Problem 13 - Algebra 2 and Trigonometry

Question

In \(3-41\) , express each product in simplest form. Variables in the radicand with an even index are non-negative. $$ (3 \sqrt{3})^{2} $$

Step-by-Step Solution

Answer

The simplest form of \((3 \sqrt{3})^{2}\) is 27.

Step 1: Understand the expression

The expression given is \((3 \sqrt{3})^{2}\). This is a product of a number and a square root raised to the power of 2. We need to simplify this expression.

Step 2: Apply the power to each factor

First, write out what it means to square the expression: \((3 \sqrt{3})^2 = (3 \sqrt{3}) \times (3 \sqrt{3})\). This is equivalent to \((3^2) \times (\sqrt{3})^2\).

Step 3: Simplify each component

Compute \(3^2\) which equals 9, and simplify \((\sqrt{3})^2\) which is 3. Therefore, the expression becomes \(9 \times 3\).

Step 4: Combine the results

Multiply the simplified components: \(9 \times 3 = 27\). Thus, \((3 \sqrt{3})^2\) simplifies to 27.

Key Concepts

Squaring Binomials

Squaring binomials refers to multiplying a two-term expression by itself. For example, the process of squaring a binomial like \( (a + b)^2 \) involves an expansion into three terms: \( a^2 + 2ab + b^2 \). However, in the given exercise, we have a monomial, which means a single term, \( (3 \sqrt{3})^2 \), rather than a binomial.

When squaring, we need to apply the square to each part of the term separately. This means squaring the numerical part (3) and squaring the square root part (\( \sqrt{3} \)). Each component, when squared and multiplied, leads us to the final simplified form of the expression.

Properties of Exponents

Understanding exponents is crucial for simplifying expressions like \( (3 \sqrt{3})^{2} \). A key property is that when you raise a product to a power, you raise each factor within the product to that power separately. Mathematically, this is shown as:

• For any numbers \(a\) and \(b\), \((a \cdot b)^n = a^n \cdot b^n\)
• This property helps us rewrite \((3 \sqrt{3})^2\) as \(3^2 \cdot (\sqrt{3})^2\)

Breaking it down, you calculate \(3^2\) which equals 9. And \((\sqrt{3})^2\), by definition, simplifies directly to 3. By understanding how exponents affect each part of the expression independently, you simplify complex expressions step-by-step.

Multiplying Radicals

Multiplying radicals involves certain rules that make the process straightforward. When dealing with radical multiplication, especially in expressions like \( (3 \sqrt{3})^2 \), knowing the essentials can help:

• The product rule for radicals states that \( \sqrt{a} \cdot \sqrt{b} = \sqrt{a \cdot b}\)
• When the indices of the radicals (square roots here) are the same, you multiply under the same radical sign

In this exercise, while simplifying \((\sqrt{3})^2\), it simplifies to 3 because \(\sqrt{3} \cdot \sqrt{3} = \sqrt{9} = 3\). This simplification is key to solving the problem efficiently. By understanding these radical rules, multiplication and simplification become much more manageable.