Question

question 3

simplify the expression.

\frac{5+2\sqrt{5}}{\sqrt{3}+7\sqrt{7}}

\bigcirc \frac{-2\sqrt{6}+6\sqrt{3}-10+15\sqrt{2}}{7}
\bigcirc \frac{-28\sqrt{7}-49\sqrt{14}+4\sqrt{3}+7\sqrt{6}}{82}
\bigcirc \frac{-5\sqrt{3}+35\sqrt{7}-2\sqrt{15}+14\sqrt{35}}{340}
\bigcirc \frac{5\sqrt{3}-2\sqrt{15}+35\sqrt{7}-14\sqrt{35}}{5}

Explanation:

⚡ Using what you learned: Rationalizing Denominators

Step 1: Identify the conjugate

To eliminate the radicals from the denominator, multiply both the numerator and the denominator by the conjugate of the denominator, which is \(\sqrt{3} - 7\sqrt{7}\).

Step 2: Simplify the denominator

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

$$ (\sqrt{3} + 7\sqrt{7})(\sqrt{3} - 7\sqrt{7}) = (\sqrt{3})^2 - (7\sqrt{7})^2 $$

$$ = 3 - 49 \cdot 7 = 3 - 343 = -340 $$

Step 3: Expand the numerator

Multiply the terms in the numerator:

$$ (5 + 2\sqrt{5})(\sqrt{3} - 7\sqrt{7}) = 5\sqrt{3} - 35\sqrt{7} + 2\sqrt{15} - 14\sqrt{35} $$

Step 4: Combine and simplify the fraction

Write the expression with the simplified denominator:

$$ \frac{5\sqrt{3} - 35\sqrt{7} + 2\sqrt{15} - 14\sqrt{35}}{-340} $$

Multiply the numerator and the denominator by \(-1\) to move the negative sign from the denominator:

$$ \frac{-5\sqrt{3} + 35\sqrt{7} - 2\sqrt{15} + 14\sqrt{35}}{340} $$

Answer:

$$ \frac{-5\sqrt{3}+35\sqrt{7}-2\sqrt{15}+14\sqrt{35}}{340} $$

(This corresponds to the third option)