Problem 13 - Intermediate Algebra

Question

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

Step-by-Step Solution

Answer

The simplified form of \(\sqrt{50}\) is \(5\sqrt{2}\).

Step 1: Identify Perfect Square Factors

The first step in simplifying a square root is to identify any perfect square factors of the number under the square root. The number 50 can be divided into the factors 25 and 2, where 25 is a perfect square.

Step 2: Apply the Square Root Property

Using the square root property \(\sqrt{a \times b} = \sqrt{a} \times \sqrt{b}\), we separate the square root of 50 into \(\sqrt{25 \times 2}\), which simplifies to \(\sqrt{25} \times \sqrt{2}\).

Step 3: Simplify the Expression

Compute the square root of the perfect square. Since \(\sqrt{25} = 5\), the expression \(\sqrt{25} \times \sqrt{2}\) simplifies to \(5 \times \sqrt{2}\).

Step 4: Write the Simplified Expression

The expression is fully simplified when expressed as \(5\sqrt{2}\), which is its simplest radical form.

Key Concepts

Perfect Square Factors

To simplify square roots, one of the first techniques involves recognizing perfect square factors. A perfect square is a number that can be expressed as the square of an integer. For instance, numbers like 1, 4, 9, 16, 25, etc., are perfect squares. Identifying perfect square factors is crucial because they simplify neatly when under a square root.

• For the number 50, if we break it into factors, we see it as 25 and 2.
• Among these, 25 is a perfect square since it equals 5 times 5.

Recognizing this perfect square helps us streamline the upcoming steps for anyone looking to simplify expressions.

Square Root Property

The square root property is a mathematical principle that simplifies the process of breaking down square roots. It states: \[ \sqrt{a \times b} = \sqrt{a} \times \sqrt{b} \] This property is instrumental in transforming a square root of a product into the product of square roots.For example, using the previous factors of 50:

• You apply this property to \( \sqrt{50} \) by expressing it as \( \sqrt{25 \times 2} \).
• With the property, it breaks down to \( \sqrt{25} \times \sqrt{2} \).

Utilizing this property makes the calculation feasible and introduces an easier way to simplify complex expressions.

Simplified Radical Form

Once we've broken down our square root expression using perfect square factors and the square root property, we arrive at our goal: the simplified radical form. This form is the most concise way to express a square root without using decimals.

• In our example, \( \sqrt{25} \times \sqrt{2} \) translates to \(5 \times \sqrt{2}\).
• Since \( \sqrt{25} \), being a perfect square, equals 5, we reduce the expression to \( 5\sqrt{2} \).

This expression, \(5\sqrt{2}\), represents the simplest form you can achieve without converting \( \sqrt{2} \) into a decimal, thereby preserving its precise mathematical integrity.