Questions: Multiply. √3(√15-3√7) Simplify your answer as much as possible.

Multiply. √3(√15-3√7) Simplify your answer as much as possible.
Transcript text: Multiply. \[ \sqrt{3}(\sqrt{15}-3 \sqrt{7}) \] Simplify your answer as much as possible.
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Solution

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Solution Steps

Step 1: Distribute the Square Root

To solve the expression \(\sqrt{3}(\sqrt{15} - 3\sqrt{7})\), we first distribute \(\sqrt{3}\) across the terms inside the parentheses:

\[ \sqrt{3} \cdot \sqrt{15} - \sqrt{3} \cdot 3\sqrt{7} \]

Step 2: Simplify Each Term

Simplify each term by multiplying the radicands:

  1. \(\sqrt{3} \cdot \sqrt{15} = \sqrt{3 \times 15} = \sqrt{45}\)
  2. \(\sqrt{3} \cdot 3\sqrt{7} = 3\sqrt{3 \times 7} = 3\sqrt{21}\)
Step 3: Further Simplify the Radicals

Break down the radicands into their square factors:

  1. \(\sqrt{45} = \sqrt{9 \times 5} = \sqrt{9} \cdot \sqrt{5} = 3\sqrt{5}\)
  2. \(\sqrt{21}\) cannot be simplified further.
Step 4: Combine the Simplified Terms

Combine the simplified terms:

\[ 3\sqrt{5} - 3\sqrt{21} \]

Final Answer

The simplified expression is:

\[ \boxed{3\sqrt{5} - 3\sqrt{21}} \]

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