Question

directions: find each product. give your answer in simplest radical form.

1. $-3\sqrt{20} \cdot 7\sqrt{8}$
2. $\sqrt{xy^2} \cdot \sqrt{x^3y^7}$
3. $\sqrt{2}(\sqrt{20} + \sqrt{8})$
4. $(8 + 2\sqrt{3})(8 - 2\sqrt{3})$

Explanation:

Problem 11

Step1: Multiply coefficients and radicands

$-3 \cdot 7 \cdot \sqrt{20 \cdot 8} = -21 \cdot \sqrt{160}$

Step2: Simplify the radical

$\sqrt{160} = \sqrt{16 \cdot 10} = 4\sqrt{10}$

Step3: Multiply coefficient and simplified radical

$-21 \cdot 4\sqrt{10} = -84\sqrt{10}$

Problem 12

Step1: Combine radicals using product rule

$\sqrt{xy^2 \cdot x^3y^7} = \sqrt{x^4y^9}$

Step2: Simplify the radical

$\sqrt{x^4y^9} = x^2y^4\sqrt{y}$

Problem 14

Step1: Distribute $\sqrt{2}$ to terms inside

$\sqrt{2} \cdot \sqrt{20} + \sqrt{2} \cdot \sqrt{8}$

Step2: Simplify each product

$\sqrt{40} + \sqrt{16} = 2\sqrt{10} + 4$

Problem 15

Step1: Apply difference of squares formula

$(a+b)(a-b)=a^2-b^2$, where $a=8$, $b=2\sqrt{3}$
$8^2 - (2\sqrt{3})^2$

Step2: Calculate each term

$64 - (4 \cdot 3) = 64 - 12$

Step3: Subtract to find the result

$64 - 12 = 52$

Answer:

1. $\boldsymbol{-84\sqrt{10}}$
2. $\boldsymbol{x^2y^4\sqrt{y}}$
3. $\boldsymbol{4 + 2\sqrt{10}}$
4. $\boldsymbol{52}$