Problem 22 - Elementary and 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 Perfect Squares

First, let's break down 50 into its prime factors to identify any perfect squares. The prime factorization of 50 is 2 × 5 × 5. Notice that 5 × 5 = 25, which is a perfect square.

Step 2: Rewrite the Radical Expression

Express the radicand (the number inside the square root) by splitting it into two components: \(\sqrt{50x^2} = \sqrt{25 \times 2 \times x^2}\). We've split 50 into 25 (a perfect square) and 2, and consider \(x^2\) separately since any whole square can be easily taken out of the square root.

Step 3: Simplify the Square Roots

Now we take the square root of each individual component: \(\sqrt{25} = 5\), \(\sqrt{2}\) remains under the radical as it is not a perfect square, and \(\sqrt{x^2} = x\).Thus, the expression simplifies to \(5x\sqrt{2}\).

Key Concepts

Perfect Squares

A perfect square is a number that can be expressed as the product of an integer with itself. In other words, if you multiply any integer by itself, the result is a perfect square. For example, numbers like 1, 4, 9, 16, 25, and so on, are perfect squares, since they are squares of the integers 1, 2, 3, 4, 5, etc.
Recognizing perfect squares is important when simplifying radical expressions.
It allows you to "pull out" the square root of the perfect square number, simplifying the expression.
For instance, in the expression \(\sqrt{25x^2}\), 25 is a perfect square because it is 5 squared.

Prime Factorization

Prime factorization is the process of breaking down a number into its basic building blocks, which are prime numbers.
A prime number is any whole number greater than 1 that cannot be divided evenly by any other number except 1 and itself.
For example, the prime factorization of 50 is different because you can break it down into 2 and 5, creating the expression \(2 \times 5 \times 5\).
Prime factorizing can help in simplifying radicals, as it allows you to see perfect squares more clearly. Identify pairs of the same factor to "take out" of a square root.
In \(\sqrt{50x^2}\), identifying \(5 \times 5\) as a perfect square leads to simplifying the expression.

Radicals

A radical expression involves a root, such as a square root, cube root, etc. Radicals are commonly represented using the symbol \(\sqrt{\ }\) for square roots.
Simplifying a radical expression usually involves pulling out any perfect squares or cubes from under the radical sign.
When simplifying a radical, separate the expression into parts that can be easily manipulated.
For the expression \(\sqrt{50x^2}\), the square root is applied to both 50 and \(x^2\). Since \(x^2\) is a perfect square, it simplifies to \(x\). The number 50 can be broken into 25 and 2, allowing 25 to be simplified out of the radical as 5. Hence, you get the simplified form \(5x\sqrt{2}\).
Working with radicals becomes straightforward when you recognize and apply these simplifications.