To simplify the expression \( \sqrt[3]{135x^8} \), we start by breaking down the components.
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Factor \( 135 \): \[ 135 = 27 \times 5 = 3^3 \times 5 \]
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Rewrite \( x^8 \): \[ x^8 = (x^3)^2 \cdot x^2 \]
Now, we can rewrite the entire expression inside the cube root:
\[ \sqrt[3]{135x^8} = \sqrt[3]{3^3 \cdot 5 \cdot (x^3)^2 \cdot x^2} \]
Next, we can apply the properties of cube roots:
\[ = \sqrt[3]{3^3} \cdot \sqrt[3]{5} \cdot \sqrt[3]{(x^3)^2} \cdot \sqrt[3]{x^2} \]
Calculating each component:
\[ = 3 \cdot \sqrt[3]{5} \cdot x^2 \cdot \sqrt[3]{x^2} \]
We can combine the cube roots:
\[ = 3x^2 \cdot \sqrt[3]{5x^2} \]
Thus, the simplified expression is:
\[ 3x^2 \sqrt[3]{5x^2} \]
Given the options provided in your question, the expression that corresponds to this result is:
3x^2 \sqrt[3]{5x^2} (which can be interpreted as \( 3x^2 \cdot \sqrt[3]{5} \cdot \sqrt[3]{x^2} \))
Please clarify or check the options again, as they might not directly match, or additional details may be needed to find the right choice among those presented.
1 answer