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Chapter 7: Problem 5

For exercises 1-66, simplify. $$ \frac{90 n^{2} p^{8}}{42 n^{5} p^{6}} $$

Short Answer

Expert verified
\[ \frac{15 p^{2}}{7 n^{3}} \]

Step by step solution

01

Write Down the Problem

Start by writing down the given expression: \[ \frac{90 n^{2} p^{8}}{42 n^{5} p^{6}} \]
02

Simplify the Coefficients

Simplify the coefficients 90 and 42 by dividing both by their greatest common divisor (GCD), which is 6. \[ \frac{90}{42} = \frac{90 \times \frac{1}{6}}{42 \times \frac{1}{6}} = \frac{15}{7} \] So, the expression becomes: \[ \frac{15 n^{2} p^{8}}{7 n^{5} p^{6}} \]
03

Simplify the Variables Individually

Simplify the variable terms by subtracting the exponents in the denominator from the exponents in the numerator for both variables n and p. \[ \text{For } n: n^{2-5} = n^{-3} = \frac{1}{n^{3}} \] \[ \text{For } p: p^{8-6} = p^{2} \] So, the expression becomes: \[ \frac{15 p^{2}}{7 n^{3}} \]

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Key Concepts

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

Greatest Common Divisor
The greatest common divisor (GCD) is a crucial concept when simplifying fractions, including algebraic ones. It refers to the largest number that can divide two (or more) numbers without leaving a remainder. In the context of the problem, we look at the coefficients 90 and 42.

To find the GCD:
1. List the factors of each number.
2. Identify the largest common factor.

For 90, factors are: 1, 2, 3, 5, 6, 9, 10, 15, 18, 30, 45, 90.
For 42, factors are: 1, 2, 3, 6, 7, 14, 21, 42.

The common factors are 1, 2, 3, and 6. Thus, the GCD is 6.

Dividing the coefficients by their GCD simplifies the expression, making complex equations more manageable and easier to solve.
Exponent Rules
Exponents, or powers, represent repeated multiplication of a number by itself. In simplifying algebraic expressions, understanding and applying exponent rules is fundamental. The essential rule used here is the division of powers with the same base.
When you have the same base being divided, you simply subtract the exponents: \(\frac{a^m}{a^n} = a^{m-n}\).

Applying this to our problem:
\(n^2\) in the numerator and \(n^5\) in the denominator simplifies as follows:
\
Fraction Simplification
Simplifying fractions is at the heart of this algebra problem. It involves reducing a fraction to its simplest form without changing its value. A fraction is fully simplified when the numerator and denominator have no common factors other than 1.

Steps to simplify a fraction:
  • Find the GCD of the numerator and the denominator.
  • Divide both the numerator and the denominator by this GCD.
  • Apply the exponent rules to simplify variable expressions.

In our problem, we simplify the algebraic expression \(\frac{90 n^2 p^8}{42 n^5 p^6}\):

1. Finding the GCD of 90 and 42, which is 6.
2. Dividing both terms by 6 yields \(\frac{15 n^2 p^8}{7 n^5 p^6}\).
3. Applying exponent rules: for \`n\`, \`n^{2-5} = n^{-3}\` which simplifies to \( \frac{1}{n^3} \); for \`p\`, \`p^{8-6} = p^2\`.

Thus, the simplified expression is \(\frac{15 p^2}{7 n^3}\).

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Most popular questions from this chapter

For exercises \(45-48\), the formula \(R=\frac{U F}{P}\) describes the glomular filtration rate by a kidney \(R\). Is the relationship of the given variables a direct variation or an inverse variation? $$ R \text { and } P \text { are constant; the relationship of } U \text { and } F \text {. } $$

In 2010, about 2,465,940 Americans died. Find the number of these deaths that were from chronic diseases. Round to the nearest hundred. (Source: www.cdc.gov, Jan. 11, 2012) 7 out of 10 deaths among Americans each year are from chronic diseases. (Source: www.cdc.gov, July 7, 2010)

$$ \text { Solve: } 0.75=\frac{k}{60} $$

For exercises \(41-44\), the formula \(R=\frac{V C}{T}\) describes the flow rate of fluid \(R\) through an intravenous drip. Is the relationship of the given variables a direct variation or an inverse variation? $$ C \text { and } T \text { are constant; the relationship of } R \text { and } V \text {. } $$

The relationship of the distance driven, \(x\), and the cost of gasoline, \(y\), is a direct variation. For a trip of \(400 \mathrm{mi}\), the cost is \(\$ 60\). a. Find the constant of proportionality. Include the units of measurement. b. Write an equation that represents this relationship. c. Find the cost of gasoline to drive \(225 \mathrm{mi}\). d. What does \(k\) represent in this equation?
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