Problem 14 - 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 $$\sqrt{18 x^{2}}$$

Step-by-Step Solution

Answer

The simplified form is \(3x \sqrt{2}\).

Step 1: Prime Factorization

Begin by finding the prime factors of the number under the square root. The number 18 can be written as a product of its prime factors: \(18 = 2 \times 3^2\).

Step 2: Use the Property of Square Roots

Apply the property of square roots: \(\sqrt{a \times b} = \sqrt{a} \times \sqrt{b}\). For \(\sqrt{18x^2}\), write it as \(\sqrt{2 \times 3^2 \times x^2}\).

Step 3: Simplify the Expression

Identify terms under the square root that are perfect squares. In \(\sqrt{2 \times 3^2 \times x^2}\), both \(3^2\) and \(x^2\) are perfect squares. Extract these from under the square root: \(3\) and \(x\).

Step 4: Compute the Final Answer

Multiply the terms outside the square root. The simplified form is \(3x \sqrt{2}\).

Key Concepts

Prime Factorization

Prime factorization is a method of expressing a number as a product of its prime numbers. Prime numbers are numbers greater than 1 that have no divisors other than 1 and themselves.
The prime factorization is like breaking down a composite number into its simplest building blocks. For example, let's take the number 18.
We divide 18 by the smallest prime number, 2, and because it is divisible, we get 18 = 2 x 9. However, 9 is not yet a prime number.
So, we keep breaking it down: 9 = 3 x 3. We now have 18 = 2 x 3 x 3 or better noted as, 18 = 2 x 3^2. This expression shows the prime factorization of 18.

• Prime factorization identifies and isolates prime numbers.
• It's essential in simplifying radical expressions.

Understanding prime factorization prepares us to manage and simplify expressions under radicals effectively.

Perfect Squares

A perfect square is a number that can be expressed as the product of an integer with itself. In the realm of radicals, identifying perfect squares simplifies our calculations significantly.
Consider numbers like 4, 9, and 16. These numbers are perfect squares because they result from 2 times 2, 3 times 3, and 4 times 4, respectively.
When simplifying radicals, if a term under the radical is a perfect square, you can "pull it out" of the radical easily.
Let's use our example of 18 = 2 x 3^2 again. Notice that 3^2 is a perfect square. Hence, it turns into 3 when brought outside the radical.

• Recognizing perfect squares simplifies calculations under the radical.
• They create straightforward opportunities to simplify expressions.

Incorporating perfect squares into your calculation can drastically reduce errors and make your final expression more manageable.

Simplifying Radicals

Simplifying radicals involves rewriting them in their simplest form. It's like cleaning up a messy room to make it more organized! In our problem, we start with \( \sqrt{18x^2} \).
First, we apply prime factorization and identify any perfect squares. In this example, 18 is broken down to 2 x 3^2 and \( x^2 \) is already a perfect square.
Next, we apply the square root property \( \sqrt{a \times b} = \sqrt{a} \times \sqrt{b} \). This lets us separate \( \sqrt{2 x 3^2 x x^2} \) into \( \sqrt{2} \times \sqrt{3^2} \times \sqrt{x^2} \).
We can "pull out" the perfect squares; \( \sqrt{3^2} = 3 \) and \( \sqrt{x^2} = x \), simplifying the expression to \( 3x \sqrt{2} \).

• Breaking down radicals into simpler components makes them more manageable.
• Understanding these techniques provides clarity in solving complex expressions.

Learning this skill can make math problems less daunting and more enjoyable to solve.