Question

multiply and simplify.

\\(\sqrt{2} + 3\sqrt{35})(\sqrt{7} - 4\sqrt{2})\\

\\(\sqrt{2} + 3\sqrt{35})(\sqrt{7} - 4\sqrt{2}) = \square\\
(simplify your answer. type an exact answer, using radicals as needed.)

Explanation:

⚡ Using what you learned: Roots and Radicals

Step 1: Expand the expression

Apply the FOIL method to multiply the binomials:

$$ (\sqrt{2} + 3\sqrt{35})(\sqrt{7} - 4\sqrt{2}) $$

$$ = \sqrt{2} \cdot \sqrt{7} - \sqrt{2} \cdot 4\sqrt{2} + 3\sqrt{35} \cdot \sqrt{7} - 3\sqrt{35} \cdot 4\sqrt{2} $$

Step 2: Multiply the terms

Simplify each product using the rule \(\sqrt{a} \cdot \sqrt{b} = \sqrt{ab}\):

• First: \(\sqrt{2} \cdot \sqrt{7} = \sqrt{14}\)
• Outer: \(-4(\sqrt{2} \cdot \sqrt{2}) = -4(2) = -8\)
• Inner: \(3\sqrt{35 \cdot 7} = 3\sqrt{245}\)
• Last: \(-12\sqrt{35 \cdot 2} = -12\sqrt{70}\)

Combine the simplified terms:

$$ \sqrt{14} - 8 + 3\sqrt{245} - 12\sqrt{70} $$

Step 3: Simplify the radicals

Simplify \(\sqrt{245}\) by finding perfect square factors:

$$ \sqrt{245} = \sqrt{49 \cdot 5} = 7\sqrt{5} $$

Substitute this back into the expression:

$$ 3\sqrt{245} = 3(7\sqrt{5}) = 21\sqrt{5} $$

The fully expanded and simplified expression is:

$$ \sqrt{14} - 8 + 21\sqrt{5} - 12\sqrt{70} $$

Rearranging the terms:

$$ -8 + 21\sqrt{5} + \sqrt{14} - 12\sqrt{70} $$

Answer:

$$ -8 + 21\sqrt{5} + \sqrt{14} - 12\sqrt{70} $$