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Answer
The simplified form is \( -24n^{2}m\sqrt{6nm} \).
Solution
Calculate or simplify the expression \( -4\sqrt{216m^{3}n^{5}} \).
Simplify the expression by following steps:
- step0: Solution:
\(-4\sqrt{216m^{3}n^{5}}\)
- step1: Simplify the root:
\(-4\times 6n^{2}m\sqrt{6nm}\)
- step2: Multiply the terms:
\(-24n^{2}m\sqrt{6nm}\)
The simplified form of \( -4 \sqrt{216 m^{3} n^{5}} \) is \( -24n^{2}m\sqrt{6nm} \).
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Simplify this solution Extra Insights
To simplify \( -4 \sqrt{216 m^{3} n^{5}} \), we first break it down. The square root of \( 216 \) can be simplified: \( 216 = 36 \times 6 \), and the square root of \( 36 \) is \( 6 \). Thus, \( \sqrt{216} = 6 \sqrt{6} \). For the variables, \( m^{3} \) simplifies to \( m^{1.5} \) or \( m \sqrt{m} \), and \( n^{5} \) simplifies to \( n^{2} \sqrt{n} \). Putting it all together: \[ -4 \sqrt{216 m^{3} n^{5}} = -4 (6 \sqrt{6}) (m \sqrt{m}) (n^{2} \sqrt{n}) = -24 n^{2} m \sqrt{6mn}. \] Thus, the final simplified result is: \[ \boxed{-24 n^{2} m \sqrt{6mn}}. \]
