Question

divide and simplify. assume that all variables are greater than zero.

\\\frac{\sqrt{150y^{10}}}{\sqrt{3y^2}}\\

\\\frac{\sqrt{150y^{10}}}{\sqrt{3y^2}} = \square\\

(type an exact answer, using radicals as needed. simplify your answer.)

Explanation:

⚡ Using what you learned: Roots and Radicals

Step 1: Apply the quotient rule for radicals

Combine the numerator and denominator under a single square root:

$$ \frac{\sqrt{150y^{10}}}{\sqrt{3y^2}} = \sqrt{\frac{150y^{10}}{3y^2}} $$

Step 2: Simplify the fraction inside the radical

Divide the coefficients and subtract the exponents of the variable \(y\):

$$ \frac{150}{3} = 50 $$

$$ \frac{y^{10}}{y^2} = y^{10-2} = y^8 $$

So, the expression becomes:

$$ \sqrt{50y^8} $$

Step 3: Simplify the radical expression

Factor \(50y^8\) into perfect squares and remaining terms:

$$ \sqrt{50y^8} = \sqrt{25 \cdot 2 \cdot y^8} = \sqrt{25} \cdot \sqrt{y^8} \cdot \sqrt{2} $$

Evaluate the square roots of the perfect squares:

$$ 5 \cdot y^4 \cdot \sqrt{2} = 5y^4\sqrt{2} $$

Answer:

$$ 5y^4\sqrt{2} $$