Problem 272 Find each product. $$ \left(... [FREE SOLUTION]

Chapter 5: Problem 272

Find each product. $$ \left(r s-\frac{2}{7}\right)\left(r s+\frac{2}{7}\right) $$

Short Answer

Expert verified

The product is $ r^2 s^2 - \frac{4}{49} $.

Step by step solution

Recognize the Pattern

Identify that the expression is of the form $(a - b)(a + b)$, which can be simplified using the difference of squares formula \[ (a - b)(a + b) = a^2 - b^2 \].

Assign Values to Variables

Assign $r s$ to $a$ and $\frac{2}{7}$ to $b$. Thus, $a = r s$ and $b = \frac{2}{7}$.

Apply the Difference of Squares Formula

Substitute $a$ and $b$ into the formula \[ a^2 - b^2 = (r s)^2 - \left( \frac{2}{7} \right)^2 \].

Simplify Each Term

Calculate each term separately: \[ (r s)^2 = r^2 s^2 \] and \[ \left( \frac{2}{7} \right)^2 = \frac{4}{49} \].

Combine and Simplify

Combine the results to get \[ r^2 s^2 - \frac{4}{49} \]. This is the simplified form of the original expression.

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Key Concepts

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

Algebraic Expressions

Algebraic expressions consist of variables, constants, and operators. They are mathematical phrases representing numbers and relationships. For example, in the expression $ r s - \frac{2}{7} $, 'r' and 's' are variables, while $ \frac{2}{7} $ is a constant. Understanding the components is crucial to manipulating and simplifying them. Each part plays a role in the broader context of solving algebraic equations and simplifying complex expressions.

Polynomial Multiplication

When multiplying polynomials, distributing each term of one polynomial to every term of the other is essential. In our exercise, the expression $(r s - \frac{2}{7})(r s + \frac{2}{7})$ is recognized as a difference of squares. This simplification involves pairing like terms and using the formula $a^2 - b^2$. Recognizing specific patterns like this helps streamline the multiplication process. It allows us to convert potentially cumbersome multiplication into simpler forms.

Simplifying Expressions

After identifying the pattern and applying the appropriate formula, the next step is to simplify the terms. For the example $r s - \frac{2}{7}$(r s + \frac{2}{7}), we simplify $a^2 - b^2$ as follows: \[ (r s)^2 = r^2 s^2 \] and \[\left( \frac{2}{7} \right)^2 = \frac{4}{49} \]. Combining these gives \[(r^2 s^2 - \frac{4}{49})\]. This form is much simpler, making it easier to understand and work with in further mathematical operations. Regular practice in recognizing and applying these steps will improve proficiency in handling algebraic expressions.

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91影视

Find each product. $$ \left(r s-\frac{2}{7}\right)\left(r s+\frac{2}{7}\right) $$

Short Answer

Step by step solution

Recognize the Pattern

Assign Values to Variables

Apply the Difference of Squares Formula

Simplify Each Term

Combine and Simplify

Key Concepts

Algebraic Expressions

Polynomial Multiplication

Simplifying Expressions

One App. One Place for Learning.

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