Problem 21 - Intermediate Algebra

Question

Simplify each radical expression. All variables represent positive real numbers. $$ \sqrt{75 a^{2}} $$

Step-by-Step Solution

Answer

The simplified form is \(5a\sqrt{3}\).

Step 1: Identify the Radicand

The radicand is the expression inside the square root. In this exercise, the radicand is \(75a^2\). We need to simplify this expression.

Step 2: Factor the Numerical Part

Factorize the numerical part of the radicand, which is 75. The prime factorization of 75 is \(3 \times 5^2\). So we write \(75 = 3 \times 5^2\).

Step 3: Apply the Square Root to Each Factor

Break down the expression under the square root into known squares: \(\sqrt{75a^2} = \sqrt{3 \times 5^2 \times a^2}\). Now, apply the square root to each component.

Step 4: Simplify Using the Product Rule for Square Roots

Use the product rule \(\sqrt{xy} = \sqrt{x} \times \sqrt{y}\) to simplify: \(\sqrt{3} \times \sqrt{5^2} \times \sqrt{a^2}\).

Step 5: Compute the Square Roots of Perfect Squares

For \(\sqrt{5^2}\), we have \(5\) because \((5^2)^{1/2} = 5\). For \(\sqrt{a^2}\), we have \(a\) because \((a^2)^{1/2} = a\). Thus, the simplified expression is \(5a\sqrt{3}\).

Step 6: Write the Final Simplified Form

Combine all the simplified parts together to get the final expression: \(5a\sqrt{3}\).

Key Concepts

Understanding the Radicand

The radicand is the term found inside the square root symbol, \(\sqrt{\cdot}\). For the expression \(\sqrt{75a^2}\), the radicand is \(75a^2\). It's essential to simplify the radicand whenever possible to simplify the entire expression. To do so, you must look at both the numerical and variable parts of the radicand individually.
For the numerical part, factor it into its prime components. This helps in recognizing perfect squares, making simplification more approachable. For example, in \(75a^2\), the numerical portion is \(75\).
For the variable part, check the exponents. If a variable is raised to an even power like \(a^2\), this indicates it is a perfect square, simplifying to \(a\) under the square root. Identifying perfect squares quickly reduces the complexity of the problem.

Prime Factorization for Simplification

Prime factorization is a key strategy in simplifying radicands because it breaks down numbers into their basic building blocks. For \(75\), determine its prime factors by recognizing that \(75 = 3 \times 25\), and further breaking it down to \(3 \times 5^2\). Doing this, we see \(5^2\) is a perfect square.
This factorization is instrumental because:

• It reveals perfect square components easily.
• Enables straightforward application of square roots.
• Simplifies the expression by focusing only on the prime factors.

Once you've fully factorized the radicand, you'll apply further simplification using the product rule for square roots.

Exploring the Product Rule for Square Roots

The product rule for square roots is a convenient property that facilitates the simplification of root expressions. If you have \(\sqrt{xy}\), this is equivalent to \(\sqrt{x} \times \sqrt{y}\). Applying this rule helps you handle complex radicands by dealing with each factor independently.
Let's apply it to simplify \(\sqrt{3 \times 5^2 \times a^2}\):

• Break it down to \(\sqrt{3} \times \sqrt{5^2} \times \sqrt{a^2}\).
• Simplify each individual term: \(\sqrt{5^2} = 5\) and \(\sqrt{a^2} = a\).

This results in the final simplified form, \(5a\sqrt{3}\). By isolating each component through the product rule, the expression becomes more manageable and understandable.