Question

multiply and simplify.

\\(\sqrt{2} + 2\sqrt{21})(\sqrt{3} - 5\sqrt{2})\\

\\(\sqrt{2} + 2\sqrt{21})(\sqrt{3} - 5\sqrt{2}) = \square\\
(simplify your answer. type an exact answer, using radicals as needed.)

Explanation:

⚡ Using what you learned: Roots and Radicals

Step 1: Distribute the terms (FOIL)

Apply the distributive property to multiply the binomials:

$$ (\sqrt{2} + 2\sqrt{21})(\sqrt{3} - 5\sqrt{2}) $$

$$ = \sqrt{2} \cdot \sqrt{3} - \sqrt{2} \cdot 5\sqrt{2} + 2\sqrt{21} \cdot \sqrt{3} - 2\sqrt{21} \cdot 5\sqrt{2} $$

Step 2: Multiply and simplify each term

• First term:

$$ \sqrt{2} \cdot \sqrt{3} = \sqrt{6} $$

• Second term:

$$ -5(\sqrt{2} \cdot \sqrt{2}) = -5(2) = -10 $$

• Third term:

$$ 2\sqrt{21 \cdot 3} = 2\sqrt{63} = 2\sqrt{9 \cdot 7} = 2 \cdot 3\sqrt{7} = 6\sqrt{7} $$

• Fourth term:

$$ -10\sqrt{21 \cdot 2} = -10\sqrt{42} $$

Step 3: Combine the simplified terms

Combine all the simplified parts together:

$$ \sqrt{6} - 10 + 6\sqrt{7} - 10\sqrt{42} $$

Rearranging the terms:

$$ -10 + \sqrt{6} + 6\sqrt{7} - 10\sqrt{42} $$

Answer:

$$ -10 + \sqrt{6} + 6\sqrt{7} - 10\sqrt{42} $$