To simplify the expression \(\sqrt{\frac{2a}{5b^3c^9}}\), follow these steps:
- Break down the square root: We can separate the expression under the square root into its components:
\[ \sqrt{\frac{2a}{5b^3c^9}} = \frac{\sqrt{2a}}{\sqrt{5b^3c^9}} \]
- Simplify the denominator: You can further simplify the denominator by splitting it up:
\[ \sqrt{5b^3c^9} = \sqrt{5} \cdot \sqrt{b^3} \cdot \sqrt{c^9} \]
- Extract roots: The square root of \(b^3\) and \(c^9\) can be simplified:
\[ \sqrt{b^3} = b^{3/2} = b^{1} \cdot \sqrt{b} = b\sqrt{b} \]
\[ \sqrt{c^9} = c^{9/2} = c^{4} \cdot \sqrt{c} = c^4\sqrt{c} \]
- Combine the terms in the denominator:
\[ \sqrt{5b^3c^9} = \sqrt{5} \cdot b\sqrt{b} \cdot c^4\sqrt{c} = \sqrt{5} \cdot bc^4 \sqrt{bc} \]
- Now substitute back to the fraction:
\[ \frac{\sqrt{2a}}{\sqrt{5} \cdot bc^4 \sqrt{bc}} = \frac{\sqrt{2a}}{\sqrt{5} b c^4 \sqrt{bc}} \]
- Combine the complete expression:
Let's summarize it as:
\[ \frac{\sqrt{2a}}{\sqrt{5} \cdot b \cdot c^4 \cdot \sqrt{bc}} \]
Thus, the simplified form of the original expression \(\sqrt{\frac{2a}{5b^3c^9}}\) is:
\[ \frac{\sqrt{2a}}{\sqrt{5} b c^4 \sqrt{bc}} \]
This is the simplest form of the expression.