/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 41 Find each product. $$ \left(... [FREE SOLUTION] | 91影视

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Chapter 4: Problem 41

Find each product. $$ \left(\frac{3}{4}-x\right)\left(\frac{3}{4}+x\right) $$

Short Answer

Expert verified
\frac{9}{16} - x^2

Step by step solution

01

- Identify the expression format

Notice that the expression \(\frac{3}{4}-x\) and \(\frac{3}{4}+x\) is in the form of \( (a - b)(a + b) \), which is a difference of squares.
02

- Recall the difference of squares formula

Recall the formula for the difference of squares: \( (a - b)(a + b) = a^2 - b^2 \)
03

- Substitute the values into the formula

In this example, \( a = \frac{3}{4} \) and \( b = x \). Substituting these values into the formula gives: \[ \frac{3}{4}^2 - x^2 \]
04

- Simplify the expression

Calculate \( (\frac{3}{4})^2 \): \[ (\frac{3}{4})^2 = \frac{9}{16} \] Thus, the expression \( \frac{9}{16} - x^2 \)

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Key Concepts

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

algebraic expressions
Algebraic expressions are combinations of variables, numbers, and operations. In the exercise given, we work with an algebraic expression of the form \(\left( \frac{3}{4} - x \right) \left( \frac{3}{4} + x \right)\). This type of expression is called a binomial because it contains two terms. Understanding these basic concepts is essential, as algebraic expressions are the foundation of algebra.
simplifying expressions
Simplifying expressions is the process of making them easier to work with. This involves reducing expressions to their simplest form. In the step-by-step solution, we start with \(\left( \frac{3}{4} - x \right) \left( \frac{3}{4} + x \right)\) and use the difference of squares formula to simplify it. The difference of squares formula, \( (a - b)(a + b) = a^2 - b^2\), allows us to transform the product of two binomials into a simpler form: \(\frac{9}{16} - x^2.\)
binomial products
Binomial products occur when multiplying two binomials together. The given exercise demonstrates a specific type of binomial product, known as the difference of squares. Using the expression \(\left( \frac{3}{4} - x \right) \left( \frac{3}{4} + x \right)\) as an example, we identify it as \( (a - b)(a + b)\), where \(a = \frac{3}{4}\) and \(b = x.\) By applying the difference of squares formula, we simplify the product to \(\frac{9}{16} - x^2,\) illustrating how binomial products can often be transformed into simpler expressions.

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Most popular questions from this chapter

If it costs \(\$ 15\) plus \(\$ 2\) per day to rent a chain saw, the binomial $$2 x+15$$ gives the cost in dollars to rent the chain saw for \(x\) days. Evaluate this binomial for \(x=6\). Use the result to fill in the blanks: If the saw is rented for_________ days, then the cost is _________ dollars.

The special product \((x+y)(x-y)=x^{2}-y^{2}\) can be used to perform some multiplications.Example: $$\begin{array}{l|l}51 \times 49 & 102 \times 98 \\\=(50+1)(50-1) & =(100+2)(100-2) \\\=50^{2}-1^{2} & =100^{2}-2^{2} \\\=2500-1 & =10,000-4 \\\=2499 & =9996\end{array}$$ Use this method to calculate each product mentally. $$ 301 \times 299 $$

Find each product. $$ (x+y)^{2}(x-y)^{2} $$

Use scientific notation to calculate the result in each expression. Write answers in scientific notation. 73\. \(\frac{650,000,000(0.0000032)}{0.00002}\)

In 2016 , the state of Minnesota had about \(7.33 \times 10^{4}\) farms with an average of \(3.53 \times 10^{2}\) acres per farm. What was the total number of acres devoted to farmland in Minnesota that year? (Data from U.S. Department of Agriculture.)
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