Problem 1 - Algebra 2 and Trigonometry

Question

Danielle said that 3\(x \sqrt{\frac{1}{3 x}}\) could be simplified by writing 3\(x \sqrt{\frac{1}{3 x}}\) as \(\sqrt{\frac{9 x^{2}}{3 x}}=\sqrt{3 x} .\) Do you agree with Danielle? Justify your answer.

Step-by-Step Solution

Answer

Yes, Danielle's simplification is correct.

Step 1: Dissect the Problem

We begin with the expression given by Danielle: \(3x \sqrt{\frac{1}{3x}}\). She claims this can be simplified and rewritten as \(\sqrt{\frac{9x^2}{3x}} = \sqrt{3x}\).

Step 2: Simplify Inside the Square Root

First, look at the expression \(\sqrt{\frac{1}{3x}}\). The reciprocal of \(3x\) is \(\frac{1}{3x}\), and taking the square root of this gives \(\sqrt{\frac{1}{3x}} = \frac{1}{\sqrt{3x}}\).

Step 3: Multiply with the 3x Term

Multiply the result from Step 2 by \(3x\):\[3x \times \frac{1}{\sqrt{3x}} = \frac{3x}{\sqrt{3x}}\].

Step 4: Simplify the Quotient

Simplify \(\frac{3x}{\sqrt{3x}}\):Divide \(3x\) by \(\sqrt{3x}\) to get \(\sqrt{3x} \times \frac{3x}{3x} = \sqrt{3x}\).Thus, \(\frac{3x}{\sqrt{3x}} = \sqrt{3x}\).

Step 5: Verify Danielle's Simplification

Danielle's simplification involves rewriting \(3x \sqrt{\frac{1}{3x}}\) as \(\sqrt{\frac{9x^2}{3x}}\). Simplify \(\frac{9x^2}{3x}\) to get \(3x\), so her expression \(\sqrt{3x}\) is indeed correct.

Key Concepts

Square Root Simplification

The square root simplification is a powerful tool in algebra that allows us to simplify expressions under square roots to make them more manageable. When we have a fraction under a square root, such as \( \sqrt{\frac{1}{3x}} \), we can simplify it by taking the square root of both the numerator and the denominator separately.
This process can be especially helpful in reducing complex expressions to simpler forms. For example, here, by recognizing that \( \sqrt{\frac{1}{3x}} \) is the same as \( \frac{1}{\sqrt{3x}} \), we can manipulate the expression more easily.
Remember, simplifying square roots can make other algebraic operations, such as multiplication and division, much easier. It helps in avoiding common mistakes and makes the expression much clearer.

Quotients in Algebra

Quotients are a fundamental concept in algebra, representing the division of one quantity by another. In the exercise example, we deal with the quotient \( \frac{3x}{\sqrt{3x}} \). Simplifying quotients is crucial for solving problems efficiently.
To simplify \( \frac{3x}{\sqrt{3x}} \), notice that both the numerator and the denominator contain the variable \( x \), which allows for easy manipulation.

• By dividing \( 3x \) by \( \sqrt{3x} \), you get that it simplifies to \( \sqrt{3x} \), utilizing the property of roots that says \( \sqrt{a^2} = a \).

This principle helps in transforming complex expressions into more manageable forms that can be easily verified or further simplified.

Verification of Simplification Steps

Verification is an essential part of the algebraic process because it ensures that each step in simplification is correctly performed, leading to a valid result. Danielle's claim was that the expression \( 3x \sqrt{\frac{1}{3x}} \) could be simplified to \( \sqrt{3x} \).
By breaking down the expression and verifying each step — from simplifying \( \sqrt{\frac{1}{3x}} \) to evaluating \( \frac{3x}{\sqrt{3x}} \) — we see that her simplification is indeed correct.
This exercise confirms the importance of verifying through systematic simplification, which helps avoid errors and ensures the correctness of the final simplified form. Always recheck each simplification step, and validate your final expression to have confidence in your results.

Algebraic Expressions

Algebraic expressions involve numbers, variables, and arithmetic operations. They form the core of algebra and help us model real-world problems mathematically.
The expression from the exercise, \( 3x \sqrt{\frac{1}{3x}} \), is an excellent example that showcases the manipulative aspect of algebra.

• Comprehending each element in the expression \( (3x) \), the operation \( (\sqrt{\frac{1}{3x}}) \), and how they interact is the key to successful simplification and transformation.
• Breaking it down into simpler components allows us to manage complexity and apply simplification rules effectively.

This understanding enables the resolution of more complex algebraic problems, highlighting the necessity of thorough comprehension and practice of these concepts.