Problem 21 - Elementary and Intermediate Algebra

Question

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

Step-by-Step Solution

Answer

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

Step 1: Identify the Perfect Square Factors

Look for perfect square numbers that can be factored out of 75. The largest perfect square factor is 25 since \( 25 \times 3 = 75 \). This allows us to write the expression inside the square root as \( \sqrt{75} = \sqrt{25 \times 3} \).

Step 2: Apply the Product Property of Square Roots

Use the property \( \sqrt{a \times b} = \sqrt{a} \times \sqrt{b} \) to simplify the expression. Thus, \( \sqrt{25 \times 3} = \sqrt{25} \times \sqrt{3} \). We know that \( \sqrt{25} = 5 \). So it simplifies to \( 5 \sqrt{3} \).

Step 3: Deal with the Variable

The expression \( \sqrt{75a^2} \) can also be split as \( \sqrt{75} \times \sqrt{a^2} \). From Step 2, we know that \( \sqrt{75} = 5\sqrt{3} \), and \( \sqrt{a^2} = a \) since \( a \) is a positive real number. Therefore, \( \sqrt{75a^{2}} = (5\sqrt{3}) \times a = 5a\sqrt{3} \).

Step 4: Combine the Results

Combine all components to obtain the final simplified expression: \( 5a\sqrt{3} \). This is the most simplified form of the given radical expression.

Key Concepts

Perfect Square

A perfect square is a number that can be expressed as the product of an integer with itself. Identifying perfect square factors can greatly simplify radical expressions. For instance, in our given exercise, we consider the number under the square root, which is 75.

• The number 75 can be broken down into two factors: 25 and 3.
• Among these, 25 is a perfect square because it is equal to 5 squared, or 5 × 5.

Recognizing 25 as a perfect square allows us to rewrite the expression, making it easier to simplify later.
When simplifying radical expressions, always check if the number can be expressed in terms of any perfect square factors. This technique helps in breaking down the problem and provides a clearer path to simplification.

Product Property of Square Roots

The product property of square roots is a very useful tool when simplifying expressions. It states that the square root of a product is the same as the product of the square roots of each factor. In mathematical terms:
\[\sqrt{a \times b} = \sqrt{a} \times \sqrt{b}\]Applying this property to our exercise, the problem \(\sqrt{75} = \sqrt{25 \times 3}\) can be further simplified by expressing it as \(\sqrt{25} \times \sqrt{3}\).

• This allows the perfect square factor, \(25\), to be extracted by simplifying \(\sqrt{25}\) into 5.
• The expression then becomes \(5 \times \sqrt{3}\).

By using this property, we make complex root expressions much easier to handle. It splits the expression into manageable parts that can be individually assessed and simplified.

Variable Simplification

When dealing with variables under a radical, simplification can often be achieved straightforwardly. In our example, we deal with \(\sqrt{75a^2}\).
Variables simplify using similar rules as numbers:

• Consider \(a^2\), which is a perfect square of \(a\).
• So, \(\sqrt{a^2} = a\) since \(a\) is a positive real number, ensuring results are also positive.

Now, we have \(\sqrt{75} \times \sqrt{a^2}\). Knowing from previous simplification that \(\sqrt{75} = 5\sqrt{3}\), we can substitute:
We find that the expression \(\sqrt{75a^2}\) simplifies to \(5a\sqrt{3}\).
By deconstructing the expression into simpler components through the use of recognition of perfect squares and the product property, we simplify even complex-seeming variables efficiently.