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Chapter 0: Problem 17

Expand the given expression. $$x y(x+y)\left(\frac{1}{x}-\frac{1}{y}\right)$$

Short Answer

Expert verified
The expanded form of the expression \(xy(x+y)\left(\frac{1}{x}-\frac{1}{y}\right)\) is \(y^2 - x^2\).

Step by step solution

01

Distribute the xy term

Distribute the \(xy\) term to both \(x\) and \(y\) in the first parentheses: \(xy(x+y) = x^2y + xy^2\) Now, the expression becomes: \((x^2y + xy^2)\left(\frac{1}{x}-\frac{1}{y}\right)\)
02

Distribute the \(\left(\frac{1}{x}-\frac{1}{y}\right)\) term

Distribute the \(\left(\frac{1}{x}-\frac{1}{y}\right)\) term to both \(x^2y\) and \(xy^2\): \(x^2y\left(\frac{1}{x}-\frac{1}{y}\right) + xy^2\left(\frac{1}{x}-\frac{1}{y}\right)\)
03

Simplify each term

Now, simplify each term within the parentheses: - For the first term: \(x^2y\left(\frac{1}{x}-\frac{1}{y}\right) = x^2y \cdot \frac{1}{x} - x^2y \cdot \frac{1}{y} = xy - x^2\) - For the second term: \(xy^2\left(\frac{1}{x}-\frac{1}{y}\right) = xy^2 \cdot \frac{1}{x} - xy^2 \cdot \frac{1}{y} = y^2 - xy\)
04

Combine the simplified terms

Combine the simplified terms to find the expanded expression: \(xy - x^2 + y^2 - xy\) Since the \(xy\) and \(-xy\) terms cancel each other, the final expression is: \(y^2 - x^2\)

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Key Concepts

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

Distributive Property
The Distributive Property is a foundational concept in algebra that allows you to multiply a single term by each term inside a parenthesis. This helps in breaking down complex equations into more manageable parts. In this exercise, the distributive property is used when the expression \(xy(x+y)\) is expanded. Here, the term \(xy\) is distributed across both terms within the parenthesis:
  • First, multiply \(xy\) by \(x\), giving \(x^2y\).
  • Then, multiply \(xy\) by \(y\), resulting in \(xy^2\).
These actions transform the original complex expression into a simpler form, ready for further manipulation. Using the distributive property is crucial as it sets up the problem for additional simplification.
Algebraic Simplification
Algebraic Simplification involves reducing expressions to their simplest form. This process often involves combining like terms and cancelling out terms, as seen in this exercise. After distributing terms, we are left with:
  • \(x^2y \cdot \frac{1}{x} - x^2y \cdot \frac{1}{y}\)
  • \(xy^2 \cdot \frac{1}{x} - xy^2 \cdot \frac{1}{y}\)
Through simplification:
  • The first expression simplifies to \(xy - x^2\).
  • The second expression simplifies to \(y^2 - xy\).
Next, combining these results results in \(xy - x^2 + y^2 - xy\). Upon identifying and cancelling the \(xy\) terms, the expression is further simplified to its final form \(y^2 - x^2\). Simplification allows us to find a cleaner and more comprehensible end result, which is especially helpful for checking the accuracy of solutions.
Binomial Multiplication
Binomial Multiplication refers to multiplying expressions that contain two terms, known as binomials. In this particular exercise, although the multiplication isn't directly labeled as binomial multiplication, the principle is applied when multiplying:
  • Each term in \((x^2y + xy^2)\) by each term in \(\left(\frac{1}{x} - \frac{1}{y}\right)\).
This series of multiplications resembles the process of expanding a binomial squared, where each term in one binomial multiplies every term in the other. The outcome is a combination of terms that are then simplified. Through multiplication, this exercise shows how understanding binomials can aid in more complex algebraic manipulations.

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Most popular questions from this chapter

Show that if \(a\) and \(b\) are real numbers such that $$|a+b|<|a|+|b|,$$ then \(a b<0\)

The intersection of two sets of numbers consists of all numbers that are in both sets. If \(A\) and \(B\) are sets, then their intersection is denoted by \(A \cap B .\) In Exercises \(47-\) \(56,\) write each intersection as a single interval. $$(-\infty,-6] \cap(-8,12)$$

Suppose shirts are on sale for $$\$ 19.99$$ each. Explain how you could use the distributive property to calculate in your head that six shirts cost $$\$ 119.94$$.

Find all numbers \(x\) satisfying the given equation. $$\left|\frac{3 x+2}{x-4}\right|=5$$

Give four examples of pairs of real numbers \(a\) and \(b\) such that \(|a+b|=2\) and \(|a|+|b|=8\).
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