Question

multiply and simplify.

\\(8\sqrt{2} + \sqrt{3})(5\sqrt{2} - 4\sqrt{3})\\

\\(8\sqrt{2} + \sqrt{3})(5\sqrt{2} - 4\sqrt{3}) = \boxed{\quad}\\
(type an exact answer, using radicals as needed.)

Explanation:

⚡ Using what you learned: Roots and Radicals

Step 1: Distribute the terms

Apply the FOIL method to multiply the two binomial expressions:

$$ (8\sqrt{2} + \sqrt{3})(5\sqrt{2} - 4\sqrt{3}) $$

$$ (8\sqrt{2})(5\sqrt{2}) + (8\sqrt{2})(-4\sqrt{3}) + (\sqrt{3})(5\sqrt{2}) + (\sqrt{3})(-4\sqrt{3}) $$

Step 2: Multiply each pair

Multiply the coefficients and the radicands:

$$ 40\sqrt{4} - 32\sqrt{6} + 5\sqrt{6} - 4\sqrt{9} $$

Step 3: Simplify the perfect squares

Simplify the square roots of perfect squares, \(\sqrt{4} = 2\) and \(\sqrt{9} = 3\):

$$ 40(2) - 32\sqrt{6} + 5\sqrt{6} - 4(3) $$

$$ 80 - 32\sqrt{6} + 5\sqrt{6} - 12 $$

Step 4: Combine like terms

Combine the constant terms and the like radical terms:

$$ (80 - 12) + (-32\sqrt{6} + 5\sqrt{6}) $$

$$ 68 - 27\sqrt{6} $$

Answer:

$$ 68 - 27\sqrt{6} $$