Expand menu menu_open Minimize Start chapters Home History history History expand_more
{{ item.displayTitle }}
navigate_next
No history yet!
Progress & Statistics equalizer Progress expand_more
Student
navigate_next
Teacher
navigate_next
{{ filterOption.label }}
{{ item.displayTitle }}
{{ item.subject.displayTitle }}
arrow_forward
No results
{{ searchError }}
search
menu_open
{{ courseTrack.displayTitle }}
{{ statistics.percent }}% Sign in to view progress
{{ printedBook.courseTrack.name }} {{ printedBook.name }}
search Use offline Tools apps
Login account_circle menu_open

Manipulating Radical Functions

Choose Course
Algebra 2
Radical Functions
close

Manipulating Radical Functions 1.4 - Solution

arrow_back Return to Manipulating Radical Functions

Given that f(x)=2x3f(x)=2x^3 and g(x)=x3,g(x)=\sqrt[3]{x}, we will define (fg)(x)(fg)(x) and (fg)(x).\left(\frac{f}{g}\right)(x). Let's begin with the product of the functions.

Multiplying Two Functions

Let's recall the definition we will use. (fg)(x)=f(x)g(x)\begin{gathered} (fg)(x)=f(x)g(x) \end{gathered} From here, we will substitute the given function rules and simplify.
(fg)(x)=f(x)g(x)(fg)(x)=f(x)g(x)
SubstituteIIf(x)=2x3f(x)={\color{#0000FF}{2x^3}}, g(x)=x3g(x)={\color{#009600}{\sqrt[3]{x}}}
(fg)(x)=2x3x3(fg)(x)={\color{#0000FF}{2x^3}}{\color{#009600}{\sqrt[3]{x}}}
Simplify right-hand side
RootToPowSLan=a1/n\sqrt[n]{a}=a^{1/n}
(fg)(x)=2x3x1/3(fg)(x)=2x^3x^{1/3}
MultPowam⋅an=am+n a^{m}\cdot{a}^{n}=a^{m+n}
(fg)(x)=2x3+1/3(fg)(x)=2x^{3+1/3}
NumberToFraca=3⋅a3a = \dfrac{3\cdot a}{3}
(fg)(x)=2x9/3+1/3(fg)(x)=2x^{9/3+1/3}
AddFracAdd fractions
(fg)(x)=2x10/3(fg)(x)=2x^{10/3}

Dividing Two Functions

Let's recall the rule for dividing functions. (fg)(x)=f(x)g(x)\begin{gathered} \left(\frac{f}{g}\right)(x)=\frac{f(x)}{g(x)} \end{gathered} We can proceed in the same way as we did for multiplying the functions.
(fg)(x)=f(x)g(x)\left(\dfrac{f}{g}\right)(x)=\dfrac{f(x)}{g(x)}
SubstituteIIf(x)=2x3f(x)={\color{#0000FF}{2x^3}}, g(x)=x3g(x)={\color{#009600}{\sqrt[3]{x}}}
(fg)(x)=2x3x3\left(\dfrac{f}{g}\right)(x)=\dfrac{{\color{#0000FF}{2x^3}}}{{\color{#009600}{\sqrt[3]{x}}}}
Simplify right-hand side
MovePartNumLefta⋅bc=a⋅bc\dfrac{a\cdot b}{c}=a\cdot\dfrac{b}{c}
(fg)(x)=2x3x3\left(\dfrac{f}{g}\right)(x)=2\dfrac{x^3}{\sqrt[3]{x}}
RootToPowSLan=a1/n\sqrt[n]{a}=a^{1/n}
(fg)(x)=2x3x1/3\left(\dfrac{f}{g}\right)(x)=2\dfrac{x^3}{x^{1/3}}
DivPowaman=am−n \dfrac{a^m}{a^n}= a^{m-n}
(fg)(x)=2x3−1/3\left(\dfrac{f}{g}\right)(x)=2x^{3-1/3}
NumberToFraca=3⋅a3a = \dfrac{3\cdot a}{3}
(fg)(x)=2x9/3−1/3\left(\dfrac{f}{g}\right)(x)=2x^{9/3-1/3}
SubFracSubtract fractions
(fg)(x)=2x8/3\left(\dfrac{f}{g}\right)(x)=2x^{8/3}