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Chapter 15: Problem 10

Simplify. $$\sqrt{50}$$

Short Answer

Expert verified
\(5\sqrt{2}\)

Step by step solution

01

Identifying Perfect Square Factors

Identify the perfect square factors of 50. The perfect square numbers that divide 50 exactly are 1 and 25. Among these, 25 is the largest perfect square factor.
02

Breaking Down the Square Root

Split the square root of 50 into the square root of its factors. This can be done using the property \(\sqrt{a*b} = \sqrt{a} * \sqrt{b}\). In this case, split \(\sqrt{50}\) into \(\sqrt{25}*\sqrt{2}\).
03

Simplify the Square Root

Now, simplify the square roots. The square root of 25 is 5, and the square root of 2 can't be simplified. Therefore, the answer is \(5*\sqrt{2}\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Understanding Perfect Square Factors
When dealing with square roots, one of the essential strategies involves identifying perfect square factors. These are numbers that can be expressed as the square of an integer. For instance, perfect squares like 1, 4, 9, 16, 25, and so on, often come in handy during simplification.

In the case of simplifying \(\sqrt{50}\), first, recognize that the largest perfect square factor of 50 is 25. Breaking down factors into perfect squares makes the simplification process easier. Here, 50 can be divided into 25 and 2, where 25 is the perfect square factor. Being able to spot such factors simplifies both the calculation and understanding.
Utilizing Square Root Properties
Square root properties make it easier to break down complex expressions. A fundamental property is \(\sqrt{a*b} = \sqrt{a} \cdot \sqrt{b}\). This property allows one to separate a square root into more manageable parts based on its factors.
  • In our scenario with \(\sqrt{50}\), apply this property to split it as \(\sqrt{25} \cdot \sqrt{2}\).
  • By doing so, you can manage each square root individually.
This property is important because it doesn't matter in what order the factors are multiplied under the square root; it yields the same result, making it universally applicable for simplification tasks.
Simplification of Radicals Made Easy
Simplification of radicals involves bringing square roots into their simplest form. Once you've identified the perfect square factors and separated the square root into parts using the properties, the next step is simplification.

For example, simplifying \(\sqrt{25} \cdot \sqrt{2}\) involves taking the square root of each factor. Because \(\sqrt{25} = 5\), this reduces the expression to \(5 \cdot \sqrt{2}\).
  • The \(\sqrt{2}\) remains unaffected as it can't be simplified further.
  • Thus, \(5\sqrt{2}\) is the simplest form of \(\sqrt{50}\).
Through this process, you ensure that your answer is concise and fully simplified, maximizing clarity in mathematical expressions.

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