Question

Simplify each expression. \(\sqrt{50}\)

Short answer

The simplified expression is \(5\sqrt{2}\).

Step-by-step solution

Identify Perfect Squares

The first step is to identify any perfect square factors within the number under the square root symbol. For 50, check if it can be broken down into smaller numbers, including a perfect square. We can write 50 as a product of 25 and 2, where 25 is a perfect square.

Rewrite the Expression

Next, rewrite the square root expression, factoring the number into the product we identified. Therefore, \(\sqrt{50} = \sqrt{25 \times 2}\).

Apply the Square Root Property

The property of square roots states \(\sqrt{a \times b} = \sqrt{a} \times \sqrt{b}\). Apply this property to separate the square root of the perfect square from the rest. Thus, \(\sqrt{50} = \sqrt{25} \times \sqrt{2}\).

Simplify the Perfect Square

Calculate the square root of the perfect square factor, \(\sqrt{25}\), which is 5. Therefore, the expression simplifies to \(5 \times \sqrt{2}\).

Final Simplified Expression

Combine the simplified components to form the final simplified expression: \(5\sqrt{2}\).

Key concepts

Perfect Square

When simplifying square roots, it helps to know about perfect squares. A perfect square is a number that comes from squaring an integer. For example, numbers like 1, 4, 9, 16, and 25 are perfect squares because they are calculated as follows:

• 1 = 1 × 1
• 4 = 2 × 2
• 9 = 3 × 3
• 16 = 4 × 4
• 25 = 5 × 5

When working with square roots, identifying perfect square factors can simplify the process. For instance, in \(\sqrt{50}\), since 25 is a perfect square and a factor of 50, it helps simplify the square root when rewriting it as \(\sqrt{25 \times 2}\). Recognizing these perfect square components means fewer calculations and less complication during simplification. The aim is to extract the perfect square root and reduce the number under the square root symbol.

Square Root Property

The square root property is a central concept in simplifying square roots. The property states that \(\sqrt{a \times b} = \sqrt{a} \times \sqrt{b}\). This property allows you to break down more complex square roots into simpler parts.
For instance, with \(\sqrt{50}\), this property lets us express it as \(\sqrt{25 \times 2}\), which can be further simplified using the property as \(\sqrt{25} \times \sqrt{2}\). One of the square roots becomes a whole number - directly simplifying the expression.
This separation is handy, especially when dealing with larger numbers or when one factor is a perfect square, as it simplifies arithmetic considerably. Understanding this property makes the process intuitive and systematic.

Factoring

Factoring is a foundational step when simplifying square roots. Factoring involves breaking down a number into its largest possible perfect square and its other factor.

• To simplify \(\sqrt{50}\), you factor 50 into 25 and 2.
• Among these, 25 is a perfect square.

So, you rewrite \(\sqrt{50}\) using these factors: it becomes \(\sqrt{25 \times 2}\).
This process of factoring makes the simplification easier because you're able to apply the square root to a known perfect square, as we did here by breaking it down into \(\sqrt{25}\times \sqrt{2}\). In essence, factoring reduces the problem, turning a complex root into a simpler and more manageable expression.