Math Problem Statement

Simplify \(\sqrt{\frac{5}{40}}\).

Solution

The mathematical expression in the image is:

540\sqrt{\frac{5}{40}}

Simplification Process:

  1. Simplify the fraction inside the square root: 540=18(divide numerator and denominator by 5).\frac{5}{40} = \frac{1}{8} \quad \text{(divide numerator and denominator by 5)}.

  2. Write the expression as: 18=18=18.\sqrt{\frac{1}{8}} = \frac{\sqrt{1}}{\sqrt{8}} = \frac{1}{\sqrt{8}}.

  3. Simplify 8\sqrt{8}: 8=42=42=22.\sqrt{8} = \sqrt{4 \cdot 2} = \sqrt{4} \cdot \sqrt{2} = 2\sqrt{2}.

  4. Substitute back: 18=122.\frac{1}{\sqrt{8}} = \frac{1}{2\sqrt{2}}.

  5. Rationalize the denominator: Multiply numerator and denominator by 2\sqrt{2}: 12222=24.\frac{1}{2\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}} = \frac{\sqrt{2}}{4}.

Final Answer:

540=24.\sqrt{\frac{5}{40}} = \frac{\sqrt{2}}{4}.

Would you like further clarifications or step-by-step elaboration?


5 Related Questions:

  1. What does it mean to "rationalize the denominator" and why is it done?
  2. How can square roots of fractions be simplified efficiently?
  3. What is the relationship between simplifying square roots and factoring?
  4. How do you simplify expressions involving square roots in the numerator and denominator?
  5. Can 540\sqrt{\frac{5}{40}} be evaluated as a decimal approximation?

Tip:

Always simplify the fraction inside the square root first, as it often reduces the complexity of calculations!

Ask a new question for Free

By Image

Drop file here or Click Here to upload

Math Problem Analysis

Mathematical Concepts

Simplification of Fractions
Square Roots
Rationalizing Denominators

Formulas

\(\sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}}\)
\(\sqrt{n \cdot m} = \sqrt{n} \cdot \sqrt{m}\)
Rationalizing: \(\frac{1}{\sqrt{n}} \cdot \frac{\sqrt{n}}{\sqrt{n}} = \frac{\sqrt{n}}{n}\)

Theorems

Properties of Square Roots
Simplification Rules for Radicals

Suitable Grade Level

Grades 9-10