Problem 22 - Intermediate Algebra

Question

Simplify each radical expression. All variables represent positive real numbers. $$ \sqrt{50 x^{2}} $$

Step-by-Step Solution

Answer

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

Step 1: Identify the Components Inside the Radical

The expression given is \( \sqrt{50x^2} \). We need to identify the components inside the radical, which are 50 and \( x^2 \).

Step 2: Factor the Radicand

Write 50 as a product of its prime factors: \( 50 = 2 \times 5^2 \). Now, the entire radicand is expressed as \( 2 \times 5^2 \times x^2 \).

Step 3: Simplify the Radical Using Properties

We know that \( \sqrt{a \times b} = \sqrt{a} \times \sqrt{b} \). Apply this to the radical: \( \sqrt{2 \times 5^2 \times x^2} = \sqrt{2} \times \sqrt{5^2} \times \sqrt{x^2} \).

Step 4: Simplify Individual Components

Simplify each component separately:- \( \sqrt{5^2} = 5 \) because the square root and square cancel each other out.- \( \sqrt{x^2} = x \) for the same reason.- \( \sqrt{2} \) remains as it is because 2 is not a perfect square.

Step 5: Combine the Simplified Parts

Combine all the simplified parts: \( \sqrt{2} \times 5 \times x = 5x\sqrt{2} \).

Key Concepts

Simplifying Radicals

Simplifying radicals involves reducing a radical expression to its simplest form. The goal is to make the expression as easy to work with as possible, often by removing any perfect squares from under the square root sign.
To start simplifying a radical:

• Identify and separate perfect squares from the numbers inside the square root.
• Use the property of square roots: \( \sqrt{a \times b} = \sqrt{a} \times \sqrt{b} \). This helps in breaking down the expression into simpler terms.
• Simplify each radical component as much as possible.

In the case of \( \sqrt{50x^2} \), we apply this process:- First, break down the number 50 into its prime factors: 50 is 2 times \(5^2\). The \(x^2\) is already a perfect square.- Then separate the components to simplify individually: \( \sqrt{50x^2} \) becomes \( \sqrt{2 \times 5^2 \times x^2} \).- Use the property: \( \sqrt{2} \times \sqrt{5^2} \times \sqrt{x^2} \).Finally, combining simplified components gets you to the simplest form \( 5x\sqrt{2} \). This is the completed simplified expression.

Prime Factorization

Prime factorization involves breaking down a number completely into prime numbers, which are numbers only divisible by 1 and themselves.
This process is pivotal in simplifying radicals because it helps identify perfect square components.
For example, finding the prime factors of 50:

• Start by dividing the number by the smallest prime, which is 2. Since 50 is even, it's divisible by 2, resulting in 25.
• Then, look at the remainder - 25. It's not divisible by 2, so we move to the next prime: 5.
• Dividing by 5 gives you 5 again, which is a perfect square \(5^2\).

This gives us: \(50 = 2 \times 5^2\). With these factors, it's straightforward to identify components that can be directly simplified under a square root, such as \(\sqrt{5^2} = 5\). This ultimately aids in reducing the entire radical expression efficiently.

Properties of Square Roots

Understanding the properties of square roots is key to simplifying radical expressions better. Square roots come with handy properties that make manipulation possible:

• The property \( \sqrt{a \times b} = \sqrt{a} \times \sqrt{b} \) allows breaking down complex radicands into more manageable parts. This is crucial when simplifying expressions like \(\sqrt{50x^2}\).
• Square root and square operations are inverse: \( \sqrt{a^2} = a \). It works unless the variable represents negative values, but we usually assume variables represent positive numbers in introductory exercises.
• If any perfect square exists in a radicand, it simplifies directly since the square root nullifies the square.

These properties let you dissect and reassemble expressions into their simplest form so effectively. By knowing these properties, we can mix and match elements under the root to get the most simplified result, as seen in the current example.