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Chapter 7: Problem 33

Simplify. Assume that all variables represent positive real numbers. \(\sqrt{\frac{x}{25}}\)

Short Answer

Expert verified
\(\frac{\root x }{5} \)

Step by step solution

01

Rewrite the Fraction Under the Radical

Express the fraction inside the square root in a simplified form: \ \( \frac{x}{25} \)
02

Apply the Square Root to the Numerator and Denominator Separately

Use the property of square roots that allows the expression to be written as: \ \( \frac{\root x }{\root 25} \)
03

Simplify the Denominator

Recognize that 25 is a perfect square. Therefore, \( \root 25 \ = 5. \) Substitute this back into the expression to get: \ \( \frac{\root x }{5} \)

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Key Concepts

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

Understanding Square Roots
Square roots are a fundamental concept in mathematics. When you see a square root symbol, like \(\root x\), it asks you to find a number which, when multiplied by itself, gives you the number x.
For example, the square root of 9 is 3 because 3 x 3 equals 9.
Square roots can be tricky, especially when dealing with variables or fractions inside the root.
Here are some key points about square roots to help you understand the process better:
  • The square root of a number is always positive, and by convention, we write the positive root.

  • You can apply square roots to both whole numbers and variables.

  • Square roots can simplify more easily if you recognize perfect squares.

In our exercise, we used the property that the square root of a fraction can be split into the square root of the numerator divided by the square root of the denominator. This property helped us to simplify the problem step by step.
Simplifying Fractions
Simplifying fractions is a useful technique when dealing with mathematical problems, especially those involving square roots. To simplify a fraction, you need to make both the numerator (the top part) and the denominator (the bottom part) as simple as possible.
Here are some steps to keep in mind:
  • Look for common factors between the numerator and the denominator.

  • Divide both parts by the greatest common divisor.

In our problem, we started with \(\frac{x}{25}\). Notice that 25 is already a simple number, but we still split it into simpler parts under the square root. Splitting and simplifying helps you in further reducing expressions and finding the simplest form.
Remember, simplifying fractions is a step-by-step process where precision matters.
What are Perfect Squares?
Perfect squares are numbers that are the product of an integer multiplied by itself. Recognizing perfect squares helps immensely while simplifying radicals. Some common perfect squares are 1, 4, 9, 16, 25, etc.
Notice that 25 is a perfect square because it is the result of 5 x 5.
Here鈥檚 why perfect squares are important:
  • They make it easier to simplify square roots.

  • Recognizing them speeds up the simplifying process.

In our exercise, we identified that 25 = 5 x 5, and hence \(\root 25 = 5\).
Applying this knowledge allows us to simplify more easily.
Always watch out for a perfect square when dealing with radical expressions.

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Most popular questions from this chapter

Rationalize each denominator. Assume that all variables represent positive real numbers and that no denominators are 0. $$ \frac{\sqrt{8}}{3-\sqrt{2}} $$

Find the distance between each pair of points. \((\sqrt{7}, 9 \sqrt{3})\) and \((-\sqrt{7}, 4 \sqrt{3})\)

The following expression occurs in a standard problem in trigonometry. $$ \frac{\sqrt{3}+1}{1-\sqrt{3}} $$ Show that it simplifies to \(-2-\sqrt{3}\). Then verify, using a calculator approximation.

Rationalize each numerator. Assume that all variables represent positive real numbers. $$ \frac{6-\sqrt{3}}{8} $$

Write each quotient in lowest terms. Assume that all variables represent positive real numbers. $$ \frac{12-9 \sqrt{72}}{18} $$
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