To simplify the given expression [tex]\(\frac{\sqrt{50 a^4 y^2}}{\sqrt{36 a^8 y^{10}}}\)[/tex], follow these steps:
1. Simplify the Radicals Separately:
- Simplify the numerator: [tex]\(\sqrt{50 a^4 y^2}\)[/tex]
Since [tex]\(50 = 25 \cdot 2\)[/tex] and [tex]\(\sqrt{25} = 5\)[/tex], we can write:
[tex]\[
\sqrt{50 a^4 y^2} = \sqrt{25 \cdot 2 \cdot a^4 \cdot y^2} = \sqrt{25} \cdot \sqrt{2} \cdot \sqrt{a^4} \cdot \sqrt{y^2}
\][/tex]
This simplifies to:
[tex]\[
5 \cdot \sqrt{2} \cdot a^2 \cdot y
\][/tex]
So, [tex]\(\sqrt{50 a^4 y^2} = 5 \sqrt{2} a^2 y\)[/tex].
- Simplify the denominator: [tex]\(\sqrt{36 a^8 y^{10}}\)[/tex]
Since [tex]\(36 = 6^2\)[/tex] and [tex]\(\sqrt{36} = 6\)[/tex], we can write:
[tex]\[
\sqrt{36 a^8 y^{10}} = \sqrt{36} \cdot \sqrt{a^8} \cdot \sqrt{y^{10}}
\][/tex]
This simplifies to:
[tex]\[
6 \cdot a^4 \cdot y^5
\][/tex]
So, [tex]\(\sqrt{36 a^8 y^{10}} = 6 a^4 y^5\)[/tex].
2. Combine the Simplified Results:
Substitute the simplified numerator and denominator back into the original expression:
[tex]\[
\frac{\sqrt{50 a^4 y^2}}{\sqrt{36 a^8 y^{10}}} = \frac{5 \sqrt{2} a^2 y}{6 a^4 y^5}
\][/tex]
3. Simplify the Fraction:
- Divide the constants: [tex]\(\frac{5 \sqrt{2}}{6}\)[/tex]
- Divide the [tex]\(a\)[/tex] terms: [tex]\(\frac{a^2}{a^4} = \frac{1}{a^2}\)[/tex]
- Divide the [tex]\(y\)[/tex] terms: [tex]\(\frac{y}{y^5} = \frac{1}{y^4}\)[/tex]
Combining all these results, we have:
[tex]\[
\frac{5 \sqrt{2} a^2 y}{6 a^4 y^5} = \frac{5 \sqrt{2}}{6} \cdot \frac{1}{a^2} \cdot \frac{1}{y^4} = \frac{5 \sqrt{2}}{6 a^2 y^4}
\][/tex]
Therefore, the simplified form of the expression is:
[tex]\[
\frac{\sqrt{50 a^4 y^2}}{\sqrt{36 a^8 y^{10}}} = \frac{5 \sqrt{2}}{6 a^2 y^4}
\][/tex]
