Question

simplify the radical expression.

1. $(sqrt{21})(sqrt{24})$

Explanation:

Step1: Combine radicals via product rule

$\sqrt{21} \times \sqrt{24} = \sqrt{21 \times 24}$

Step2: Factor into prime components

$21 \times 24 = 3 \times 7 \times 2^3 \times 3 = 2^3 \times 3^2 \times 7$

Step3: Simplify the radical

$\sqrt{2^3 \times 3^2 \times 7} = \sqrt{2^2 \times 2 \times 3^2 \times 7} = 2 \times 3 \times \sqrt{2 \times 7}$

Step4: Calculate remaining product

$2 \times 3 \times \sqrt{14} = 6\sqrt{14}$

Answer:

$6\sqrt{14}$