
\(\displaystyle \displaystyle |2-x|\)
\(\displaystyle \displaystyle \frac{x-2}{|x-2|}\)
\(\displaystyle \displaystyle \frac{|2-x|}{x-2}\)
\(\displaystyle \displaystyle \frac{x^2-4}{|x-2|}\)
Answer .
\(\displaystyle \displaystyle |2-x| = \begin{cases}
2-x, & \text{if $x \lt 2$}\\
x-2, & \text{if $x \geq 2$}\end{cases}\)
\(\displaystyle \displaystyle \frac{x-2}{|x-2|} = \begin{cases}
-1, & \text{if $x \lt 2$} \\
1, & \text{if $x \gt 2$} \\
\text{undef.}, & \text{if $x = 2$} \end{cases}\)
\(\displaystyle \displaystyle \frac{|2-x|}{x-2} = \begin{cases}
-1, & \text{if $x \lt 2$} \\
1, & \text{if $x \gt 2$} \\
\text{undef.}, & \text{if $x = 2$} \end{cases}\)
\(\displaystyle \displaystyle \frac{x^2-4}{|2-x|} = \begin{cases}
x+2, & \text{if $x \lt 2$} \\
-x-2, & \text{if $x \gt 2$} \\
\text{undef.}, & \text{if $x = 2$} \end{cases}\)
10. Simplifying Difference Quotients.
Observe that none of the expressions below are defined when \(h=0\text{;}\) don’t attempt to evaluate at this value. Instead, simplify as much as possible. Show your work carefully as a sequence of equal expressions seperated with the equal sign. Then try evaluating the resulting simplified expression at \(h=0\text{.}\)
\(\displaystyle \displaystyle \frac{(2+h)^2-(4+h)}{h}\)
\(\displaystyle \displaystyle \frac{\frac{1}{h+4}-\frac{1}{4}}{h}\)
\(\displaystyle \displaystyle \frac{\frac{1}{(h+3)^2}-\frac{1}{9}}{h}\)
\(\displaystyle \displaystyle \frac{\sqrt{h+2}-\sqrt{2}}{h} \cdot \frac{\sqrt{h+2}+\sqrt{2}}{\sqrt{h+2}+\sqrt{2}}\)
These kind of expressions are called difference quotients and play an important role in the development of calculus.
Answer .
\(\displaystyle \frac{(2+h)^2-(4+h)}{h} = 3+h\) which evaluates to \(3\) when \(h = 0\text{.}\)
\(\displaystyle \frac{\frac{1}{h+4}-\frac{1}{4}}{h} = \frac{-1}{4(4+h)}\) which evaluates to \(-1/16\) when \(h = 0\text{.}\)
\(\displaystyle \frac{\frac{1}{(h+3)^2}-\frac{1}{9}}{h} = -\frac{h+6}{9 (h+3)^2}\) which evaluates to \(-2/27\) when \(h = 0\text{.}\)
\(\displaystyle \frac{\sqrt{h+2}-\sqrt{2}}{h}\cdot \frac{\sqrt{h+2}+\sqrt{2}}{\sqrt{h+2}+\sqrt{2}} = \frac{1}{\sqrt{h+2}+\sqrt{2}}\) which evaluates to \(\frac{1}{2\sqrt{2}}\) when \(h=0\text{.}\)
Working with Exponents.
11. Rewriting Powers.
Write each term as a power of \(x\) using negative or rational exponents as necessary.
\(\displaystyle \displaystyle 3x^4-\frac{1}{x^2} + \frac{2}{x}+1\)
\(\displaystyle \displaystyle 4\sqrt{x}-5\sqrt[3]{x}+6\sqrt[5]{x^2}\)
\(\displaystyle \displaystyle \frac{1}{x^5} + \frac{1}{\sqrt{x^5}} - \frac{1}{\sqrt[5]{x}}\)
\(\displaystyle \displaystyle \frac{2}{x^3}-\frac{x^3}{2} + \frac{1}{2x^3}\)
Answer .
\(\displaystyle \displaystyle 3x^4-\frac{1}{x^2} + \frac{2}{x}+1 = 3x^4-x^{-2}+2x^{-1}+1\)
\(\displaystyle \displaystyle 4\sqrt{x}-5\sqrt[3]{x}+6\sqrt[5]{x^2} = 4x^{1/2}-5 x^{1/3} + 6 x^{2/5}\)
\(\displaystyle \displaystyle \frac{1}{x^5} + \frac{1}{\sqrt{x^5}} - \frac{1}{\sqrt[5]{x}} = x^{-5}+x^{-5/2}-x^{-1/5}\)
\(\displaystyle \displaystyle \frac{2}{x^3}-\frac{x^3}{2} + \frac{1}{2x^3} = 2x^{-3}-\frac{1}{2} x^3 +\frac{1}{2} x^{-3}\)
12. Rewriting Powers.
Rewrite each term without any negative or rational exponents.
\(\displaystyle \displaystyle -5x^{1/3}-4x^{-3}+\frac{1}{x^{-3}}\)
\(\displaystyle \displaystyle 4x^{-5}+\frac{x^{-4}}{5}-\frac{4}{x^{-5}}\)
\(\displaystyle \displaystyle 4x^{-3/2}+5x^{-2/3}-6x^{4/2}\)
Answer .
