To determine which options are equivalent to $\sqrt{108}$ feet, we simplify $\sqrt{108}$ and compare it to the given choices.

Step 1: Simplify $\sqrt{108}$

Prime factorize $108$: $108 = 2^2 \cdot 3^3$

Using the square root property: $\sqrt{108} = \sqrt{2^2 \cdot 3^3} = 2 \cdot 3 \cdot \sqrt{3} = 6\sqrt{3}.$

Thus, $\sqrt{108} = 6\sqrt{3}$.

Step 2: Compare with the options

We need to determine which of the choices simplify to $6\sqrt{3}$:

• A. $6\sqrt{3}$: This is identical to $\sqrt{108}$. Correct.

• B. $12\sqrt{3}$: This is twice as large as $6\sqrt{3}$. Incorrect.

• C. $3\sqrt{12}$: Simplify $\sqrt{12}$: $\sqrt{12} = \sqrt{4 \cdot 3} = 2\sqrt{3}.$ Substituting: $3\sqrt{12} = 3 \cdot 2\sqrt{3} = 6\sqrt{3}.$ This matches $\sqrt{108}$. Correct.

• D. $9\sqrt{3}$: This is larger than $6\sqrt{3}$. Incorrect.

• E. $3\sqrt{9}$: Simplify $\sqrt{9}$: $\sqrt{9} = 3.$ Substituting: $3\sqrt{9} = 3 \cdot 3 = 9.$ This does not match $6\sqrt{3}$. Incorrect.

Final Answer:

The correct choices are: A. $6\sqrt{3}$ and C. $3\sqrt{12}$.

Would you like further clarification or a breakdown of any specific step?

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5 Related Questions:

1. How do you simplify square roots involving prime factorization?
2. Why is $3\sqrt{12}$ equivalent to $6\sqrt{3}$?
3. What are other examples of simplifying square root expressions like $\sqrt{108}$?
4. How can you compare expressions like $\sqrt{108}$ and $9\sqrt{3}$ without a calculator?
5. What is the importance of factoring in simplifying square roots?

Tip:

When simplifying square roots, always look for perfect squares in the factorization to simplify efficiently.