Math Problem Statement

Simplify \(\sqrt{\frac{1}{2}}\) and \(\sqrt{\frac{1}{3}}\).

Solution

The expressions in the image are:

  1. 12\sqrt{\frac{1}{2}}
  2. 13\sqrt{\frac{1}{3}}

Let’s simplify each of these.

  1. Simplifying 12\sqrt{\frac{1}{2}}: 12=12=12\sqrt{\frac{1}{2}} = \frac{\sqrt{1}}{\sqrt{2}} = \frac{1}{\sqrt{2}} To rationalize the denominator: 12×22=22\frac{1}{\sqrt{2}} \times \frac{\sqrt{2}}{\sqrt{2}} = \frac{\sqrt{2}}{2} So, 12=22\sqrt{\frac{1}{2}} = \frac{\sqrt{2}}{2}.

  2. Simplifying 13\sqrt{\frac{1}{3}}: 13=13=13\sqrt{\frac{1}{3}} = \frac{\sqrt{1}}{\sqrt{3}} = \frac{1}{\sqrt{3}} To rationalize the denominator: 13×33=33\frac{1}{\sqrt{3}} \times \frac{\sqrt{3}}{\sqrt{3}} = \frac{\sqrt{3}}{3} So, 13=33\sqrt{\frac{1}{3}} = \frac{\sqrt{3}}{3}.

Let me know if you would like further clarification on these steps.


Here are five related questions for further exploration:

  1. How would you simplify 14\sqrt{\frac{1}{4}} or other similar fractions?
  2. What happens if you take ab\sqrt{\frac{a}{b}} for different values of aa and bb?
  3. How do you rationalize the denominator of 15\frac{1}{\sqrt{5}}?
  4. How would these expressions change if the square root were cubed?
  5. Can you apply this simplification technique to complex fractions?

Tip: Rationalizing the denominator is a common technique to simplify expressions, making them easier to interpret in both exact and decimal forms.

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Math Problem Analysis

Mathematical Concepts

Algebra
Square Roots
Rationalization

Formulas

Square root of a fraction: \(\sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}}\)
Rationalization of \(\frac{1}{\sqrt{b}}\): \(\frac{1}{\sqrt{b}} \times \frac{\sqrt{b}}{\sqrt{b}} = \frac{\sqrt{b}}{b}\)

Theorems

Properties of square roots

Suitable Grade Level

Grades 8-10