Problem 44
Question
For exercises 39-82, simplify. $$ \frac{4 c}{7} \div \frac{8}{21 c^{2}} $$
Step-by-Step Solution
Verified Answer
\( \frac{3c^{3}}{2} \)
1Step 1: Rewrite the division as multiplication
Change the division operation to multiplication by taking the reciprocal of the second fraction. \( \frac{4 c}{7} \times \frac{21 c^{2}}{8} \)
2Step 2: Multiply the numerators
Multiply the numerators together.\( 4c \times 21c^{2} = 84c^{3} \)
3Step 3: Multiply the denominators
Multiply the denominators together.\( 7 \times 8 = 56 \)
4Step 4: Simplify the fraction
Simplify \( \frac{84c^{3}}{56} \). Notice that both 84 and 56 have a common factor of 28.\( \frac{84c^{3}}{56} = \frac{84 ÷ 28}{56 ÷ 28} = \frac{3c^{3}}{2} \)
Key Concepts
Fraction DivisionReciprocalMultiplication of FractionsSimplification
Fraction Division
When dividing algebraic fractions, the key idea is to convert the division problem into a multiplication problem. Instead of dividing by a fraction, you multiply by its reciprocal. It may seem tricky at first, but this method can simplify the process greatly.
Suppose you have a division problem involving fractions, such as \( \frac{a}{b} \div \frac{c}{d} \). This can be rewritten as \( \frac{a}{b} \times \frac{d}{c} \). This step is crucial and forms the foundation of simplifying fraction problems.
Remember to always flip the second fraction and change the division sign to multiplication. Once you master this step, solving these problems becomes straightforward.
Suppose you have a division problem involving fractions, such as \( \frac{a}{b} \div \frac{c}{d} \). This can be rewritten as \( \frac{a}{b} \times \frac{d}{c} \). This step is crucial and forms the foundation of simplifying fraction problems.
Remember to always flip the second fraction and change the division sign to multiplication. Once you master this step, solving these problems becomes straightforward.
Reciprocal
A reciprocal is simply a flipped version of a fraction. For instance, the reciprocal of \( \frac{3}{4} \) is \( \frac{4}{3} \). When dealing with division of fractions, taking the reciprocal of the second fraction and multiplying is how you solve it.
In our example with \( \frac{4c}{7} \div \frac{8}{21c^2} \), we take the reciprocal of \( \frac{8}{21c^2} \), which gives us \( \frac{21c^2}{8} \). This lets us rewrite the expression as a multiplication problem: \( \frac{4c}{7} \times \frac{21c^2}{8} \).
Practicing finding reciprocals until it becomes second nature will help greatly with algebraic fractions.
In our example with \( \frac{4c}{7} \div \frac{8}{21c^2} \), we take the reciprocal of \( \frac{8}{21c^2} \), which gives us \( \frac{21c^2}{8} \). This lets us rewrite the expression as a multiplication problem: \( \frac{4c}{7} \times \frac{21c^2}{8} \).
Practicing finding reciprocals until it becomes second nature will help greatly with algebraic fractions.
Multiplication of Fractions
Multiplying fractions is simpler than it initially seems. Here's a step-by-step approach to guide you:
For the fractions from our example, multiplying \( \frac{4c}{7} \times \frac{21c^2}{8} \) entails:
1. Multiplying the numerators: \( 4c \times 21c^2 \) results in \( 84c^3 \).
2. Multiplying the denominators: \( 7 \times 8 \) equals \( 56 \).
Now, we have the fraction \( \frac{84c^3}{56} \). The next step is simplifying this fraction.
- Multiply the numerators (top numbers).
- Multiply the denominators (bottom numbers).
- Combine the results into a single fraction.
For the fractions from our example, multiplying \( \frac{4c}{7} \times \frac{21c^2}{8} \) entails:
1. Multiplying the numerators: \( 4c \times 21c^2 \) results in \( 84c^3 \).
2. Multiplying the denominators: \( 7 \times 8 \) equals \( 56 \).
Now, we have the fraction \( \frac{84c^3}{56} \). The next step is simplifying this fraction.
Simplification
Simplifying a fraction involves reducing it to its simplest form. This step ensures that the fraction is as simple as possible, making it easier to understand and work with.
To simplify \( \frac{84c^3}{56} \), identify the greatest common factor (GCF) of the numerator and denominator. Here, the GCF of 84 and 56 is 28.
Divide both the numerator and denominator by this GCF:
\[\frac{84 \div 28}{56 \div 28} = \frac{3c^3}{2} \]
Our final simplified fraction is \( \frac{3c^3}{2} \). Always check whether further simplification is possible, especially for more complex algebraic fractions.
To simplify \( \frac{84c^3}{56} \), identify the greatest common factor (GCF) of the numerator and denominator. Here, the GCF of 84 and 56 is 28.
Divide both the numerator and denominator by this GCF:
\[\frac{84 \div 28}{56 \div 28} = \frac{3c^3}{2} \]
Our final simplified fraction is \( \frac{3c^3}{2} \). Always check whether further simplification is possible, especially for more complex algebraic fractions.
Other exercises in this chapter
Problem 44
For exercises \(25-68\), evaluate or simplify. $$ \frac{\frac{2}{x-2}+\frac{3}{x+3}}{\frac{2}{x-2}+\frac{1}{x+3}} $$
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For exercises \(5-48\), simplify. $$ \frac{8 p^{2}}{4 p^{2}+32 p+64}-\frac{128}{4 p^{2}+32 p+64} $$
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For exercises 1-66, simplify. $$ \frac{p^{2}-6 p-27}{p^{2}-2 p-15} $$
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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
View solution