Question

quiz-- section 9.6

question 1

simplify the expression.

\\\frac{3\sqrt{12}}{10\sqrt{20}}\\

• \\(\frac{3\sqrt{15}}{50}\\)
• \\(\frac{3\sqrt{240}}{20}\\)
• \\(\frac{3\sqrt{240}}{200}\\)

Explanation:

⚡ Using what you learned: Roots and Radicals · Rationalizing Denominators

Step 1: Simplify the radicals in the numerator and denominator

$$ \sqrt{12} = \sqrt{4 \cdot 3} = 2\sqrt{3} $$

$$ \sqrt{20} = \sqrt{4 \cdot 5} = 2\sqrt{5} $$

Substitute these back into the expression:

$$ \frac{3\sqrt{12}}{10\sqrt{20}} = \frac{3(2\sqrt{3})}{10(2\sqrt{5})} = \frac{6\sqrt{3}}{20\sqrt{5}} $$

Reduce the fraction by dividing the coefficients by \(2\):

$$ \frac{3\sqrt{3}}{10\sqrt{5}} $$

Step 2: Rationalize the denominator

Multiply the numerator and the denominator by \(\sqrt{5}\):

$$ \frac{3\sqrt{3} \cdot \sqrt{5}}{10\sqrt{5} \cdot \sqrt{5}} = \frac{3\sqrt{15}}{10 \cdot 5} = \frac{3\sqrt{15}}{50} $$

Answer:

$$ \frac{3\sqrt{15}}{50} $$