To simplify the expression \(3\sqrt{5a}\cdot 8\sqrt{35a^2}\), we will first combine the coefficients and the square roots separately.
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Multiply the coefficients: \[ 3 \cdot 8 = 24 \]
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Combine the square roots: \[ \sqrt{5a} \cdot \sqrt{35a^2} = \sqrt{(5a)(35a^2)} = \sqrt{175a^3} \]
Now we compute \(175a^3\) to simplify the square root:
- Factor \(175\): \[ 175 = 25 \cdot 7 = 5^2 \cdot 7 \] Therefore, \[ 175a^3 = 25 \cdot 7 \cdot a^3 = 5^2 \cdot 7 \cdot a^2 \cdot a = (5^2)(7)(a^2)(a) \]
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Simplify the square root: \[ \sqrt{175a^3} = \sqrt{5^2 \cdot 7 \cdot a^2 \cdot a} = \sqrt{5^2} \cdot \sqrt{7} \cdot \sqrt{a^2} \cdot \sqrt{a} \] This simplifies to: \[ 5 \cdot \sqrt{7} \cdot a \cdot \sqrt{a} = 5a\sqrt{7a} \]
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Combine the results: Now we combine the coefficient \(24\) with the simplified square root: \[ 24 \sqrt{175a^3} = 24 \cdot 5a \sqrt{7a} = 120a \sqrt{7a} \]
Thus, the final simplified expression is: \[ \boxed{120a\sqrt{7a}} \]
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