\(\displaystyle \displaystyle -5x^{1/3}-4x^{-3}+\frac{1}{x^{-3}} = -5 \sqrt[3]{x}-\frac{4}{x^3} + x^3\)
\(\displaystyle \displaystyle 4x^{-5}+\frac{x^{-4}}{5}-\frac{4}{x^{-5}} = \frac{4}{x^{5}}+\frac{1}{5x^4} - 4x^{5}\)
\(\displaystyle \displaystyle 4x^{-3/2}+5x^{-2/3}-6x^{4/2} = \frac{4}{\sqrt{x^3}} + \frac{5}{\sqrt[3]{x^2}} - 6 x^2\)
13. Laws of Exponents.
Simplify each expression, including rewriting any negative or rational exponents. You may assume all variables are positive real numbers.
\(\displaystyle \displaystyle \left(x^4 y^3\right)^{2}\)
\(\displaystyle \displaystyle \left(x^4 y^3\right)^{-2}\)
\(\displaystyle \displaystyle \left(x^{-4} y^{3/2}\right)^{-2}\)
\(\displaystyle \displaystyle \left(x^4 y^3\right)^{2} \left(x^{-2} y^{1/2}\right)^4\)
\(\displaystyle \displaystyle \left(\frac{2x^2 y}{x^4 y^0}\right)^{-2}\)
\(\displaystyle \displaystyle \left(\frac{-2x^2 y^{3/2}}{x^4 y^1}\right)^{2}\)
Answer .
\(\displaystyle \displaystyle \left(x^4 y^3\right)^{2} = x^8 y^6\)
\(\displaystyle \displaystyle \left(x^4 y^3\right)^{-2} = \frac{1}{x^8 y^6}\)
\(\displaystyle \displaystyle \left(x^{-4} y^{3/2}\right)^{-2} = \frac{x^8}{y^3}\)
\(\displaystyle \displaystyle \left(x^4 y^3\right)^{2} \left(x^{-2} y^{1/2}\right)^4 = y^8\)
\(\displaystyle \displaystyle \left(\frac{2x^2 y}{x^4 y^0}\right)^{-2} = \frac{x^4}{4y^2}\)
\(\displaystyle \displaystyle \left(\frac{-2x^2 y^{3/2}}{x^4 y^1}\right)^{2} = \frac{4y}{x^4}\)
14. Laws of Exponents.
Simplify each expression. You may assume all variables are positive real numbers.
\(\displaystyle \displaystyle \sqrt{64 x^6 y^5}\)
\(\displaystyle \displaystyle \sqrt[3]{64 x^6 y^5}\)
\(\displaystyle \displaystyle \sqrt{64 x^6 y^5} \sqrt{9 x^3 y^4}\)
\(\displaystyle \displaystyle \frac{\sqrt[3]{27 x^6 y^5}}{\sqrt[3]{-8 x^3 y^4}}\)
Answer .
\(\displaystyle \displaystyle \sqrt{64 x^6 y^5} = 8 x^3 y^2 \sqrt{y}\)
\(\displaystyle \displaystyle \sqrt[3]{64 x^6 y^5} = 4 x^2 y \sqrt[3]{y^2}\)
\(\displaystyle \displaystyle \sqrt{64 x^6 y^5} \sqrt{9 x^3 y^4} = 24 x^4 y^4 \sqrt{xy}\)
\(\displaystyle \displaystyle \frac{\sqrt[3]{27 x^6 y^5}}{\sqrt[3]{-8 x^3 y^4}} = -\frac{3x \sqrt[3]{y}}{2}\)
15. Laws of Exponents.
Simplify each expression, including rewriting any negative or rational exponents. You may assume all variables are positive real numbers.
\(\displaystyle \displaystyle x^{-2}\left(x^2-x^{-2}\right)\)
\(\displaystyle \displaystyle \left(x^2+x^{-2}\right)\left(x^2-x^{-2}\right)\)
\(\displaystyle \displaystyle \left(x^2-x^{-2}\right)^2\)
\(\displaystyle \displaystyle \frac{x^2-x^{-2}}{x^2}\)
\(\displaystyle \displaystyle x^{1/2}\left(x^{-1/2}-x^{1/2}\right)\)
\(\displaystyle \displaystyle \frac{x^{1/2}-x^{-1/2}}{x^{1/2}}\)
Answer .
\(\displaystyle \displaystyle x^{-2}\left(x^2-x^{-2}\right) = 1-\frac{1}{x^4} = \frac{x^4-1}{x^4}\)
\(\displaystyle \displaystyle \left(x^2+x^{-2}\right)\left(x^2-x^{-2}\right) = x^4-\frac{1}{x^4} = \frac{x^8-1}{x^4}\)
\(\displaystyle \displaystyle \left(x^2-x^{-2}\right)^2 = x^4+\frac{1}{x^4}-2 = \frac{x^8-2 x^4+1}{x^4}\)
\(\displaystyle \displaystyle \frac{x^2-x^{-2}}{x^2} = 1-\frac{1}{x^4} = \frac{x^4-1}{x^4}\)
\(\displaystyle \displaystyle x^{1/2}\left(x^{-1/2}-x^{1/2}\right) = 1-x\)
\(\displaystyle \displaystyle \frac{x^{1/2}-x^{-1/2}}{x^{1/2}} = 1-\frac{1}{x} = \frac{x-1}{x}\)
