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Chapter 8: Problem 46

Perform the operations and simplify, if possible. See Example 5 $$\frac{16 c^{3}}{5 d^{2}} \cdot \frac{25 d}{12 c}$$

Short Answer

Expert verified
The expression simplifies to \( \frac{20c^2}{3d} \).

Step by step solution

01

Multiply the Numerators

Write down the product of the numerators: \( 16c^3 \times 25d \). Multiply the coefficients and the variables separately:\[16 \times 25 = 400,\]and the variables:\[c^3 \cdot d = c^3d.\]The resulting expression is \( 400c^3d \).
02

Multiply the Denominators

Write down the product of the denominators: \( 5d^2 \times 12c \). Multiply the coefficients and the variables separately:\[5 \times 12 = 60,\]and the variables:\[d^2 \cdot c = d^2c.\]The resulting expression is \( 60d^2c \).
03

Divide the Products

Take the expressions from Steps 1 and 2 to form the fraction:\[\frac{400c^3d}{60d^2c}\]Divide the coefficients and the variables separately:\[\frac{400}{60} = \frac{40}{6} = \frac{20}{3},\]For the variables, cancel out like terms:\[\frac{c^3}{c} = c^2,\]and\[\frac{d}{d^2} = \frac{1}{d}.\]
04

Simplify the Expression

Combine the results from Step 3 to write the simplified expression:\[\frac{20}{3} \cdot c^2 \cdot \frac{1}{d} = \frac{20c^2}{3d}.\]This is the final simplified form of the expression.

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Key Concepts

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

Understanding Multiplication of Fractions
Multiplication of fractions is a fundamental operation in algebra. When multiplying fractions, you multiply both the numerators and the denominators. In the given exercise, two algebraic fractions are multiplied: \( \frac{16 c^{3}}{5 d^{2}} \) and \( \frac{25 d}{12 c} \).
To multiply these:
  • Multiply the numerators: \( 16c^3 \times 25d = 400c^3d \). Here, you multiply the coefficients and the variables separately. The coefficients are 16 and 25, resulting in 400. The variables \( c^3 \) and \( d \) give \( c^3d \).
  • Multiply the denominators: \( 5d^2 \times 12c = 60d^2c \). Similarly, the coefficients 5 and 12 multiply to give 60, and the variables \( d^2 \) and \( c \) give \( d^2c \).
This process results in a new fraction: \( \frac{400c^3d}{60d^2c} \). Taking these steps ensures that every variable and coefficient is correctly accounted for.
Simplifying Algebraic Expressions
Simplifying algebraic expressions means reducing them into their simplest form. This often makes calculations easier and expressions clearer. In our case, we simplify \( \frac{400c^3d}{60d^2c} \) by reducing both the coefficients and the variables.
Here鈥檚 how you do it:
  • Begin with the coefficients from the fraction \( \frac{400}{60} \). Dividing both by their greatest common divisor (GCD), which is 20, gives \( \frac{20}{3} \).
  • Next, handle the variables. For \( \frac{c^3}{c} \), simplify by canceling common \( c \) terms, resulting in \( c^2 \). For \( \frac{d}{d^2} \), one \( d \) cancels out, leaving \( \frac{1}{d} \).
The simplified expression then becomes \( \frac{20c^2}{3d} \). This process is crucial as it helps in working with less complex expressions in future calculations.
Exploring Variables and Coefficients
Variables and coefficients are the building blocks of algebraic expressions. Understanding their roles is crucial in performing mathematical operations correctly.
  • **Variables** - These are symbols, like \( c \) and \( d \), representing numbers that can vary or change. In algebra, they allow for the expression of general patterns and relationships.
  • **Coefficients** - These are the numerical parts of terms. For instance, in \( 16c^3 \), the coefficient is 16. Coefficients give terms their size or magnitude but not their direction.
When multiplying algebraic fractions, each coefficient and variable pair plays a critical role. Knowing how to handle them separately and how they interact in terms like \( 16c^3d \) or \( 60d^2c \) ensures proper algebraic manipulation. This knowledge is foundational for any algebraic operation.

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Most popular questions from this chapter

Rain Gutters. A rectangular sheet of metal will be used to make a rain gutter by bending up its sides, as shown. If the ends are covered, the capacity \(f(x)\) of the gutter is a polynomial function of \(x: f(x)=-240 x^{2}+1,440 x .\) Find the capacity of the gutter if \(x\) is 3 inches. (GRAPH CANNOT COPY)

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