Questions: √36x^3y^9

Solution Steps

To simplify the expression \(\sqrt{36 x^{3} y^{9}}\), we need to break it down into its prime factors and then apply the square root to each factor separately. The square root of a product is the product of the square roots of the factors.

We start with the expression \( \sqrt{36 x^{3} y^{9}} \). We can factor this into its components: \( 36 \), \( x^{3} \), and \( y^{9} \).

Next, we apply the square root to each factor separately: \[ \sqrt{36} = 6, \quad \sqrt{x^{3}} = \sqrt{x^{2} \cdot x} = x \sqrt{x}, \quad \sqrt{y^{9}} = \sqrt{(y^{4})^{2} \cdot y} = y^{4} \sqrt{y} \]

Combining these results, we have: \[ \sqrt{36 x^{3} y^{9}} = 6 \cdot x \sqrt{x} \cdot y^{4} \sqrt{y} = 6 y^{4} x \sqrt{xy} \]

Final Answer