Questions: Perform the indicated operation and simplify. Assume the variable y represents a nonnegative real number sqrt(8y) * sqrt(2y^4) + 5y^2 sqrt(y) (Simplify your answer. Type an exact answer, using radicals as needed.)

Solution Steps

To simplify the given expression, we need to use the properties of square roots and exponents. Specifically, we will:

1. Simplify the product of the square roots.
2. Combine like terms if possible.

We start with the expression: \[ \sqrt{8y} \cdot \sqrt{2y^4} + 5y^2 \sqrt{y} \]

Using the property of square roots \(\sqrt{a} \cdot \sqrt{b} = \sqrt{ab}\), we can simplify the product of the square roots: \[ \sqrt{8y} \cdot \sqrt{2y^4} = \sqrt{(8y) \cdot (2y^4)} = \sqrt{16y^5} \]

Next, we simplify \(\sqrt{16y^5}\): \[ \sqrt{16y^5} = \sqrt{16} \cdot \sqrt{y^5} = 4 \cdot y^{5/2} \]

Now, we substitute back into the original expression: \[ 4y^{5/2} + 5y^2 \sqrt{y} \]

Since \(\sqrt{y} = y^{1/2}\), we can rewrite the second term: \[ 5y^2 \sqrt{y} = 5y^2 \cdot y^{1/2} = 5y^{2 + 1/2} = 5y^{5/2} \]

Finally, we add the like terms: \[ 4y^{5/2} + 5y^{5/2} = 9y^{5/2} \]

Final Answer