Problem 37
Question
Multiply and simplify. All variables represent positive real numbers. $$ (\sqrt{3 x}-\sqrt{2 y})(\sqrt{3 x}+\sqrt{2 y}) $$
Step-by-Step Solution
Verified Answer
The result is \(3x - 2y\).
1Step 1: Identify the formula
The expression \((\sqrt{3x} - \sqrt{2y})(\sqrt{3x} + \sqrt{2y})\) fits the form of the difference of squares formula, \((a - b)(a + b) = a^2 - b^2\), with \(a = \sqrt{3x}\) and \(b = \sqrt{2y}\).
2Step 2: Apply the formula
Using the difference of squares formula, substitute \(a = \sqrt{3x}\) and \(b = \sqrt{2y}\). This results in the expression \(a^2 - b^2\), which becomes \((\sqrt{3x})^2 - (\sqrt{2y})^2\).
3Step 3: Simplify the squares
Compute the squares: \((\sqrt{3x})^2 = 3x\) and \((\sqrt{2y})^2 = 2y\). Substitute these values back into the expression to get \(3x - 2y\).
Key Concepts
Multiplication of RadicalsSimplifying ExpressionsPositive Real Numbers
Multiplication of Radicals
When dealing with radicals, multiplication can initially appear daunting, but it follows simple rules. Radicals are symbols for roots, such as square roots or cube roots. Multiplying them involves an essential principle: the product of two square roots is the square root of the product of their radicands (the numbers or expressions inside the root notation).
In mathematical terms, \[\sqrt{a} \times \sqrt{b} = \sqrt{a \times b}\]This rule applies as long as both numbers under the radicals are non-negative. It simplifies the process of manipulating expressions with radicals, making calculations more straightforward. In the original exercise, you see an expression like \[(\sqrt{3x} - \sqrt{2y})(\sqrt{3x} + \sqrt{2y})\]which can be addressed by multiplying the radicals using the difference of squares formula.
Knowing this makes the multiplication part smoother. With this knowledge, the seemingly complex task of combining radicals becomes more digestible.
In mathematical terms, \[\sqrt{a} \times \sqrt{b} = \sqrt{a \times b}\]This rule applies as long as both numbers under the radicals are non-negative. It simplifies the process of manipulating expressions with radicals, making calculations more straightforward. In the original exercise, you see an expression like \[(\sqrt{3x} - \sqrt{2y})(\sqrt{3x} + \sqrt{2y})\]which can be addressed by multiplying the radicals using the difference of squares formula.
Knowing this makes the multiplication part smoother. With this knowledge, the seemingly complex task of combining radicals becomes more digestible.
Simplifying Expressions
Simplifying expressions often involves reducing them to their simplest form for better understanding and easier calculations. It generally means removing parentheses, combining like terms, and executing basic arithmetic operations where possible.
In the context of operations involving radicals, simplification often includes applying specific strategies to make expressions cleaner. For instance, when dealing with our exercise, we used the difference of squares formula: \[(a - b)(a + b) = a^2 - b^2\]Once the formula is applied, the expression can be further simplified by squaring and reducing terms:
Simplifying expressions in this way helps clarify the result and makes it easier to manage in any subsequent operations.
In the context of operations involving radicals, simplification often includes applying specific strategies to make expressions cleaner. For instance, when dealing with our exercise, we used the difference of squares formula: \[(a - b)(a + b) = a^2 - b^2\]Once the formula is applied, the expression can be further simplified by squaring and reducing terms:
- Square the radical expressions such as \((\sqrt{3x})^2\) which simplifies to \(3x\)
- Similarly, \((\sqrt{2y})^2\) simplifies to \(2y\)
Simplifying expressions in this way helps clarify the result and makes it easier to manage in any subsequent operations.
Positive Real Numbers
Positive real numbers are those greater than zero, including all possible fractional and decimal values, but not including imaginary numbers. They represent all quantities that have a real value in the positive side of the number line.
In algebra, specifying that variables represent positive real numbers is crucial because it often affects the legality and operation of certain mathematical rules, like taking square roots, as negative numbers do not have real square roots. Thus, when multiplying and simplifying expressions, especially those involving radicals—as in our exercise—stipulating that the variables are positive real numbers ensures valid results and avoids complex or imaginary numbers.
Since we deal with square roots in the exercise, asserting that variables are positive real numbers confirms that the square roots are well-defined without dealing in complex numbers which often complicate the real number operations.
This condition simplifies calculations, providing assurance that the results are valid and rooted in the real number system.
In algebra, specifying that variables represent positive real numbers is crucial because it often affects the legality and operation of certain mathematical rules, like taking square roots, as negative numbers do not have real square roots. Thus, when multiplying and simplifying expressions, especially those involving radicals—as in our exercise—stipulating that the variables are positive real numbers ensures valid results and avoids complex or imaginary numbers.
Since we deal with square roots in the exercise, asserting that variables are positive real numbers confirms that the square roots are well-defined without dealing in complex numbers which often complicate the real number operations.
This condition simplifies calculations, providing assurance that the results are valid and rooted in the real number system.
Other exercises in this chapter
Problem 37
Simplify each expression. Assume that the variables can be any real number, and use absolute value symbols See Example 2. $$ \left(-64 x^{8}\right)^{1 / 8} $$
View solution Problem 37
Perform the operations. Write all answers in the form \(a+b i.\) $$ (3+4 i)+(5-6 i) $$
View solution Problem 37
Simplify each radical expression. All variables represent positive real numbers. $$ \sqrt{\frac{11}{9}} $$
View solution Problem 38
Use a calculator to find each square root. Give each answer to four decimal places. See Objective 1. $$ \sqrt{0.0063} $$
View solution