Answer
$$\frac{54\sqrt{54}}{\sqrt{15}^3}=\frac{54\sqrt{10}}{25}$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}\frac{54\sqrt{54}}{\sqrt{15}^3}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{ 54\cdot \sqrt{ 9 \cdot 6 } }{ 15\sqrt{15} } \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}\frac{ 54\cdot \sqrt{ 9 } \cdot \sqrt{ 6 } }{ 15\sqrt{15} } \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle5}{\textcircled {5}} \htmlClass{explanationCircle explanationCircle6}{\textcircled {6}} } }}}\frac{ 54\cdot3 \sqrt{ 6 } }{ 15\sqrt{15} } \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle7}{\textcircled {7}} } }}}\frac{162\sqrt{6}}{15\sqrt{15}} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle8}{\textcircled {8}} } }}}\frac{162\sqrt{6}}{15\sqrt{15}}\frac{\sqrt{15}}{\sqrt{15}} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle9}{\textcircled {9}} } }}}\frac{486\sqrt{10}}{225} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle10}{\textcircled {10}} } }}}\frac{ 486 \sqrt{ 10 } : \color{blue}{ 9 } } { 225 : \color{blue}{ 9 }} \xlongequal{ } \\[1 em] & \xlongequal{ }\frac{54\sqrt{10}}{25}\end{aligned} $$ | |
| ① | Factor out the largest perfect square of 54. ( in this example we factored out $ 9 $ ) |
| ② | $$ \sqrt{15}^3 =
\sqrt{15} ^2 \cdot \sqrt{15} =
\lvert 15 \rvert \cdot \sqrt{15} =
15\sqrt{15} $$ |
| ③ | Rewrite $ \sqrt{ 9 \cdot 6 } $ as the product of two radicals. |
| ④ | $$ \sqrt{15}^3 =
\sqrt{15} ^2 \cdot \sqrt{15} =
\lvert 15 \rvert \cdot \sqrt{15} =
15\sqrt{15} $$ |
| ⑤ | The square root of $ 9 $ is $ 3 $. |
| ⑥ | $$ \sqrt{15}^3 =
\sqrt{15} ^2 \cdot \sqrt{15} =
\lvert 15 \rvert \cdot \sqrt{15} =
15\sqrt{15} $$ |
| ⑦ | $$ \sqrt{15}^3 =
\sqrt{15} ^2 \cdot \sqrt{15} =
\lvert 15 \rvert \cdot \sqrt{15} =
15\sqrt{15} $$ |
| ⑧ | Multiply the numerator and denominator by the conjugate of the denominator . $$\color{blue}{ \sqrt{15}} $$. |
| ⑨ | Multiply in a numerator. $$ \color{blue}{ 162 \sqrt{6} } \cdot \sqrt{15} = 486 \sqrt{10} $$ Simplify denominator. $$ \color{blue}{ 15 \sqrt{15} } \cdot \sqrt{15} = 225 $$ |
| ⑩ | Divide numerator and denominator by $ \color{blue}{ 9 } $. |
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