Problem 19 - 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 $$\sqrt{50 x^{3}}$$

Step-by-Step Solution

Answer

The simplified expression is \(5x \sqrt{2x}\).

Step 1: Factor the number inside the radical

To simplify the square root, start by factoring the number 50 under the radical. The prime factorization of 50 is \( 50 = 2 imes 5^2 \). Now the expression under the square root is \( \sqrt{2 imes 5^2 \times x^3} \).

Step 2: Apply properties of square roots

The property \( \sqrt{a imes b} = \sqrt{a} \cdot \sqrt{b} \) allows us to separate the expression under the square root into \( \sqrt{2} \cdot \sqrt{5^2} \cdot \sqrt{x^3} \).

Step 3: Simplify each radical term separately

Calculate each term: \( \sqrt{2} \) remains as it is because 2 is not a perfect square. Calculate \( \sqrt{5^2} = 5 \) and \( \sqrt{x^3} = \sqrt{x^2}\cdot\sqrt{x} = x\sqrt{x} \) since \(x^2\) is a perfect square.

Step 4: Combine the simplified terms

Combine all the simplified parts: \( 5x \sqrt{2} \sqrt{x} \), and express the expression as \( 5x \sqrt{2x} \) which is the simplified form of the original expression.

Key Concepts

Understanding Prime Factorization

Prime factorization involves breaking down a number into its basic building blocks, which are the prime numbers that multiply together to give the original number. This is an essential step in simplifying square roots. For example, the number 50 can be factored into its prime components as shown:

• 50 divides evenly by 2 (the smallest prime number), resulting in 25. So, we write 50 as \( 2 \times 25 \).
• Next, 25 is 5 multiplied by itself, or \( 5 \times 5 \), which can be expressed as \( 5^2 \).

Therefore, the prime factorization of 50 is \( 2 \times 5^2 \). Prime factorization helps identify perfect squares in an expression, making it easier to simplify square roots by pulling numbers outside the radical.

Exploring the Properties of Square Roots

The properties of square roots are incredibly useful for breaking down complex square root problems. One primary property to remember is that the square root of a product can be expressed as the product of square roots: \[ \sqrt{a \times b} = \sqrt{a} \cdot \sqrt{b} \] This allows you to split the factors under the square root into individual roots. For instance, in the expression \( \sqrt{2 \times 5^2 \times x^3} \), you can separate it into:

• \( \sqrt{2} \)
• \( \sqrt{5^2} \)
• \( \sqrt{x^3} \)

This step simplifies the expression because it is easier to calculate the square root of each term separately. Specifically, if a factor is a perfect square, such as 5 in \( 5^2 \), it simplifies to just the base number. Additionally, applying this property to \( x^3 \) allows us to break it into perfect squares: \( x^2 \) and \( x \), resulting in \( x \sqrt{x} \).

Perfect Squares and Simplification

Perfect squares are numbers that have an integer as their square root. Recognizing perfect squares in an expression is crucial when simplifying square roots because it allows these pieces to be taken "out" of the square root.

• For example, in \( \sqrt{5^2} \), since 5 is a perfect square, it simplifies directly to 5.
• Similarly, \( x^2 \) inside \( \sqrt{x^3} \) also counts as a perfect square, which becomes \( x \) when simplified.

These simplifications help reduce the complexity of the original expression as much as possible. The final form that only leaves non-perfect square elements under the square root is preferred for simplicity. In the original problem, combining perfect square simplifications results in obtaining \( 5x \sqrt{2x} \), where the perfect squares have been extracted and simplified, leaving a clear and concise expression.