Question

simplify.
remove all perfect squares from inside the square root. assume y is positive.
\sqrt{39y^{9}} =

Explanation:

Step1: Factor the radicand

We can factor \( y^9 \) as \( y^8 \cdot y \), since \( y^8=(y^4)^2 \) is a perfect square. So, \( \sqrt{39y^9}=\sqrt{39\times y^8\times y} \).

Step2: Use the property of square roots

Using the property \( \sqrt{ab}=\sqrt{a}\cdot\sqrt{b} \) (where \( a\geq0,b\geq0 \)), we can split the square root: \( \sqrt{39\times y^8\times y}=\sqrt{39}\times\sqrt{y^8}\times\sqrt{y} \).

Step3: Simplify the perfect square root

Since \( \sqrt{y^8}=y^4 \) (because \( (y^4)^2 = y^8 \) and \( y \) is positive), we have \( \sqrt{39}\times y^4\times\sqrt{y}=y^4\sqrt{39y} \).

Answer:

\( y^4\sqrt{39y} \)