Question

multiply and simplify.

\\((7-2\sqrt{6})(6+6\sqrt{6})\\)

\\((7-2\sqrt{6})(6+6\sqrt{6}) = \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:

$$ (7 - 2\sqrt{6})(6 + 6\sqrt{6}) = 7(6) + 7(6\sqrt{6}) - 2\sqrt{6}(6) - 2\sqrt{6}(6\sqrt{6}) $$

Step 2: Multiply individual terms

Calculate each product:

• First: \( 7 \cdot 6 = 42 \)
• Outer: \( 7 \cdot 6\sqrt{6} = 42\sqrt{6} \)
• Inner: \( -2\sqrt{6} \cdot 6 = -12\sqrt{6} \)
• Last: \( -2\sqrt{6} \cdot 6\sqrt{6} = -12 \cdot 6 = -72 \)

Combine these results:

$$ 42 + 42\sqrt{6} - 12\sqrt{6} - 72 $$

Step 3: Combine like terms

Group the integer terms and the radical terms:

• Integers: \( 42 - 72 = -30 \)
• Radicals: \( 42\sqrt{6} - 12\sqrt{6} = 30\sqrt{6} \)

Combine them to get the final simplified expression:

$$ -30 + 30\sqrt{6} $$

Answer:

$$-30 + 30\sqrt{6}$$