Math Problem Statement

Simplify \(\frac{\sqrt{18}}{\sqrt{7}}\).

Solution

The image shows the mathematical expression 187\frac{\sqrt{18}}{\sqrt{7}}.

To simplify this expression:

  1. Combine the square roots: 187=187\frac{\sqrt{18}}{\sqrt{7}} = \sqrt{\frac{18}{7}}

  2. Simplify 187\frac{18}{7} (if possible). Since 1818 and 77 have no common factors other than 1, the fraction is already in simplest form.

  3. The expression simplifies to: 187\sqrt{\frac{18}{7}}

Alternatively, if you want to rationalize the denominator (expressing it without a square root in the denominator):

  1. Multiply both the numerator and the denominator by 7\sqrt{7}: 187×77=18×77=1267\frac{\sqrt{18}}{\sqrt{7}} \times \frac{\sqrt{7}}{\sqrt{7}} = \frac{\sqrt{18 \times 7}}{7} = \frac{\sqrt{126}}{7}

  2. Simplify 126\sqrt{126}. Since 126=2×32×7126 = 2 \times 3^2 \times 7, we have: 126=2×32×7=314\sqrt{126} = \sqrt{2 \times 3^2 \times 7} = 3\sqrt{14}

  3. The expression becomes: 3147\frac{3\sqrt{14}}{7}

Thus, 187=3147\frac{\sqrt{18}}{\sqrt{7}} = \frac{3\sqrt{14}}{7}.

Would you like further details or have any questions?

Here are 5 related questions to explore further:

  1. How do you rationalize a denominator with a square root?
  2. What are the steps for simplifying square roots of non-perfect squares?
  3. How can you express 18\sqrt{18} in simplest radical form?
  4. What are some methods to simplify complex fractions involving square roots?
  5. How do you multiply and simplify expressions involving square roots?

Tip: When simplifying square roots, always look for perfect square factors to simplify the radical expression.

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Math Problem Analysis

Mathematical Concepts

Simplification of square roots
Rationalizing the denominator

Formulas

\(\frac{\sqrt{a}}{\sqrt{b}} = \sqrt{\frac{a}{b}}\)
Rationalizing \(\frac{\sqrt{a}}{\sqrt{b}} = \frac{\sqrt{a} \cdot \sqrt{b}}{b}\)

Theorems

Properties of square roots

Suitable Grade Level

Grades 9-10