1.4.1: Exercises 1.4

Terms and Concepts

Exercise $\PageIndex{1}$

Do exponent rules apply to root functions? Explain.

Answer

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

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

Exponential form

Exercise $\PageIndex{4}$

Is $\displaystyle \frac{3x}{\sqrt[3]{(2x+3)^5}}$ in radical or exponential form?

Answer

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

$\displaystyle \frac{1}{\sqrt[3]{8x_1-5x_2+11}}$

Exercise $\PageIndex{6}$

$\displaystyle (-2x+y)^{-1/5}$

Answer

$\displaystyle \frac{1}{\sqrt[5]{-2x+y}}$

Exercise $\PageIndex{7}$

$\displaystyle (5x-2)^{1/4}$

Answer

$\displaystyle \sqrt[4]{5x-2}$

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

$\displaystyle x + 2 + \frac{1}{x}$

Exercise $\PageIndex{9}$

$\displaystyle (\sqrt{x})^2 + \Bigg(\frac{1}{\sqrt{x}} \Bigg)^2$

Answer

$\displaystyle x + \frac{1}{x}$

Exercise $\PageIndex{10}$

$\displaystyle \Bigg(\sqrt[3]{x} +1 \Bigg)^3$

Answer

$\displaystyle x + 3x^{2/3} + 3x^{1/3} + 1$

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

$\displaystyle \frac{25y^5}{x}, y \neq 0$

Exercise $\PageIndex{12}$

$\displaystyle \Bigg( \frac{-2x^{2/3}y^2}{x^{-2}y^{1/2}}\Bigg)^6$

Answer

$\displaystyle 64x^{16}y^9; x,y \neq 0$

Exercise $\PageIndex{13}$

$\displaystyle \Bigg( \frac{-3s^{2/3}t^2}{4s^3t^{5/3}}\Bigg)^3$

Answer

$\displaystyle \frac{-27t}{64s^7}, t\neq 0$

Exercise $\PageIndex{14}$

$\displaystyle -3(x^2+4x+4)^{-4}(2x+4)$

Answer

$\displaystyle \frac{-6}{(x+2)^7}$

Exercise $\PageIndex{15}$

$\displaystyle \frac{1}{3}(x^4)^{-2/3}(4x^3)$

Answer

$\displaystyle \frac{4x^{1/3}}{3}$

Exercise $\PageIndex{16}$

$\displaystyle \frac{(e^{x+3})^2}{e^{-x}}$

Answer

$\displaystyle e^{3x+6}$

Exercise $\PageIndex{17}$

$\displaystyle \frac{e^{x^2-1}}{e^{x+1}}$

Answer

$\displaystyle e^{x^2-x-2}$

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

$\displaystyle (4x-1)(3x+2)^{-2/3}$

Exercise $\PageIndex{19}$

$\displaystyle \sqrt[3]{\Bigg(\frac{e^{4\theta-6}y^2}{e^{\theta}y^{-4}}\Bigg)}$

Answer

$\displaystyle e^{\theta-2}y^2, y \neq 0$

Exercise $\PageIndex{20}$

$\displaystyle \sqrt[4]{\Bigg(\frac{x^2y^5}{y^{-3}}\Bigg)^2}$

Answer

$\displaystyle xy^4, y\neq 0$