Question
Answer
$$\dfrac{5^3\sqrt{54}}{3^3\sqrt{40}}=\dfrac{25\sqrt{15}}{18}$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}\frac{5^3\sqrt{54}}{3^3\sqrt{40}}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{ 125 \cdot \sqrt{ 9 \cdot 6 } }{ 27 \cdot \sqrt{ 4 \cdot 10 } } \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}\frac{ 125 \cdot \sqrt{ 9 } \cdot \sqrt{ 6 } }{ 27 \cdot \sqrt{ 4 } \cdot \sqrt{ 10 } } \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle5}{\textcircled {5}} \htmlClass{explanationCircle explanationCircle6}{\textcircled {6}} } }}}\frac{125\cdot3\sqrt{6}}{27\cdot2\sqrt{10}} \xlongequal{ } \\[1 em] & \xlongequal{ }\frac{375\sqrt{6}}{54\sqrt{10}} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle7}{\textcircled {7}} } }}}\frac{375\sqrt{6}}{54\sqrt{10}}\frac{\sqrt{10}}{\sqrt{10}} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle8}{\textcircled {8}} } }}}\frac{750\sqrt{15}}{540} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle9}{\textcircled {9}} } }}}\frac{ 750 \sqrt{ 15 } : \color{blue}{ 30 } } { 540 : \color{blue}{ 30 }} \xlongequal{ } \\[1 em] & \xlongequal{ }\frac{25\sqrt{15}}{18}\end{aligned} $$ | |
| ① | Factor out the largest perfect square of 54. ( in this example we factored out $ 9 $ ) |
| ② | Factor out the largest perfect square of 40. ( in this example we factored out $ 4 $ ) |
| ③ | Rewrite $ \sqrt{ 9 \cdot 6 } $ as the product of two radicals. |
| ④ | Rewrite $ \sqrt{ 4 \cdot 10 } $ as the product of two radicals. |
| ⑤ | The square root of $ 9 $ is $ 3 $. |
| ⑥ | The square root of $ 4 $ is $ 2 $. |
| ⑦ | Multiply the numerator and denominator by the conjugate of the denominator . $$\color{blue}{ \sqrt{10}} $$. |
| ⑧ | Multiply in a numerator. $$ \color{blue}{ 375 \sqrt{6} } \cdot \sqrt{10} = 750 \sqrt{15} $$ Simplify denominator. $$ \color{blue}{ 54 \sqrt{10} } \cdot \sqrt{10} = 540 $$ |
| ⑨ | Divide numerator and denominator by $ \color{blue}{ 30 } $. |
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