Problem 13 - 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 $$\sqrt{12 x^{2}}$$

Step-by-Step Solution

Answer

\(2x\sqrt{3}\)

Step 1: Factor the expression inside the square root

We start by factoring the expression inside the square root, \(12x^2\). The number 12 can be factored into its prime components: \(12 = 2^2 imes 3\). Therefore, the expression becomes \(\sqrt{2^2 \times 3 \times x^2}\).

Step 2: Apply the square root property

The square root property allows us to simplify the expression by removing perfect squares from under the radical. We recognize \(2^2\) and \(x^2\) as perfect squares. So, we can take these outside the radical: \( \sqrt{2^2 \times 3 \times x^2} = 2x\sqrt{3} \).

Step 3: Simplify the expression

After taking the perfect squares out of the square root in the previous step, our expression is \(2x\sqrt{3}\). There's no further simplification available since \(3\) is not a perfect square.

Key Concepts

Prime Factorization

The journey to simplifying radicals often begins with prime factorization. Prime factorization is the process of breaking down a number into its fundamental building blocks, known as prime numbers. A prime number is simply a number that only has two factors: one and itself. This process is crucial, as it helps us identify any perfect squares that can be removed from under a square root.

To understand how prime factorization works in practical scenarios, take the number 12 as an example. When we factor 12 into primes, we proceed as follows:

• 12 can be divided by 2 to get 6 since 2 is a prime number.
• Then, 6 can be divided by 2 again to get 3.
• At last, 3 is a prime number.

So, the prime factorization of 12 is written as \(2^2 \times 3\). By breaking down numbers this way, you prepare the expression to be simplified using the properties of square roots in the following steps.

Square Root Property

The square root property is a mathematical tool that helps in simplifying radicals by freeing perfect squares. It states that the square root of a product is the product of the square roots of each factor. Simply put, if you have a number inside a square root that can be broken down into perfect squares — a number whose square root is an integer — you can "take it out" of the radical.

Let's apply this property to the expression \(\sqrt{2^2 \times 3 \times x^2}\):

• Notice that \(2^2\) and \(x^2\) are perfect squares.
• The square root of \(2^2\) is 2, and the square root of \(x^2\) is \(x\).
• These can be removed from under the radical, simplifying the expression to \(2x\sqrt{3}\).

Being able to identify and apply this property is significant because it reduces the complexity of a radical expression, allowing for simpler calculations and often a cleaner expression overall.

Perfect Squares

Recognizing perfect squares plays an integral role in simplifying radical expressions. A perfect square is simply an integer that is the square of another integer, such as 1, 4, 9, 16, and so on. Identifying perfect squares under a square root allows you to remove them from the radical sign, which simplifies the expression.

When you're simplifying radicals like \(\sqrt{12x^2}\), understanding what a perfect square looks like helps you spot terms that can be fully simplified. Consider each part of the factored expression \(\sqrt{2^2 \times 3 \times x^2}\):

• \(2^2\) is a perfect square, as it equals 4.
• \(x^2\) is a perfect square because it's the square of \(x\).

You take these perfect squares out of the radical, which gives you the simplified result of \(2x\sqrt{3}\). Identifying and leveraging perfect squares makes the simplification process both quicker and more accurate.