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Chapter 1: Problem 86

Simplify. $$ \frac{\left(u^{3} v w^{2}\right)^{2}}{9\left(u^{2} w\right)^{2}} $$

Short Answer

Expert verified
The simplified expression is \(\frac{u^2 v^2 w^2}{9}\).

Step by step solution

01

Expand the Numerator

First, expand the numerator \((u^3 v w^2)^2\). To do this, apply the power rule \((a^m b^n)^p = a^{mp} b^{np}\): \((u^3 v w^2)^2 = u^{6} v^2 w^4\).
02

Expand the Denominator

Now, expand the denominator \(9(u^2 w)^2\). Apply the power rule and multiplication to get: \(9(u^2 w)^2 = 9 imes u^4 imes w^2 = 9u^4w^2\).
03

Simplify the Expression

Combine the expanded numerator with the denominator: \[\frac{u^6 v^2 w^4}{9 u^4 w^2}\]. Now simplify by dividing each term in the numerator by the corresponding term in the denominator: \[u^{6-4} v^2 w^{4-2} = \frac{u^2 v^2 w^2}{9}\].
04

Final Simplification

Ensure that the expression is fully simplified. After confirming, the simplified expression is \[\frac{u^2 v^2 w^2}{9}\].

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Key Concepts

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

Exponent Rules
Exponent rules are essential tools in algebra, enabling us to manage powers efficiently. In the exercise, we use a key rule:
  • Power of a Power: \( (a^m)^n = a^{m \cdot n} \)
This rule helps us simplify expressions by distributing the external exponent across each of the factors inside the parentheses. Let's look at how these rules apply when simplifying algebraic expressions:
First, expand the expression \( (u^3 v w^2)^2 \). Apply the \'power of a power\' rule to each part inside:
  • \( u^3 \rightarrow (u^3)^2 = u^{3\cdot2} = u^6 \)
  • \( v \rightarrow (v)^2 = v^2 \)
  • \( w^2 \rightarrow (w^2)^2 = w^{2\cdot2} = w^4 \)
The expanded form for the numerator becomes \( u^6 v^2 w^4 \). Recognizing and correctly applying exponent rules leads to accurate and simplified expressions, every step of the way.
Numerator and Denominator
In any fraction, the numerator sits on top and the denominator on the bottom. Understanding this position is crucial for proper algebraic manipulation. Let's explore how we work with the numerator and denominator in this exercise.
Starting with the numerator \( (u^3 v w^2)^2 \), we expanded it using exponent rules to \( u^6 v^2 w^4 \). This is placed over the denominator \( 9(u^2 w)^2 \).
Now, to simplify the denominator, we expand and simplify:
  • \( 9 \rightarrow \text{stays the same as it\'s a constant} \)
  • \( (u^2)^2 = u^{2\cdot2} = u^4 \)
  • \( (w)^2 = w^2 \)
This expansion leads to the denominator becoming \( 9u^4w^2 \). With both the numerator and the denominator clearly expanded, we proceed to the next step of simplification.
Algebraic Manipulation
Algebraic manipulation is the process of reshaping algebraic expressions into more manageable forms. In this exercise, one key action was reducing the fraction.
We began with the fraction \[\frac{u^6 v^2 w^4}{9 u^4 w^2}\]. The goal is to cancel common factors from the numerator and denominator, making the expression simpler. Here's how:
  • Divide \( u^6 \) by \( u^4 \) to get \( u^{6-4} = u^2 \)
  • \( v^2 \) stays unchanged since \( v \) is not in the denominator
  • Divide \( w^4 \) by \( w^2 \) to get \( w^{4-2} = w^2 \)
Thus, after careful reduction of like terms, the expression simplifies to \[\frac{u^2 v^2 w^2}{9}\].
Algebraic manipulation, especially factor cancellation, transforms expressions into their simplest forms, making them easier to interpret and use, for future calculations or solving equations.

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Most popular questions from this chapter

GENERAL: Earthquakes The sizes of major earthquakes are measured on the Moment Magnitude Scale, or MMS, although the media often still refer to the outdated Richter scale. The MMS measures the total energy relensed by an earthquake, in units denoted \(M_{W}\) (W for the work accomplished). An increase of \(1 M_{W}\) means the energy increased by a factor of \(32,\) so an increase from \(A\) to \(B\) means the energy increased by a factor of \(32^{B-A}\). Use this formula to find the increase in energy between the following earthquakes: The 2001 earthquake in India that measured \(7.7 M_{W}\) and the 2011 earthquake in Japan that measured \(9.0 M_{W}\). (The earthquake in Japan generated a 28 -foot tsunami wave that traveled six miles inland, killing 24,000 and causing an estimated \(\$ 300\) billion in damage, making it the most expensive natural disaster ever recorded.)

ENVIRONMENTAL SCIENCE: Biodiversity It is well known that larger land areas can support larger numbers of species. According to one study, multiplying the land area by a factor of \(x\) multiplies the number of species by a factor of \(x^{0.239}\). Use a graphing calculator to graph \(y=x^{0.239}\). Use the window [0,100] by [0,4]. Find the multiple \(x\) for the land area that leads to triple the number of species. That is, find the value of \(x\) such that \(x^{0.239}=3\). [Hint: Either use TRACE or find where \(y_{1}=x^{0.239}\) INTERSECTs \(\left.y_{2}=3 .\right]\)

Can the graph of a function have more than one \(x\) -intercept? Can it have more than one \(y\) -intercept?

Simplify. $$ \frac{\left(2 u^{2} v w^{3}\right)^{2}}{4\left(u w^{2}\right)^{2}} $$

An electronics company's research budget is \(R(p)=3 p^{0.25},\) where \(p\) is the company's profit, and the profit is predicted to be \(p(t)=55+4 t,\) where \(t\) is the number of years from now. (Both \(R\) and \(p\) are in millions of dollars.) Express the research expenditure \(R\) as a function of \(t,\) and evaluate the function at \(t=5\).
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