Without using a calculator, find the square root of 3power4times5power2times11power6, leaving your answer in standard form

We are asked to find the square root of:

\[
3^4 \times 5^2 \times 11^6
\]

### Step 1: Apply the square root to the product
We know that the square root of a product is the product of the square roots, so:

\[
\sqrt{3^4 \times 5^2 \times 11^6} = \sqrt{3^4} \times \sqrt{5^2} \times \sqrt{11^6}
\]

### Step 2: Simplify each square root
- \(\sqrt{3^4} = 3^{\frac{4}{2}} = 3^2 = 9\)
- \(\sqrt{5^2} = 5^{\frac{2}{2}} = 5^1 = 5\)
- \(\sqrt{11^6} = 11^{\frac{6}{2}} = 11^3 = 1331\)

### Step 3: Multiply the results
Now, multiply the simplified terms:

\[
9 \times 5 \times 1331
\]

Let's break it down:
- \(9 \times 5 = 45\)
- \(45 \times 1331 = 59895\)

### Final Answer:
The square root of \(3^4 \times 5^2 \times 11^6\) is:

\[
\boxed{59895}
\]

Would you like any further clarification or details on any of the steps?

Here are 5 related questions to expand your understanding:
1. What is the square root of \(2^6 \times 7^4\)?
2. How would you simplify \(\sqrt{4^5 \times 6^2}\)?
3. How do you calculate powers of a product under a square root?
4. What is the cube root of \(2^6 \times 3^3 \times 5^9\)?
5. How do you multiply large numbers mentally or by hand?

**Tip:** When taking square roots of products of powers, simplify the exponent by halving it to make calculation easier!

Without using a calculator, find the square root of 3^4 * 5^2 * 11^6, leaving your answer in standard form.

Learn how to find the square root of 3^4 * 5^2 * 11^6 without a calculator. This solution covers the key mathematical concepts of exponents, square roots, and multiplication of powers, providing a detailed explanation suitable for Grades 9-11.

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