Questions: Simplify the following, assuming x>0 and y>0 : sqrt((25 x^8)/(81 y^12))

Simplify the following, assuming x>0 and y>0 : sqrt((25 x^8)/(81 y^12))
Transcript text: Simplify the following, assuming $x>0$ and $y>0$ : \[ \sqrt{\frac{25 x^{8}}{81 y^{12}}} \]
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Solution

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Solution Steps

To simplify the given expression, we need to apply the properties of square roots and exponents. Specifically, we can use the property that the square root of a fraction is the fraction of the square roots, and the property that the square root of a power is the power of the square root.

  1. Separate the square root of the fraction into the square root of the numerator and the square root of the denominator.
  2. Simplify the square root of each part by applying the square root to the constants and the exponents.
Step 1: Separate the Square Root of the Fraction

We start by separating the square root of the fraction into the square root of the numerator and the square root of the denominator: \[ \sqrt{\frac{25 x^{8}}{81 y^{12}}} = \frac{\sqrt{25 x^{8}}}{\sqrt{81 y^{12}}} \]

Step 2: Simplify the Square Root of the Numerator

Next, we simplify the square root of the numerator: \[ \sqrt{25 x^{8}} = \sqrt{25} \cdot \sqrt{x^{8}} = 5 \cdot x^{4} = 5x^{4} \]

Step 3: Simplify the Square Root of the Denominator

Then, we simplify the square root of the denominator: \[ \sqrt{81 y^{12}} = \sqrt{81} \cdot \sqrt{y^{12}} = 9 \cdot y^{6} = 9y^{6} \]

Step 4: Combine the Simplified Parts

Finally, we combine the simplified parts to get the simplified expression: \[ \frac{\sqrt{25 x^{8}}}{\sqrt{81 y^{12}}} = \frac{5x^{4}}{9y^{6}} \]

Final Answer

\[ \boxed{\frac{5x^{4}}{9y^{6}}} \]

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