Question
Write the following expression in simplest form. \( \sqrt{3^{2} \cdot 11^{4}} \) \( \sqrt{3^{2} \cdot 11^{4}}=\square \) (Simplify your answer. Type an exact answer, using radicals as needed.)
Ask by Allan Sullivan. in the United States
Jan 24,2025
Real Tutor Solution
Tutor-Verified Answer
Answer
\( \sqrt{3^{2} \cdot 11^{4}} = 363 \)
Solution
To simplify the expression \( \sqrt{3^{2} \cdot 11^{4}} \), follow these steps:
1. **Break down the square root:**
\[
\sqrt{3^{2} \cdot 11^{4}} = \sqrt{3^{2}} \cdot \sqrt{11^{4}}
\]
2. **Simplify each square root:**
\[
\sqrt{3^{2}} = 3
\]
\[
\sqrt{11^{4}} = 11^{2} = 121
\]
3. **Multiply the simplified terms:**
\[
3 \cdot 121 = 363
\]
So, the simplest form of the expression is:
\[
\sqrt{3^{2} \cdot 11^{4}} = 363
\]
**Answer:**
\( 363 \)
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Simplify this solution Beyond the Answer
To simplify the expression \( \sqrt{3^{2} \cdot 11^{4}} \), we can use the property of square roots that states \( \sqrt{a \cdot b} = \sqrt{a} \cdot \sqrt{b} \). So we have: \[ \sqrt{3^{2} \cdot 11^{4}} = \sqrt{3^{2}} \cdot \sqrt{11^{4}} \] Calculating each square root gives: \[ \sqrt{3^{2}} = 3 \quad \text{and} \quad \sqrt{11^{4}} = 11^{2} = 121 \] Therefore, \[ \sqrt{3^{2}} \cdot \sqrt{11^{4}} = 3 \cdot 121 = 363 \] Thus, the simplest form of the expression is: \[ \sqrt{3^{2} \cdot 11^{4}} = 363 \]
