Question

multiply and simplify.

\\((4 - 6\sqrt{5})(3 + 3\sqrt{5})\\)

\\((4 - 6\sqrt{5})(3 + 3\sqrt{5}) = \square\\)
(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 two binomials:

$$ (4 - 6\sqrt{5})(3 + 3\sqrt{5}) = (4)(3) + (4)(3\sqrt{5}) - (6\sqrt{5})(3) - (6\sqrt{5})(3\sqrt{5}) $$

Step 2: Multiply individual terms

Calculate each product:

$$ (4)(3) = 12 $$

$$ (4)(3\sqrt{5}) = 12\sqrt{5} $$

$$ -(6\sqrt{5})(3) = -18\sqrt{5} $$

$$ -(6\sqrt{5})(3\sqrt{5}) = -18 \cdot 5 = -90 $$

Substitute these values back into the expression:

$$ 12 + 12\sqrt{5} - 18\sqrt{5} - 90 $$

Step 3: Combine like terms

Combine the integer terms and the radical terms:

$$ (12 - 90) + (12\sqrt{5} - 18\sqrt{5}) $$

$$ -78 - 6\sqrt{5} $$

Answer:

$$ -78 - 6\sqrt{5} $$