Problem 20 - Prealgebra

Question

Simplify each expression by taking as much out from under the radical as possible. You may assume that all variables represent positive numbers $$\sqrt{45 x^{3}}$$

Step-by-Step Solution

Answer

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

Step 1: Simplify the Radicand

Start by factoring the number inside the square root. The number 45 can be factored as 9 and 5, knowing that 9 is a perfect square. Thus, rewrite the expression: \[ \sqrt{45 x^3} = \sqrt{9 \cdot 5 \cdot x^3} \] as \( \sqrt{9 \cdot x^2 \cdot 5 \cdot x} \)

Step 2: Apply the Square Root Property

Apply the property \( \sqrt{a \cdot b} = \sqrt{a} \cdot \sqrt{b} \). We can separate the perfect squares and the remaining factors:\[ \sqrt{9} \cdot \sqrt{x^2} \cdot \sqrt{5x} \]

Step 3: Simplify Each Square Root

Simplify the parts of the expression by taking the square root of each perfect square:\[ \sqrt{9} = 3 \quad \text{and} \quad \sqrt{x^2} = x \]So now the expression becomes:\[ 3x \cdot \sqrt{5x} \]

Step 4: Combine the Simplified Factors

The expression \( 3x \cdot \sqrt{5x} \) cannot be simplified further inside the radical, as 5 is not a perfect square and \( x \) inside the square root is already in its simplest form. Thus, the fully simplified expression is:\[ 3x\sqrt{5x} \]

Key Concepts

Square Roots Simplified

Square roots are incredibly useful as they allow us to reverse the process of squaring a number. When you encounter a square root, you're often looking for a value which, when multiplied by itself, gives you the number inside the root, known as the radicand. For instance, in the expression \( \sqrt{45} \), the square root is asking for a number \( y \) such that \( y \times y = 45 \). Although 45 isn’t a perfect square, it can be broken down into factors to find parts of it that *are* perfect squares.

When simplifying expressions under a square root, these are the steps to follow:

• Begin by factoring the radicand into its basic components.
• Identify and separate any perfect square factors.

This can greatly simplify your calculations and allow you to "take out" as much from under the root as possible, making the expression easier to handle.

The Art of Factoring

Factoring is an essential skill in algebra that involves breaking down numbers or expressions into multiples, which are its factors. For example, the number 45 can be factored into 9 and 5. Here, 9 is particularly special because it is a perfect square. Factoring transforms complicated expressions into more straightforward components that can be easier to work with.
To factor effectively:

• Identify factors that are perfect squares first, as these simplify neatly when under a square root.
• Factor completely, meaning down to the smallest prime numbers or terms, like in our example where \( 45 = 9 \times 5 \).

In algebra, factoring also applies to variables, as seen in \( x^3 = x^2 \times x \). By factoring out powers of variables, it becomes possible to pull them out from the square root as well, vastly simplifying expressions.

Understanding Perfect Squares

Perfect squares are numbers which, when square rooted, yield an integer value. Examples include 1, 4, 9, 16, and so forth. Recognizing and utilizing perfect squares is crucial in simplifying expressions with square roots.
When dissecting an expression under a square root:

• Look for any perfect square factors first; these can be addressed more straightforwardly in calculations.
• Every perfect square, such as 9 or \( x^2 \), can be extracted from under the square root as its base—9 becomes 3.

This technique simplifies the square root drastically. It turns your expression from being cumbersome to being neat and manageable. Perfect squares serve as markers that simplify and streamline algebraic expressions.