Problem 27 - College Algebra with Modeling and Visualization

Question

Simplify the expression. Assume that all variables are positive. $$ \sqrt{3 x} \cdot \sqrt{12 x} $$

Step-by-Step Solution

Answer

The simplified expression is \( 6x \).

Step 1: Use the Product Property of Square Roots

The expression \( \sqrt{3x} \cdot \sqrt{12x} \) can be simplified using the product property of square roots, which states \( \sqrt{a} \cdot \sqrt{b} = \sqrt{a \cdot b} \). Therefore, combine the expressions inside the square roots as follows: \( \sqrt{3x \cdot 12x} \). This simplifies to \( \sqrt{36x^2} \).

Step 2: Simplify the Square Root

Now, simplify \( \sqrt{36x^2} \). Since 36 is a perfect square, and \( x^2 \) also has a perfect square root, take the square root of each separately: \( \sqrt{36} = 6 \) and \( \sqrt{x^2} = x \). Thus, \( \sqrt{36x^2} = 6x \).

Key Concepts

Square Roots

Square roots are a fundamental concept in mathematics. They allow you to determine a number which, when multiplied by itself, yields the original number. For instance, the square root of 9 is 3, because 3 multiplied by 3 gives 9. Square roots are denoted using the radical symbol \( \sqrt{} \). Understanding square roots is essential for simplifying expressions, particularly when you have terms under the square root that can be simplified further.

In algebra, it is common to operate with variables alongside numbers. For example, \( \sqrt{36x^2} \) combines numerical values and variables. To simplify such an expression, finding the square root involves considering both the constant (like 36) and the variable component (like \( x^2 \)). This is especially practical when using the square root to simplify complex equations or expressions.

Product Property

The product property of square roots is a powerful tool in simplifying expressions. It states: \( \sqrt{a} \cdot \sqrt{b} = \sqrt{a \cdot b} \). Essentially, this property allows you to combine two separate square root expressions into one by multiplying what's inside the radicals together.

This property is particularly useful when faced with expressions like \( \sqrt{3x} \cdot \sqrt{12x} \). According to the product property, this can be rewritten as \( \sqrt{3x \cdot 12x} \), which further simplifies the problem-solving process. Thus, you only need to take one square root after you've multiplied the contents together, reducing the steps necessary to reach your solution.

Using the product property correctly can simplify the task of dealing with more complicated expressions, as it allows you to handle multiple square roots efficiently. Apply this property whenever you encounter a situation with products under separate square roots to streamline your calculations.

Perfect Squares

Perfect squares are numbers whose square roots are integers. For example, 16 is a perfect square since \( \sqrt{16} = 4 \), and 25 is another perfect square because \( \sqrt{25} = 5 \). Recognizing perfect squares is crucial when simplifying square root expressions as they help identify terms that can be easily reduced.

In variables, the principle remains the same. For instance, \( x^2 \) is a perfect square because its square root is simply \( x \). When simplifying expressions like \( \sqrt{36x^2} \), understanding that 36 and \( x^2 \) are both perfect squares allows you to break them down into their components: \( \sqrt{36} = 6 \) and \( \sqrt{x^2} = x \).
This knowledge turns what might seem like a complex expression into something straightforward as each part reduces to its simplest form—making algebraic manipulations and simplifications much more straightforward.