Problem 21 - 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. $$ \sqrt{\frac{a}{3}} \cdot \sqrt{\frac{a^{2}}{4}} $$

Step-by-Step Solution

Answer

The product simplifies to \( \frac{a \sqrt{a}}{2 \sqrt{3}} \).

Step 1: Apply the Product Rule of Square Roots

Use the rule \( \sqrt{m} \cdot \sqrt{n} = \sqrt{m \cdot n} \) to combine the two square roots: \( \sqrt{\frac{a}{3}} \cdot \sqrt{\frac{a^2}{4}} = \sqrt{\frac{a}{3} \cdot \frac{a^2}{4}} \).

Step 2: Simplify Inside the Square Root

Multiply the numerators and the denominators inside the square root. This gives \( \sqrt{\frac{a \cdot a^2}{3 \cdot 4}} = \sqrt{\frac{a^3}{12}} \).

Step 3: Simplify the Square Root

Rewrite \( \sqrt{\frac{a^3}{12}} \) as \( \frac{\sqrt{a^3}}{\sqrt{12}} \). Simplify \( \sqrt{a^3} \) as \( a \sqrt{a} \) (since \( a^3 = a^2 \cdot a \) and \( \sqrt{a^2} = a \)).

Step 4: Simplify \( \sqrt{12} \)

The number \( 12 \) factors as \( 4 \times 3 \). So, \( \sqrt{12} = \sqrt{4 \cdot 3} = \sqrt{4} \cdot \sqrt{3} = 2\sqrt{3} \).

Step 5: Combine Results into Simplest Form

Combine the results: \( \frac{a \sqrt{a}}{2 \sqrt{3}} \). This is the product in its simplest form.

Key Concepts

Product Rule of Square Roots

The Product Rule of Square Roots is a foundational concept when dealing with square roots and is incredibly useful for simplifying expressions. This rule allows you to combine two square roots into one by multiplying the numbers inside the radicals, assuming they're both positive. The principle is straightforward:

• Given two square roots, \( \sqrt{m} \cdot \sqrt{n} \), they can be expressed as a single square root: \( \sqrt{m \cdot n} \).
• This is tremendously helpful when simplifying complex expressions, as it helps reduce the number of square roots you must deal with.

In the problem above, we applied the Product Rule of Square Roots to simplify the expression \( \sqrt{\frac{a}{3}} \cdot \sqrt{\frac{a^2}{4}} \) into \( \sqrt{\frac{a}{3} \cdot \frac{a^2}{4}} \), leading us towards a more manageable calculation. Using this rule efficiently combines square roots and streams lines complex operations, ultimately simplifying the process of finding solutions.

Simplifying Radicals

When dealing with square roots, it's often helpful to simplify radicals whenever possible to make calculations easier. Simplifying radicals involves expressing the square root in its simplest and most concise form.

• A good starting point is to multiply the numbers inside the radicals and then break them down into their simplest components.
• In our problem, after using the product rule, we ended up with \( \sqrt{\frac{a^3}{12}} \). This expression itself wasn't in its simplest form yet.
• We can break it down further by expressing \( \sqrt{\frac{a^3}{12}} \) as \( \frac{\sqrt{a^3}}{\sqrt{12}} \).

Another key aspect of simplifying radicals is recognizing perfect squares and factoring these out. For instance, \( a^3 = a^2 \cdot a \), and through the use of square root rules, \( \sqrt{a^2} = a \). Similarly, recognizing that \( 12 = 4 \times 3 \) and \( \sqrt{4} = 2 \) leads us to \( \sqrt{12} = 2\sqrt{3} \), further simplifying our expression.

Radicand

The term "radicand" refers to the number or expression inside a radical symbol, like the square root. Understanding the concept of a radicand is crucial in simplifying expressions, as it often dictates how the expression can be simplified.

• For any square root \( \sqrt{x} \), \( x \) is known as the radicand.
• The key to effectively working with and simplifying square roots lies in manipulating the radicands.

In the given problem, the original radicands were \( \frac{a}{3} \) and \( \frac{a^2}{4} \). Upon applying the Product Rule of Square Roots, they were combined into a single radicand: \( \frac{a^3}{12} \).The simplification of this combined radicand involved recognizing its divisible factors and expressing them in terms of perfect squares, which is essential for breaking down the square root into products of simpler square roots. Effectively managing radicands allows one to streamline the calculation process, making it much easier to reach the simplest form of the expression.