remove all perfect squares from the square root of 108a to the power of 6 assume a is positive

Let's simplify the square root of \( 108a^6 \) by removing all perfect squares from inside the square root.

Given expression:
\[
\sqrt{108a^6}
\]

1. **Factor the radicand \( 108a^6 \)**:
   - \( 108 \) can be factored into prime factors: \( 108 = 2^2 \times 3^3 \).
   - \( a^6 \) is already a perfect square because the exponent is even.

   So, the expression inside the square root becomes:
   \[
   \sqrt{2^2 \times 3^3 \times a^6}
   \]

2. **Separate the perfect squares from the square root**:
   - \( 2^2 = 4 \) is a perfect square.
   - \( a^6 \) is also a perfect square (as \( \sqrt{a^6} = a^3 \)).

   So, we can rewrite the expression as:
   \[
   \sqrt{2^2} \times \sqrt{3^3} \times \sqrt{a^6}
   \]

3. **Simplify each square root**:
   - \( \sqrt{2^2} = 2 \)
   - \( \sqrt{3^3} = \sqrt{9 \times 3} = 3\sqrt{3} \)
   - \( \sqrt{a^6} = a^3 \)

   So, the expression simplifies to:
   \[
   2 \times 3 \times a^3 \times \sqrt{3} = 6a^3\sqrt{3}
   \]

The final simplified expression is:
\[
6a^3\sqrt{3}
\]

Would you like more details or have any questions?

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Here are 8 related questions that might interest you:
1. How do you determine which parts of an expression under a square root are perfect squares?
2. Can you simplify the square root of a product of variables and constants?
3. What is the process for simplifying the square root of an expression with even exponents?
4. How do prime factorization and square roots relate to simplifying expressions?
5. What is the square root of a variable raised to an even power?
6. Can you simplify square roots involving odd exponents on variables?
7. How do you simplify a square root involving multiple factors?
8. What are some common mistakes to avoid when simplifying square roots?

**Tip:** Always look for perfect squares when simplifying square roots, as this allows you to remove terms from under the square root for a simpler expression.

Learn how to simplify the square root of 108a^6 by removing perfect squares. This problem involves understanding square roots, perfect squares, exponentiation, and prime factorization. Suitable for high school students studying algebra and advanced arithmetic.

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