/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 60 Exercise 59 with \(f(x)=3 \sqrt[... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 3: Problem 60

Exercise 59 with \(f(x)=3 \sqrt[3]{x^{2}}\).

Short Answer

Expert verified
The simplified form of the function \(f(x) = 3 \sqrt[3]{x^{2}}\) is \(f(x) = 3x^{2/3}\).

Step by step solution

01

Identify the Power and Root within the Function

The function given is \(f(x) = 3 \sqrt[3]{x^{2}}\). Here, we identify that the function involves a cubed root, represented by \( \sqrt[3]{ }\), and a power of 2, represented by \({x^{2}}\).
02

Simplify the Function Using Exponent Rules

The cube root of a quantity squared can be simplified by using exponent rules. Specifically, \(\sqrt[3]{a^{2}}\) can be written as \(a^{2/3}\). Thus, the function \(f(x)\) can be rewritten as \(f(x) = 3x^{2/3}\).
03

Conclusion

After simplifying the root and the power within the function, we find that the function \(f(x) = 3 \sqrt[3]{x^{2}}\) simplifies to \(f(x)=3x^{2/3}\).

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Exponent Rules
When dealing with exponent rules, it's important to remember that they help us simplify expressions that involve powers of numbers or variables. These rules include product of powers, power of a power, and power of a product, among others. In the context of this exercise, we are specifically leveraging the fractional exponent rule.

This rule allows roots and powers to be expressed as exponents. For instance, when you encounter a cube root, it can be expressed as a fractional exponent. A general form of expressing any root as a fractional exponent is given as:
  • The nth root of a number or expression, a , raised to the m power is written as a^{m/n}.
  • In this particular exercise, since we have a cube root and a square power, the cube root of x^2 can be expressed more simply as x^{2/3}.
By applying this property, you simplify the process, making it more straightforward to work with the expression.
Cube Root
The cube root of a number or expression involves finding a value that, when raised to the power of three, equals the original number or expression. In terms of exponents, taking the cube root is equivalent to raising a number to the power of 1/3. Therefore, \(sqrt[3]{x^2} \) is transforming x^2 using the exponent form.
  • Cube roots are particularly useful when simplifying powers because they allow us to express complex roots using fractional exponents.
  • In our exercise, \({(sqrt[3]{x^2} = x^{2/3})}\) shows how the cube root transitions into a relatively simpler fractional exponent form.
This transformation is crucial for operations that involve multiplying or dividing powers since it converts roots to exponents, which are easier to handle mathematically.
Power of a Function
A function like \(f(x) = 3 \sqrt[3]{x^2}\)involves applying both powers and roots on x. When you have a function like f(x),it means that an operation is being performed on x to produce a result. Here, the power of x is adjusted to simplify the function.
  • Consider that x^2 is initially a power applied to x within the cubic root.
  • Once converted using exponent rules to x^{2/3}, we essentially have a function with a single power, making it more straightforward for further calculations or evaluations.
Understanding the power of a function and how to manipulate it is key for solving more complex calculus or algebraic problems, as it directly impacts how easily you can derive or integrate such functions. By simplifying the power, calculations are made easier and more intuitive.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Let \(u, v, w\) be differentiable functions of \(x .\) Express the derivative of tile product \(uvw\)in in terms of the functions \(u, v, w\) and their derivatives.

Let \(f(x)=\sin x-\cos 2 x\) for \(0 \leq x \leq 2 \pi\) (a) Use a graphing utility to estimate the points on the graph where the tangent is horizontal. (b) Use a CAS to estimate the numbers \(x\) at which \(f^{\prime}(x)=0\) (c) Reconcile your results in (a) and (b).

Determine the values of \(x\) for which (a) \(f^{\prime}(x)=0 ;(b) f^{\prime}(x)>0 ;(c) f^{\prime}(x)<0\). $$f(x)=\left(1-x^{2}\right)^{2}$$

A triangle has sides of length \(a\) and \(b\), and the angle between them is \(x\) radians. Given that \(a\) and \(b\) are \(\mathrm{kcpt}\) constant, find the rate of change of the third side \(c\) with respect to \(x .\) HINT: Use the law of cosines.

Use a CAS to find the slope of the line tangent to the curve at the given point. Use a graphing utility to draw the curve and the tangent line together in one figure. \(\sqrt[1]{x^{2}}+\sqrt[3]{y^{2}}=4 ; \quad P(1,3 \sqrt{3})\).
See all solutions

Recommended explanations on Math Textbooks

Probability and Statistics

Read Explanation

Geometry

Read Explanation

Logic and Functions

Read Explanation

Applied Mathematics

Read Explanation

Calculus

Read Explanation

Mechanics Maths

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.