3 Square Root of 2 Squared: A Comprehensive Guide

When working with expressions involving square roots, such as 3√2, it's essential to understand how to simplify and manipulate these terms. Below, we delve into the details of simplifying and squaring the expression 3√2.

Simplifying 3√2

The term 3√2 represents three times the square root of 2. This expression is already in its simplest form, but it can be evaluated to a decimal if needed:

• Exact form: 3√2
• Decimal form: 3 * 1.414 ≈ 4.242

Squaring 3√2

To square the expression 3√2, follow these steps:

1. Square the coefficient and the square root separately.
2. Combine the results.

Mathematically, this process looks like:

$$


3

2


2

=

3
2

·


2

2

=
9
·
2
=
18

Example Problem: Simplify and Square (3 + √2)²

Let's simplify and square the expression (3 + √2)2. We use the distributive property (also known as the FOIL method for binomials):

$$

(
3
+

2

)

2

=
3

2

+
2
·
3
·

2

+


2

2

=
9
+
6

2

+
2
=
11
+
6

2

Therefore, (3 + √2)2 = 11 + 6√2.

Conclusion

Understanding how to simplify and manipulate expressions involving square roots is a fundamental skill in algebra. The expression 3√2 can be easily squared to get 18, and more complex expressions, such as (3 + √2)2, can be expanded and simplified using the distributive property.

For further practice, consider using online tools like the Square Root Calculator from , , and .

The expression \(3\sqrt{2}\) squared can be a bit confusing at first glance, but it follows a straightforward process in mathematics. When you square the expression \(3\sqrt{2}\), you are essentially multiplying \(3\sqrt{2}\) by itself. This can be broken down step by step to simplify the calculation.

First, recall that squaring a product involves squaring each factor separately and then multiplying the results. So, squaring \(3\sqrt{2}\) means squaring 3 and squaring \(\sqrt{2}\). Here's how it works:

• \((3\sqrt{2})^2 = 3^2 \cdot (\sqrt{2})^2\)
• Calculate \(3^2\) which equals 9.
• Calculate \((\sqrt{2})^2\) which simplifies to 2 because the square and the square root cancel each other out.
• Multiply the results: \(9 \cdot 2 = 18\).

Therefore, \((3\sqrt{2})^2 = 18\). This method of breaking down the expression helps in understanding how the operations are performed and simplifies the expression step by step.

The expression \(3 \sqrt{2}\) squared involves both multiplication and exponentiation. Let's break it down step by step:

1. First, we start with the expression \( (3 \sqrt{2})^2 \).
2. According to the rules of exponents, we need to square both the coefficient and the square root separately:
• \( 3^2 \) = 9
• \( (\sqrt{2})^2 \) = 2
3. Next, we multiply these results together:
• 9 * 2 = 18
4. Therefore, \( (3 \sqrt{2})^2 = 18 \).

This step-by-step process demonstrates the basic principles of handling expressions with both coefficients and square roots. By squaring each component individually and then combining the results, we simplify the expression to its final form.

Understanding the calculation of square roots involves several steps, particularly when dealing with complex expressions like \(3 \sqrt{2}^2\). This section provides a detailed, step-by-step explanation to make the process clear.

1. Identify the Expression: Start with the given mathematical expression: \(3 \sqrt{2}^2\).

2. Understand the Components: The expression involves the square root of a squared term. Begin by focusing on the inner part: \(\sqrt{2}^2\).

3. Calculate the Square: Squaring a number and then taking its square root is a straightforward process. Here, \(\sqrt{2}^2\) simplifies to 2, since the square root and the square cancel each other out.

4. Multiply the Result: Now multiply the simplified result by 3. So, \(3 \times 2 = 6\).

5. Conclusion: Therefore, the final result of \(3 \sqrt{2}^2\) is 6.

This process illustrates the straightforward nature of dealing with square roots and exponents, providing a clear path to simplifying and solving such expressions.

To simplify the expression \((3\sqrt{2})^2\), follow these detailed steps:

1. First, recognize the expression \((3\sqrt{2})^2\) involves both the coefficient (3) and the square root (√2).

2. Apply the power rule: \((a \cdot b)^n = a^n \cdot b^n\). Thus, \((3\sqrt{2})^2 = 3^2 \cdot (\sqrt{2})^2\).

3. Square the coefficient: \(3^2 = 9\).

4. Square the square root: \((\sqrt{2})^2 = 2\).

5. Multiply the results: \(9 \cdot 2 = 18\).

Therefore, \((3\sqrt{2})^2\) simplifies to \(18\).

The expression \((3\sqrt{2})^2\) can be simplified and applied in various fields. Here, we will explore some examples and applications to understand its significance and usage.

• Geometry: Calculating the area of a square with a side length of \(3\sqrt{2}\).
  1. Side length = \(3\sqrt{2}\)
  2. Area = \((3\sqrt{2})^2 = 18\)
• Physics: Determining the resultant force in vector problems where magnitudes are expressed with square roots.
  1. Force magnitude = \(3\sqrt{2}\) N
  2. Resultant = \((3\sqrt{2})^2 = 18\) N²
• Finance: Calculating volatility in stock prices using square roots.
  1. Variance = \(18\)
  2. Volatility = \(\sqrt{18} \approx 4.24\)
• Engineering: Evaluating natural frequencies in structural analysis.
  1. Frequency factor = \(3\sqrt{2}\)
  2. Frequency = \((3\sqrt{2})^2 = 18\) Hz
• Computer Science: Using in algorithms for encryption and data processing.
  1. Key factor = \(3\sqrt{2}\)
  2. Processed key = \((3\sqrt{2})^2 = 18\)

These examples demonstrate the versatility of the expression \((3\sqrt{2})^2\) in various mathematical and practical applications, illustrating its importance in different fields.

Understanding and calculating the expression \((3\sqrt{2})^2\) can be greatly simplified using various online calculator tools. These tools provide step-by-step solutions, visual aids, and additional resources to deepen your understanding of the mathematical concepts involved.

• Mathway: This tool offers a comprehensive algebra calculator that can evaluate expressions like \((3\sqrt{2})^2\). It provides detailed steps, helping you understand the process of simplifying and solving the expression.
• CalculatorSoup: A versatile calculator that not only computes square roots but also offers explanations for complex or imaginary solutions for negative numbers. It highlights the principal square root and its properties.
• Symbolab: This platform features an extensive range of calculators, including those for square roots and radicals. It also provides explanations and additional learning resources to help you grasp the underlying mathematics.

These tools are valuable for students and anyone needing quick, accurate computations. By using them, you can ensure precision in your calculations and enhance your mathematical proficiency.