Problem 13 - Elementary and Intermediate Algebra

Question

Simplify each expression. $$ \sqrt{50} $$

Step-by-Step Solution

Answer

\( \sqrt{50} = 5\sqrt{2} \).

Step 1: Identify Prime Factors

The first step is to identify the prime factors of the number under the square root. For 50, let's find its prime factors. Start by dividing 50 by the smallest prime number 2: 50 ÷ 2 = 25 Next, divide 25 by the smallest prime number that goes into it, which is 5: 25 ÷ 5 = 5 Since 5 is a prime number, we stop here. Therefore, the prime factorization of 50 is 2 × 5^2.

Step 2: Apply Square Root to Factors

Now that we have the prime factorization of 50 as 2 × 5^2, we can apply the square root to the factors.The square root of a product can be found by taking the square root of each factor:\[\sqrt{50} = \sqrt{2 \times 5^2} = \sqrt{2} \times \sqrt{5^2}\]

Step 3: Simplify the Expression

Since the square root of a square number like 5^2 is simply 5, we can simplify the expression further:\[\sqrt{5^2} = 5\]So, simplify the expression to:\[\sqrt{50} = \sqrt{2} \times 5 = 5\sqrt{2}\]

Key Concepts

Prime Factorization

Prime factorization is a fundamental mathematical technique used to simplify square roots. Whenever you encounter a number, breaking it down into prime numbers can make things much clearer. Prime numbers are those that only have two divisors: 1 and the number itself. For example, 2, 3, 5, 7, and 11 are considered prime numbers. When simplifying square roots like \(\sqrt{50}\), the first step is to break down 50 into its prime components. This involves dividing the number by the smallest prime number possible, which is usually 2, and continuing with subsequent smaller numbers as needed.Here's how to perform the prime factorization of 50:

• Divide 50 by 2, the smallest prime: 50 ÷ 2 = 25.
• Next, divide 25 by 5: 25 ÷ 5 = 5. Since 5 is a prime number, we stop here.

Thus, 50 factors into \(2 \times 5^2\). Keeping it in mind makes further simplification easier.

Square Root Properties

Understanding the properties of square roots is key to simplifying expressions. One of the most useful properties is that the square root of a product is the product of the square roots of the factors. For instance, knowing that \(\sqrt{ab} = \sqrt{a} \times \sqrt{b}\) helps break down complex roots into simpler parts. When you have the prime factorization like \(2 \times 5^2\) from the previous section, you can apply square roots to each factor separately. This means:

• \(\sqrt{50} = \sqrt{2 \times 5^2}\)
• Use the property: \(\sqrt{2 \times 5^2} = \sqrt{2} \times \sqrt{5^2}\)

By recognizing that \(\sqrt{5^2} = 5\) (since squaring and square rooting cancel each other), it simplifies the expression significantly. This property is invaluable for simplifying radical expressions efficiently.

Radical Expressions

Radical expressions involve roots, typically square roots, cube roots, etc. They can appear daunting, but breaking them down using prime factors and square root properties makes them easier to work with. Simplifying a radical expression like \(\sqrt{50}\) requires combining several techniques:

• Identify prime factors: 50 becomes \(2 \times 5^2\).
• Use square root properties to split the factors: \(\sqrt{2} \times \sqrt{5^2}\).
• Simplify further using known square roots: \(\sqrt{5^2} = 5\).

Hence, \(\sqrt{50}\) simplifies to \(5\sqrt{2}\). Not only does this process streamline solving problems, but it also makes comprehending radical math much simpler for anyone once mastered.