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Chapter 0: Problem 91

Multiply and simplify. Assume that all variable expressions represent positive real numbers. $$ (4 x \sqrt{y}-2 y \sqrt{x})(4 x \sqrt{y}+2 y \sqrt{x}) $$

Short Answer

Expert verified
The simplified expression is \(16x^2 y - 4y^2 x\).

Step by step solution

01

Identify the expression

The given expression is \((4x\sqrt{y}-2y\sqrt{x})(4x\sqrt{y}+2y\sqrt{x})\).
02

Recognize the pattern

Identify the expression as a difference of squares: \(a^2 - b^2 = (a-b)(a+b)\).Here, \(a = 4x\sqrt{y}\) and \(b = 2y\sqrt{x}\).
03

Apply the difference of squares formula

Using \((a-b)(a+b) = a^2 - b^2\), substitute the identified values: \((4x\sqrt{y})^2 - (2y\sqrt{x})^2\).
04

Simplify each term

Calculate \((4x\sqrt{y})^2\) and \((2y\sqrt{x})^2\): \((4x\sqrt{y})^2 = 16x^2 y\)\((2y\sqrt{x})^2 = 4y^2 x\)
05

Subtract the simplified terms

Combine the terms to get \(16x^2 y - 4y^2 x\).

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Key Concepts

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

Difference of Squares
The difference of squares is a unique algebraic formula used to simplify expressions. One of the hallmark steps to solving such problems is recognizing the pattern. The formula is \(a^2 - b^2 = (a - b)(a + b)\). This means that if you have a difference of two squares, you can factor it into the product of two binomials. In our case, \( (4x鈭歿y} - 2y鈭歿x})(4x鈭歿y} + 2y鈭歿x}) \) can be seen as \(a = 4x鈭歿y}\) and \(b = 2y鈭歿x} \). So the expression follows the form \( (a - b)(a + b) \).
Simplifying Expressions
Simplifying expressions involves breaking down the components to their simplest form. For instance, recognizing elements like \( (4x鈭歿y})^2 \) and \( (2y鈭歿x})^2 \) helps in simplifying the terms. Let's break them down:
  • For \( (4x鈭歿y})^2 \), multiply out: \ (4x鈭歿y})(4x鈭歿y}) = 16x^2y. \
  • For \( (2y鈭歿x})^2 \), multiply out: \ (2y鈭歿x})(2y鈭歿x}) = 4y^2x. \
Now substituting back, the simplified result is \( 16x^2y - 4y^2x \).
Multiplying Radicals
Multiplying radicals is a critical skill in algebra. When you encounter an expression involving radicals, like \( 4x鈭歿y}\), it means you are dealing with the square root of numbers or variables. For example, in our original expression \( (4x鈭歿y} - 2y鈭歿x})(4x鈭歿y} + 2y鈭歿x})\), each component involves radicals.

To multiply these, you multiply the coefficients (numbers in front) and then the radicals separately:
  • \ (4x鈭歿y})^2 = (4x)^2(鈭歿y})^2 = 16x^2y \
  • \ (2y鈭歿x})^2 = (2y)^2(鈭歿x})^2 = 4y^2x \
This allows you to effectively simplify the terms and combine them, providing a clean and understandable result.

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Most popular questions from this chapter

A polynomial in the variable \(x\) is defined as an expression of the form \(a_{n} x^{n}+a_{n-1} x^{n-1}+a_{n-2} x^{n-2}+\cdots+a_{1} x+a_{0} .\) Explain what this means.

Determine the sign of the expression. Assume that \(a, b,\) and \(c\) are real numbers and \(a<0, b>0,\) and \(c<0\). $$\frac{a^{2} c}{b^{4}}$$

The total national expenditure for health care has been increasing since the year 2000 . For privately insured individuals in the United States, the following models give the total amount spent for health insurance premiums \(I\) (in \$ billions) and the total amount spent on other out-of-pocket health- related expenses \(P\) (in \$ billions). (Source: U.S. Centers for Medicare \& Medicaid Services, www.census.gov) \(I=45.58 x+460.1 \quad\) Total spent on health insurance premiums \(x\) years since 2000. \(P=10.86 x+191.5 \quad\) Other out-of-pocket health-related expenses \(x\) years since 2000. a. Write and simplify the polynomial representing \(I+P\). b. Interpret the meaning of the polynomial from part (a). c. Evaluate the polynomial \(I+P\) for \(x=6,\) and interpret the meaning of this value.

Determine if the statement is true or false. The sum of two polynomials each of degree 5 will be less than or equal to degree 5 .

Simplify each expression. $$ (y+7)^{2}-2(y-3)^{2} $$
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