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Chapter 7: Problem 84

Perform the operations and simplify, if possible. \(33 r\left(\frac{5 r+4}{11 r}\right)\)

Short Answer

Expert verified
First distribute 33r across the terms and simplify.

Step by step solution

01

Distribute the 33r Across the Fraction

Multiply the term outside the parentheses, which is \(33r\), by each term inside the parentheses individually. This means: \[ 33r \times \left( \frac{5r}{11r} + \frac{4}{11r} \right) \]Distribute to get:\[ 33r \left( \frac{5r}{11r} \right) + 33r \left( \frac{4}{11r} \right) \]

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Key Concepts

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

Distribution in Algebra
Distribution in algebra might sound intimidating, but with a simple breakdown, it becomes much easier to understand. In essence, when you see an expression like \( a(b + c) \), distribution helps you expand this by multiplying \( a \) with all terms inside the parentheses. To apply this to our exercise, think of \( 33r \) as \( a \), and the contents inside the parentheses, \( \frac{5r}{11r} + \frac{4}{11r} \), as \( b + c \). The distribution process involves:
  • Multiplying \(33r\) by \(\frac{5r}{11r}\)
  • Multiplying \(33r\) by \(\frac{4}{11r}\)
Breaking it down step-by-step:
  • \(33r \times \frac{5r}{11r} = \frac{33r \times 5r}{11r}\)
  • \(33r \times \frac{4}{11r} = \frac{33r \times 4}{11r}\)
This step sets the stage for simplification, which involves canceling common terms.
Simplification of Fractions
Once distribution is done, the next step is simplifying fractions. Simplification makes expressions tidier and easier to work with by reducing fractions to their simplest form.Let's simplify the fractions from the distributed terms:
  • For \( \frac{33r \times 5r}{11r} \):
    • Cancel the \(11r\) from both numerator and denominator where possible.
    • You're left with \(3 \times 5r = 15r\).
  • For \( \frac{33r \times 4}{11r} \):
    • Again, cancel \(11r\) from both numerator and denominator.
    • This leads to \(3 \times 4 = 12\).
After simplifying, the expression becomes \(15r + 12\). This step is about identifying and eliminating common factors, a fundamental technique in algebra.
Polynomial Expressions
Polynomial expressions consist of variables and coefficients, combined using addition, subtraction, multiplication, and non-negative integer exponents. In our scenario, the simplified form \(15r + 12\) is a polynomial.Here鈥檚 what makes it a polynomial:
  • Each term contains a coefficient and a variable. E.g., \(15r\) has a coefficient of 15 and a variable \(r\).
  • The terms are connected by addition/subtraction, in this case, addition.
  • In its simplest form, there are no divisions by variables, only straightforward multiplication.
Understanding polynomials is crucial since they form the basis for many algebraic operations and are used widely in both pure and applied mathematics. Each part of a polynomial, like the coefficients and variables, plays a specific role in its structure and behavior.

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