Problem 47 - Elementary and Intermediate Algebra

Question

Square or cube each quantity and simplify the result. $$ (3 \sqrt{2})^{2} $$

Step-by-Step Solution

Answer

(3 \sqrt{2})^{2} = 18

Step 1: Understand the Expression

The given expression is \((3 \sqrt{2})^{2}\). We are required to square the entire quantity inside the parentheses.

Step 2: Square the Coefficient

First, square the coefficient 3. \(3^2 = 9\).

Step 3: Square the Radical

Now, square \(\sqrt{2}\). The square of a square root cancels out the square root, leaving the number itself. \((\sqrt{2})^2 = 2\).

Step 4: Multiply the Results

Multiply the results from Step 2 and Step 3. Multiply the squared coefficient and the squared radical: \(9 \times 2 = 18\).

Step 5: Simplify

Combine the results. Hence, \((3 \sqrt{2})^{2} = 18\).

Key Concepts

Squaring Numbers

Squaring a number means multiplying it by itself. This is a fundamental concept in mathematics and is often represented as a number raised to the power of 2, like this: \(n^2\). When you square a number, you perform the following steps:

• Identify the number you need to square.
• Multiply the number by itself.

Let's apply this to the expression \((3 \sqrt{2})^{2}\). First, there is a coefficient, 3. To square it, we multiply it: \(3 \times 3 = 9\).
Squaring numbers becomes even more straightforward with practice. It's an essential skill in algebra because it helps you solve equations and simplify expressions.

Radicals in Algebra

Radicals, often known as roots, help us deal with numbers in algebra that aren't perfect squares. In our case, we are specifically focusing on square roots. A square root of a number is a value that, when multiplied by itself, gives the original number.
For example, \(\sqrt{4} = 2\) because \(2 \times 2 = 4\). When squaring a radical, the square and the square root operations cancel each other out.

• Find the radical part of your expression.
• Square this radical; it removes the radical sign.

In the expression \(\sqrt{2}\), squaring simplies it to 2 because \((\sqrt{2})^2 = 2\). This concept allows us to deal with more complex algebraic expressions involving square roots.

Simplifying Expressions

Simplifying expressions is like cleaning up the math by combining and reducing terms to their simplest form. It's a skill you'll use often in algebra.

• Identify which terms can be combined or simplified.
• Apply mathematical operations such as multiplication.

Once we've squared the number 3 and the radical \(\sqrt{2}\), we're left with separate results. Now, we multiply these results together. In our example:
\(9 \times 2 = 18\).
That’s how you simplify the expression \((3 \sqrt{2})^{2}\). Simplifying isn't just limited to squaring; it applies to various algebraic manipulations. It makes your final answer neat, manageable, and easier to understand.