To simplify the expression \( \sqrt{135x^8} \), we can break it down as follows:
First, simplify \( 135 \):
\[ 135 = 9 \times 15 = 3^2 \times 15 \]
Next, we simplify \( x^8 \):
Now, rewriting the square root:
\[ \sqrt{135x^8} = \sqrt{(3^2 \times 15) \times x^8} \]
Using the property of square roots \( \sqrt{a \times b} = \sqrt{a} \times \sqrt{b} \):
\[ \sqrt{135x^8} = \sqrt{3^2} \times \sqrt{15} \times \sqrt{x^8} \]
Now, calculate each part:
\[ \sqrt{3^2} = 3, \quad \sqrt{15} \text{ remains as } \sqrt{15}, \quad \text{and } \sqrt{x^8} = x^4 \]
Putting it all together:
\[ \sqrt{135x^8} = 3 \times x^4 \times \sqrt{15} = 3x^4\sqrt{15} \]
From the options given, it doesn't seem like we have a match directly for \( 3x^4\sqrt{15} \) based on the simplifications made.
Let's evaluate the options:
- \( 3x^4 \cdot 35 \) (not correct)
- \( 3x\sqrt{7x^2} \)
- \( 3x^2\sqrt{5x^2} \)
- \( 5x^2 \cdot 3x \) (not correct)
The most closely related option, based on our simplification, isn't explicitly listed here. However, if we consider possible errors in the question or formatting, the correct output of our simplification remains \( 3x^4\sqrt{15} \).
Thus, the answer to simplify \( \sqrt{135x^8} \) is \( 3x^4\sqrt{15} \).
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