Question

for this quiz, answer each question then s

you have 20 minutes to complete the quiz.

you have only one attempt.

question 5

simplify the expression.

\frac{\sqrt{6}+5\sqrt{2}}{-2+7\sqrt{2}}

\bigcirc \frac{\sqrt{6}-5\sqrt{2}-7\sqrt{3}+35}{22}
\bigcirc \frac{-6\sqrt{6}-6\sqrt{14}-5\sqrt{15}-5\sqrt{35}}{4}
\bigcirc \frac{\sqrt{6}+7\sqrt{3}+5\sqrt{2}+35}{47}
\bigcirc \frac{150+35\sqrt{10}-30\sqrt{5}-35\sqrt{2}}{82}

Explanation:

⚡ Using what you learned: Rationalizing Denominators

Step 1: Multiply by the conjugate

To rationalize the denominator of \(\frac{\sqrt{6} + 5\sqrt{2}}{-2 + 7\sqrt{2}}\), multiply the numerator and the denominator by the conjugate of the denominator, which is \(-2 - 7\sqrt{2}\):

$$ \frac{(\sqrt{6} + 5\sqrt{2})(-2 - 7\sqrt{2})}{(-2 + 7\sqrt{2})(-2 - 7\sqrt{2})} $$

Step 2: Simplify the denominator

Use the difference of squares formula \((a-b)(a+b) = a^2 - b^2\):

$$ (-2)^2 - (7\sqrt{2})^2 = 4 - (49 \cdot 2) = 4 - 98 = -94 $$

Step 3: Expand the numerator

Distribute the terms in the numerator:

$$ (\sqrt{6} + 5\sqrt{2})(-2 - 7\sqrt{2}) = -2\sqrt{6} - 7\sqrt{12} - 10\sqrt{2} - 35(2) $$

Simplify \(\sqrt{12} = 2\sqrt{3}\):

$$ -2\sqrt{6} - 7(2\sqrt{3}) - 10\sqrt{2} - 70 = -2\sqrt{6} - 14\sqrt{3} - 10\sqrt{2} - 70 $$

Step 4: Combine and simplify the fraction

Combine the simplified numerator and denominator:

$$ \frac{-2\sqrt{6} - 14\sqrt{3} - 10\sqrt{2} - 70}{-94} $$

Divide every term in the numerator and denominator by \(-2\):

$$ \frac{\sqrt{6} + 7\sqrt{3} + 5\sqrt{2} + 35}{47} $$

Answer:

$$ \frac{\sqrt{6} + 7\sqrt{3} + 5\sqrt{2} + 35}{47} $$

(This corresponds to the third option)