To simplify \(\sqrt{48b^7}\), we start by factoring the expression under the square root.
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Factor \(48\) into its prime factors: \[ 48 = 16 \times 3 = 4^2 \times 3 = 2^4 \times 3 \] Thus, we can write: \[ \sqrt{48} = \sqrt{16 \times 3} = \sqrt{16} \cdot \sqrt{3} = 4\sqrt{3} \]
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Next, we simplify \(\sqrt{b^7}\). Notice that \(b^7 = (b^3)^2 \cdot b\): \[ \sqrt{b^7} = \sqrt{(b^3)^2 \cdot b} = \sqrt{(b^3)^2} \cdot \sqrt{b} = b^3\sqrt{b} \]
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Now we can combine these results to simplify \(\sqrt{48b^7}\): \[ \sqrt{48b^7} = \sqrt{48} \cdot \sqrt{b^7} = (4\sqrt{3}) \cdot (b^3\sqrt{b}) = 4b^3\sqrt{3b} \]
Therefore, the simplified expression is: \[ \boxed{4b^3\sqrt{3b}} \]
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