Question

example 4 multiply radical expressions
simplify $5\sqrt{3} \cdot 4\sqrt{6}$.
$5\sqrt{3} \cdot 4\sqrt{6} = (5 \cdot 4)(\sqrt{3} \cdot \sqrt{6})$
$= \\_\\_\\_\\_\\_\\_$
$= 20(3\sqrt{2})$
$= \\_\\_\\_\\_\\_\\_
check
simplify $3\sqrt{10} \cdot (-9\sqrt{6})$
$\\_\\_\\_\\_\\_\\_$

Explanation:

Step1: Multiply coefficients + radicals

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

Step2: Simplify products

$= 20\sqrt{18}$

Step3: Simplify radical $\sqrt{18}$

$\sqrt{18} = \sqrt{9 \cdot 2} = 3\sqrt{2}$, so $20(3\sqrt{2})$

Step4: Multiply remaining terms

$= 20 \cdot 3 \cdot \sqrt{2} = 60\sqrt{2}$

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Step1: Multiply coefficients + radicals (check problem)

$3\sqrt{10} \cdot (-9\sqrt{6}) = (3 \cdot -9)(\sqrt{10 \cdot 6})$

Step2: Simplify products

$= -27\sqrt{60}$

Step3: Simplify radical $\sqrt{60}$

$\sqrt{60} = \sqrt{4 \cdot 15} = 2\sqrt{15}$, so $-27(2\sqrt{15})$

Step4: Multiply remaining terms

$= -27 \cdot 2 \cdot \sqrt{15} = -54\sqrt{15}$

Answer:

For $5\sqrt{3} \cdot 4\sqrt{6}$:

1. Missing blank 1: $20\sqrt{18}$
2. Missing blank 2: $60\sqrt{2}$

For $3\sqrt{10} \cdot (-9\sqrt{6})$:
$ -54\sqrt{15}$