To simplify [tex]\(\sqrt{27 x^6 y^9}\)[/tex] to the form [tex]\(a \sqrt{b}\)[/tex], let's break it down step-by-step:
1. Factor Inside the Square Root:
Break down the expression inside the square root into its prime factors and simpler components:
[tex]\[
27 x^6 y^9 = 3^3 \cdot x^6 \cdot y^9
\][/tex]
2. Separate the Square Root:
Use the property that [tex]\(\sqrt{a \cdot b} = \sqrt{a} \cdot \sqrt{b}\)[/tex]:
[tex]\[
\sqrt{27 x^6 y^9} = \sqrt{3^3 \cdot x^6 \cdot y^9}
\][/tex]
3. Simplify Each Component:
- For [tex]\(3^3 = 27\)[/tex]:
[tex]\[
\sqrt{3^3} = \sqrt{27} = \sqrt{3 \cdot 3 \cdot 3} = 3 \sqrt{3}
\][/tex]
- For [tex]\(x^6\)[/tex]:
[tex]\[
\sqrt{x^6} = x^3 \quad \text{(since } x^6 = (x^3)^2\text{)}
\][/tex]
- For [tex]\(y^9\)[/tex]:
[tex]\[
\sqrt{y^9} = \sqrt{y^8 \cdot y} = \sqrt{(y^4)^2 \cdot y} = y^4 \sqrt{y}
\][/tex]
4. Combine The Results:
Now, combine all simplified components:
[tex]\[
\sqrt{27 x^6 y^9} = 3 \sqrt{3} \cdot x^3 \cdot y^4 \sqrt{y}
\][/tex]
5. Final Form:
Group similar terms to express in the form [tex]\(a \sqrt{b}\)[/tex]:
[tex]\[
\sqrt{27 x^6 y^9} = 3 x^3 y^4 \sqrt{3 y}
\][/tex]
Hence, the simplified form of [tex]\(\sqrt{27 x^6 y^9}\)[/tex] is:
[tex]\[
3 x^3 y^4 \sqrt{3 y}
\][/tex]
