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Chapter 7: Problem 11

Find each product and simplify if possible. $$ \frac{(m+n)^{2}}{m-n} \cdot \frac{m}{m^{2}+m n} $$

Short Answer

Expert verified
The product simplifies to \( \frac{m^2 + 2mn + n^2}{(m-n)(m+n)} \).

Step by step solution

01

Expand the Squared Term

When we see \((m+n)^2\), we should recognize it's a binomial squared. This means we expand it as \((m+n)(m+n) = m^2 + 2mn + n^2\).
02

Simplify the Product

The product given in the problem is \( \frac{(m+n)^2}{m-n} \cdot \frac{m}{m^2+mn} \). Substitute the expanded form from Step 1: \[ \frac{m^2 + 2mn + n^2}{m-n} \cdot \frac{m}{m^2+mn} \]
03

Factor the Denominator of the Second Fraction

Notice in the second fraction, the denominator \(m^2+mn\) can be factored as \(m(m+n)\). So, we replace it to get: \[ \frac{m^2 + 2mn + n^2}{m-n} \cdot \frac{m}{m(m+n)} \]
04

Combine the Fractions

Multiply the two fractions. The numerators multiply together and the denominators do too. We get:\[ \frac{m(m^2 + 2mn + n^2)}{(m-n)m(m+n)}\]Notice \(m\) simplifies out, resulting in:\[ \frac{m^2 + 2mn + n^2}{(m-n)(m+n)}\]
05

Recognize Cancellation Possibilities

Re-examine each term for any possible simplifications. Notice if there are any terms in the numerator that are multiples of terms in the denominator, however, in this case, \((m^2 + 2mn + n^2)\) doesn't allow for such simplification.

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Key Concepts

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

Binomial Expansion
The concept of binomial expansion is crucial for understanding how to expand expressions like \((m+n)^2\). A binomial is a polynomial with two terms. In our example, we are expanding the binomial squared, which is \((m+n)(m+n)\).
This results in \((m+n)(m+n) = m^2 + 2mn + n^2\).
  • First Term: Square the first term of the binomial: \(m^2\).
  • Middle Term: Multiply the two terms of the binomial, double the product: \(2mn\).
  • Last Term: Square the second term: \(n^2\).
Understanding this expansion process is fundamental when simplifying algebraic fractions that involve squared binomials.
Factoring
Factoring is the process of identifying components that multiply together to form an expression. It's like breaking down a complex structure into simpler parts.
In our problem, we encounter a denominator \(m^2 + mn\) in one of the fractions.
This expression can be factored as \(m(m+n)\).
  • Identify the common factor, which is \(m\).
  • Rewrite the expression as a product of the common factor and another expression.
Factoring plays a key role in simplifying algebraic fractions because it allows us to identify common elements that can cancel out with elements in the numerator.
Simplification
Simplification involves reducing expressions to their simplest form. By expanding, factoring, and identifying cancellation opportunities, we make the problem easier to solve.
For instance, after combining fractions, we encountered an expression \(\frac{m(m^2 + 2mn + n^2)}{(m-n)m(m+n)}\).
We simplified this by recognizing that the \(m\) in the numerator and denominator cancels each other out.
  • Examine both numerator and denominator for common factors.
  • Cancel common factors to simplify the expression.
Always double-check for further simplification possibilities to arrive at the most concise form.
Polynomial Multiplication
Multiplying polynomials involves distributing each term in the first polynomial to each term in the second polynomial.
This method is used during the step where we multiply fractions together.
Consider our multiplication of the expanded and factored expressions.
  • Sequential Multiplication: Multiply each term in the numerator by each term in the denominator.
  • Manage Complexity: Simplify terms across the expression to avoid overly complicated polynomials.
Understanding polynomial multiplication helps build strong foundations in algebraic manipulation, allowing you to tackle complex problems with ease.

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Most popular questions from this chapter

Solve. See the Concept Check in the Section. Which of the following are equivalent to \(\frac{\frac{a}{7}}{\frac{b}{13}} ?\) a. \(\frac{a}{7} \cdot \frac{b}{13}\) b. \(\frac{a}{7} \div \frac{b}{13}\) c. \(\frac{a}{7} \div \frac{13}{b}\) d. \(\frac{a}{7} \cdot \frac{13}{b}\)

Simplify. $$ \frac{\frac{2}{y^{2}}-\frac{5}{x y}-\frac{3}{x^{2}}}{\frac{2}{y^{2}}+\frac{7}{x y}+\frac{3}{x^{2}}} $$

One of the great algebraists of ancient times a man named Diophantus. Litle is known of his life other than the lived and worked in Alexandria. Some historians believe he lived during the first century of the Christian era, about the time of Nero. The only che to his personal life is the following epigram found in a collection called the Palatine A nthology. God granted him youth for a sixth of his life and added a twelfth part to this. He clothed his cheeks in down. He lit him the light of wedlock after a seventh part and five years after his marriage. He granted him a son. Alas, lateborn wretched child. After attaining the measure of half his father's life, cruel fate overtook him, thus leaving Diophantus during the last four years of his life only such consolation as the science of numbers. How old was Diophantus at his death?" We are looking for Diophantus' age when he died, so let x represent that age. If we sum the parts of his life, we should get the total age. Parts of his life \(\left\\{\begin{array}{l}{\frac{1}{6} x+\frac{1}{12} x \text { is the time of his youth. }} \\ {\frac{1}{7} x \text { is the time between his youth and when }} \\ {\text { he married. }} \\ {5 \text { years is the time between his marriage }} \\ {\text { and the birth of his son. }} \\\ {\frac{1}{2} x \text { is the time Diophantus had with his son. }} \\ {4 \text { years is the time between his son's death }} \\ {\text { and his own. }}\end{array}\right.\) The sum of these parts should equal Diophantus' age when he died. $$ \frac{1}{6} \cdot x+\frac{1}{12} \cdot x+\frac{1}{7} \cdot x+5+\frac{1}{2} \cdot x+4=x $$ How old was Diophantus when his son was born? How old was the son when he died?

Perform each indicated operation. In your own words, explain how to add two rational expressions with different denominators.

Perform each indicated operation. See Section 1.3. $$ \frac{1}{3} \cdot \frac{9}{11} $$
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