/*! 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 29 Evaluate each expression without... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 1: Problem 29

Evaluate each expression without using a calculator. $$ \left(\frac{27}{125}\right)^{2 / 3} $$

Short Answer

Expert verified
The expression evaluates to \(\frac{9}{25}\).

Step by step solution

01

Understanding the Expression

We are given \[\left(\frac{27}{125}\right)^{2/3}\] This expression involves a fractional exponent, indicating both a power and a root. The base \(\frac{27}{125}\) is raised to the power of \(\frac{2}{3}\).
02

Rewriting the Fraction and Exponent

The expression can be rewritten as:\[\left(\frac{27}{125}\right)^{2/3} = \left({27}^{1/3} \cdot {125}^{-1/3}\right)^2\]This structure shows that we will first take the cube roots of the numerator and denominator separately, then square the result.
03

Taking the Cube Roots

Calculate the cube roots:* The cube root of 27 is 3 because 3 cubed is 27 (\(3^3 = 27\)).* The cube root of 125 is 5 because 5 cubed is 125 (\(5^3 = 125\)).So, we have:\[\left(\frac{27}{125}\right)^{1/3} = \frac{3}{5}\]
04

Squaring the Result

Now, square the result of the cube roots:\[\left(\frac{3}{5}\right)^2 = \frac{3^2}{5^2} = \frac{9}{25}\]Thus, the squared result is \(\frac{9}{25}\).
05

Final Answer

The value of the given expression is \(\frac{9}{25}\).

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.

Cube Roots
Cube roots are special because they tell us what number, when multiplied by itself three times, equals a given number. Unlike square roots, which are concerned with a pair of multipliers, cube roots deal with a trio. Understanding this concept is crucial when working with expressions like \( \left(\frac{27}{125}\right)^{2/3} \).

Let's break it down:
  • For 27, the cube root is 3 because \( 3 \times 3 \times 3 = 27 \).
  • For 125, the cube root is 5 since \( 5 \times 5 \times 5 = 125 \).
When you see a fractional exponent like \( \theta/3 \), the denominator tells you to take the cube root. This means each part of the fraction (numerator and denominator separately) needs to be analyzed in terms of cube roots before any other calculations.
Numerator and Denominator
The numerator and the denominator are the two key parts of a fraction. The numerator is the number on top, while the denominator is the number on the bottom. When given a fraction like \( \frac{27}{125} \), 27 is the numerator, and 125 is the denominator.

In expressions with fractional exponents, you work on the numerator and the denominator separately to simplify the operation. In our case:
  • The numerator 27 is simplified by taking its cube root to get 3.
  • The denominator 125 is simplified by taking its cube root to get 5.
This separation allows us to manage complex operations with ease, because once we have dealt with the roots, we can then move to more familiar calculations such as squaring.
Squaring Fractions
Squaring fractions can be simple if you remember that both the numerator and the denominator must be squared separately. When given a fractional outcome from cube roots, like the \( \frac{3}{5} \) from \( \left(\frac{27}{125}\right)^{1/3} \), the process of squaring follows these straightforward steps:

To square \( \frac{3}{5} \):
  • Square the numerator: \( 3^2 = 9 \).
  • Square the denominator: \( 5^2 = 25 \).
This gives you \( \frac{9}{25} \), the final answer of the original expression. Squaring a fraction separately ensures accuracy and keeps the calculations ordered. Always remember to apply the square to both parts of the fraction to avoid mistakes.

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

If a linear function is such that \(f(4)=7\) and \(f(6)=11\), then \(f(5)=? \quad[\) Hint \(:\) No work necessary.]

For the quadratic function \(f(x)=a x^{2}+b x+c\), what condition on one of the coefficients will guarantee that the function has a highest value? A lowest value?

GENERAL: Impact Time of a Projectile If an object is thrown upward so that its height (in feet) above the ground \(t\) seconds after it is thrown is given by the function \(h(t)\) below, find when the object hits the ground. That is, find the positive value of \(t\) such that \(h(t)=0\). Give the answer correct to two decimal places. [Hint: Enter the function in terms of \(x\) rather than t. Use the ZERO operation, or TRACE and ZOOM IN, or similar operations.] $$ h(t)=-16 t^{2}+40 t+4 $$

The Apocryphal Manufacturing Company makes blivets out of widgets. If a linear function \(f(x)=m x+b\) gives the number of widgets that can be made from \(x\) blivets, what are the units of the slope \(m\) (widgets per blivet or blivets per widget)?

i. Show that the general linear equation \(a x+b y=c\) with \(b \neq 0\) can be written as \(y=-\frac{a}{b} x+\frac{c}{b}\) which is the equation of a line in slope-intercept form. ii. Show that the general linear equation \(a x+b y=c\) with \(b=0\) but \(a \neq 0\) can be written as \(x=\frac{c}{a}\), which is the equation of a vertical line. [Note: Since these steps are reversible, parts (i) and (ii) together show that the general linear equation \(a x+b y=c\) (for \(a\) and \(b\) not both zero) includes vertical and nonvertical lines.]
See all solutions

Recommended explanations on Math Textbooks

Probability and Statistics

Read Explanation

Applied Mathematics

Read Explanation

Discrete Mathematics

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Pure Maths

Read Explanation

Statistics

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.