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Chapter 7: Problem 60

For exercises 39-82, simplify. $$ \frac{40 n^{5}}{21} \div 8 n $$

Short Answer

Expert verified
\[\frac{5 n^{4}}{21}\]

Step by step solution

01

Write the Division as a Multiplication

Rewrite the division expression as a multiplication by the reciprocal. \[\frac{40 n^{5}}{21} \times \frac{1}{8 n}\]
02

Multiply the Numerators and Denominators

Multiply the numerators together and the denominators together. \[\frac{40 n^{5} \times 1}{21 \times 8 n}\]
03

Simplify the Fractions

Divide the numbers in the numerator and the denominator by their greatest common divisor. \[\frac{40 n^{5}}{168 n}\]
04

Reduce the Coefficients

Simplify the coefficients by dividing 40 and 168 by their greatest common divisor, which is 8. \[\frac{5 n^{5}}{21 n}\]
05

Simplify the Exponents

Reduce the exponents in the fraction by subtracting the exponent in the denominator from the exponent in the numerator. \[\frac{5 n^{4}}{21}\]

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Key Concepts

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

Division of Fractions
To divide one fraction by another, you need to multiply by the reciprocal of the fraction you're dividing by. The reciprocal simply means flipping the numerator and the denominator.
If you have \(\frac{a}{b} 梅 \frac{c}{d}\), you can rewrite this as \(\frac{a}{b} \times \frac{d}{c}\).
This transforms the division problem into a multiplication one, which is often easier to handle.
Remember to treat complex expressions carefully, ensuring each term is properly flipped when making the reciprocal.
Multiplication of Fractions
Multiplying fractions involves a straightforward process. You multiply the numerators (top numbers) together and the denominators (bottom numbers) together. For example, \(\frac{a}{b} \times \frac{c}{d} = \frac{ac}{bd}\).
This rule also applies when fractions include variables and coefficients, such as in our given problem.
Combine all terms in the numerators and denominators before simplifying, ensuring all multiplications are complete before moving to the next step.
Greatest Common Divisor
The greatest common divisor (GCD) of two numbers is the largest positive integer that divides both numbers without leaving a remainder.
For instance, the GCD of 40 and 168 is 8. To find the GCD, you can use several methods, including:
  • Prime factorization: Break down each number into prime factors and multiply the common factors.
  • Euclidean algorithm: Repeatedly apply division and take remainders until reaching zero.
Using the GCD helps simplify fractions and coefficients, making the expressions easier to manage.
Simplifying Exponents
Simplifying exponents in algebraic expressions involves basic rules. When you divide terms with the same base, subtract the exponents.
The general rule is \(x^a / x^b = x^{a-b}\). Applying this to our expression, you reduce the exponent in the numerator by subtracting the exponent of the denominator.
In our example, \(n^{5}/ n\), we get \(n^{5-1}=n^{4}\). Always ensure the bases are the same before performing the subtraction.

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Most popular questions from this chapter

For exercises 37-38, \(T=\frac{R}{A}\) represents the relationship of the asset turnover ratio, \(T\); the sales revenue of a company, \(R\); and the total revenues of a company, \(A\). Is the relationship of the given variables a direct variation or an inverse variation? $$ A \text { is constant; the relationship of } R \text { and } T \text {. } $$

The relationship of the number of tickets sold, \(x\), and the total ticket receipts for an outdoor concert, \(y\), is a direct variation. When 11,000 tickets are sold, the total ticket receipts are \(\$ 495,000\). a. Find the constant of proportionality, \(k\). Include the units of measurement. b. Write an equation that represents this relationship. c. Find the number of tickets sold when the total ticket receipts are \(\$ 562,500\). d. Find the total ticket receipts from the sale of 7575 tickets. e. What does \(k\) represent in this equation?

The relationship of the radius of a circle, \(x\), and the circumference of the circle, \(y\), is a direct variation. The radius of a circle is \(10 \mathrm{~cm}\), and the circumference is \(62.8 \mathrm{~cm}\). a. Find the constant of proportionality, \(k\). b. Write an equation that represents this relationship. c. Find the circumference of a circle with a radius of \(20 \mathrm{~cm}\).

The relationship of the taxable value of a property, \(x\), and the annual property tax, \(y\), is a direct variation. When the taxable value of a property is \(\$ 250,000\), the annual property tax bill is \(\$ 5375\). a. Find the constant of proportionality, \(k\). b. Write an equation that represents this relationship. c. Find the taxable value of a property with an annual property tax bill of \(\$ 8062.50\). d. Find the tax owed for a property with an assessed value of \(\$ 185,000\). Round to the nearest whole number. e. What does \(k\) represent in this equation?

The relationship of the amount of weed killer concentrate, \(x\), and the amount of mixed weed killer spray, \(y\), is a direct variation. A gardener uses \(2 \mathrm{oz}\) of concentrate to make 1 gal of weed killer spray. a. Find the constant of proportionality, \(k\). Include the units of measurement. b. Write an equation that represents this relationship. c. Find the amount of mixed weed killer spray that can be made with \(8 \mathrm{oz}\) of concentrate. d. Use slope-intercept graphing to graph this equation. e. Use the graph to find the amount of mixed weed killer spray that can be made with \(6 \mathrm{oz}\) of concentrate.
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