To simplify the expression [tex]\(\sqrt{39 y^9}\)[/tex] and remove all perfect squares from inside the square root, follow these steps:
1. Express the radicand (the expression inside the square root) as a product of perfect squares and any remaining factors:
[tex]\[
\sqrt{39 y^9}
\][/tex]
2. Separate the expression into two parts: one part that contains the perfect squares and the other that does not:
Since [tex]\(y^9\)[/tex] can be written as [tex]\((y^4)^2 \cdot y\)[/tex], we separate the perfect square part:
[tex]\[
\sqrt{39 \cdot (y^4)^2 \cdot y}
\][/tex]
3. Apply the property of square roots that states [tex]\(\sqrt{a \cdot b} = \sqrt{a} \cdot \sqrt{b}\)[/tex]:
[tex]\[
\sqrt{39 \cdot (y^4)^2 \cdot y} = \sqrt{39} \cdot \sqrt{(y^4)^2} \cdot \sqrt{y}
\][/tex]
4. Simplify the square root of the perfect square:
Since [tex]\(\sqrt{(y^4)^2} = y^4\)[/tex], we get:
[tex]\[
\sqrt{39} \cdot y^4 \cdot \sqrt{y}
\][/tex]
5. Combine the simplified parts:
[tex]\[
\sqrt{39} \cdot y^4 \cdot \sqrt{y} = \sqrt{39} \cdot y^4 \cdot y^{1/2}
\][/tex]
6. Combine the exponents of [tex]\(y\)[/tex]:
Using the property [tex]\(y^a \cdot y^b = y^{a+b}\)[/tex], we add the exponents [tex]\(4\)[/tex] and [tex]\(\frac{1}{2}\)[/tex]:
[tex]\[
y^4 \cdot y^{1/2} = y^{4 + \frac{1}{2}} = y^{\frac{8}{2} + \frac{1}{2}} = y^{\frac{9}{2}}
\][/tex]
7. Write the final simplified form:
[tex]\[
\sqrt{39} \cdot y^{\frac{9}{2}}
\][/tex]
Therefore, the simplified expression is:
[tex]\[
\sqrt{39 y^9} = \sqrt{39} \cdot y^{\frac{9}{2}}
\][/tex]
