1.4.1: Exercises 1.4
- Page ID
- 62181
Terms and Concepts
Exercise \(\PageIndex{1}\)
Do exponent rules apply to root functions? Explain.
- Answer
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Yes, a root function is just a power function with a fractional exponent.
Exercise \(\PageIndex{2}\)
Explain why a negative exponents moves the term to the denominator and gives it a positive exponent.
- Answer
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A positive exponent means we are multiplying that term repeatedly, a negative exponent means we are dividing by that term repeatedly.
Exercise \(\PageIndex{3}\)
Is \(3x(2x+3)^{-5/3}\) in radical or exponential form?
- Answer
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Exponential form
Exercise \(\PageIndex{4}\)
Is \(\displaystyle \frac{3x}{\sqrt[3]{(2x+3)^5}}\) in radical or exponential form?
- Answer
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Radical form
Problems
In exercises \(\PageIndex{5}\) - \(\PageIndex{7}\), write the given term without using exponents.
Exercise \(\PageIndex{5}\)
\(\displaystyle (8x_1-5x_2+11)^{-1/3}\)
- Answer
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\(\displaystyle \frac{1}{\sqrt[3]{8x_1-5x_2+11}}\)
Exercise \(\PageIndex{6}\)
\(\displaystyle (-2x+y)^{-1/5}\)
- Answer
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\(\displaystyle \frac{1}{\sqrt[5]{-2x+y}}\)
Exercise \(\PageIndex{7}\)
\(\displaystyle (5x-2)^{1/4}\)
- Answer
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\(\displaystyle \sqrt[4]{5x-2}\)
In exercises \(\PageIndex{8}\) - \(\PageIndex{10}\), simplify and write the given term without using radicals.
Exercise \(\PageIndex{8}\)
\(\displaystyle \Bigg( \sqrt{x} + \frac{1}{\sqrt{x}} \Bigg)^2\)
- Answer
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\(\displaystyle x + 2 + \frac{1}{x}\)
Exercise \(\PageIndex{9}\)
\(\displaystyle (\sqrt{x})^2 + \Bigg(\frac{1}{\sqrt{x}} \Bigg)^2\)
- Answer
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\(\displaystyle x + \frac{1}{x}\)
Exercise \(\PageIndex{10}\)
\(\displaystyle \Bigg(\sqrt[3]{x} +1 \Bigg)^3\)
- Answer
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\(\displaystyle x + 3x^{2/3} + 3x^{1/3} + 1\)
In exercises \(\PageIndex{11}\) - \(\PageIndex{17}\), simplify the given term and write your answer without negative exponents.
Exercise \(\PageIndex{11}\)
\(\displaystyle \Bigg( \frac{-5x^{-1/4}y^3}{x^{1/4}y^{1/2}}\Bigg)^2\)
- Answer
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\(\displaystyle \frac{25y^5}{x}, y \neq 0\)
Exercise \(\PageIndex{12}\)
\(\displaystyle \Bigg( \frac{-2x^{2/3}y^2}{x^{-2}y^{1/2}}\Bigg)^6\)
- Answer
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\(\displaystyle 64x^{16}y^9; x,y \neq 0\)
Exercise \(\PageIndex{13}\)
\(\displaystyle \Bigg( \frac{-3s^{2/3}t^2}{4s^3t^{5/3}}\Bigg)^3\)
- Answer
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\(\displaystyle \frac{-27t}{64s^7}, t\neq 0\)
Exercise \(\PageIndex{14}\)
\(\displaystyle -3(x^2+4x+4)^{-4}(2x+4)\)
- Answer
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\(\displaystyle \frac{-6}{(x+2)^7}\)
Exercise \(\PageIndex{15}\)
\(\displaystyle \frac{1}{3}(x^4)^{-2/3}(4x^3)\)
- Answer
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\(\displaystyle \frac{4x^{1/3}}{3}\)
Exercise \(\PageIndex{16}\)
\(\displaystyle \frac{(e^{x+3})^2}{e^{-x}}\)
- Answer
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\(\displaystyle e^{3x+6}\)
Exercise \(\PageIndex{17}\)
\(\displaystyle \frac{e^{x^2-1}}{e^{x+1}}\)
- Answer
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\(\displaystyle e^{x^2-x-2}\)
In exercises \(\PageIndex{18}\) - \(\PageIndex{20}\), simplify and write the given term in exponential form.
Exercise \(\PageIndex{18}\)
\(\displaystyle \frac{4x-1}{\sqrt[3]{(3x+2)^2}}\)
- Answer
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\(\displaystyle (4x-1)(3x+2)^{-2/3}\)
Exercise \(\PageIndex{19}\)
\(\displaystyle \sqrt[3]{\Bigg(\frac{e^{4\theta-6}y^2}{e^{\theta}y^{-4}}\Bigg)}\)
- Answer
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\(\displaystyle e^{\theta-2}y^2, y \neq 0\)
Exercise \(\PageIndex{20}\)
\(\displaystyle \sqrt[4]{\Bigg(\frac{x^2y^5}{y^{-3}}\Bigg)^2}\)
- Answer
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\(\displaystyle xy^4, y\neq 0\)