Question

multiply.
$(2\sqrt{2} + 1)(4 + 5\sqrt{10})$
simplify your answer as much as possible.

Explanation:

Step1: Apply distributive property (FOIL)

Multiply each term in the first binomial by each term in the second binomial:
$(2\sqrt{2})(4) + (2\sqrt{2})(5\sqrt{10}) + (1)(4) + (1)(5\sqrt{10})$

Step2: Simplify each product

• First term: $(2\sqrt{2})(4) = 8\sqrt{2}$
• Second term: $(2\sqrt{2})(5\sqrt{10}) = 10\sqrt{20} = 10\sqrt{4 \cdot 5} = 10 \cdot 2\sqrt{5} = 20\sqrt{5}$ (simplify $\sqrt{20}$)
• Third term: $(1)(4) = 4$
• Fourth term: $(1)(5\sqrt{10}) = 5\sqrt{10}$

Step3: Combine like terms (no like radicals, so combine constants)

$8\sqrt{2} + 20\sqrt{5} + 4 + 5\sqrt{10}$
Rearrange: $4 + 8\sqrt{2} + 5\sqrt{10} + 20\sqrt{5}$

Answer:

$4 + 8\sqrt{2} + 5\sqrt{10} + 20\sqrt{5}$