Questions: Simplify. 9 sqrt(56 x^7 y^12)

Solution Steps

We start with the expression: \[ 9 \sqrt{56 x^{7} y^{12}} \] First, we factor \(56\) into its prime factors: \[ 56 = 4 \cdot 14 = 2^2 \cdot 14 \] Thus, we can rewrite the expression as: \[ 9 \sqrt{4 \cdot 14 \cdot x^{7} \cdot y^{12}} \]

Next, we separate the perfect squares from the non-perfect squares inside the square root: \[ \sqrt{4} = 2, \quad \sqrt{x^{6}} = x^{3}, \quad \text{and} \quad \sqrt{y^{12}} = y^{6} \] This allows us to express the square root as: \[ \sqrt{4 \cdot 14 \cdot x^{7} \cdot y^{12}} = \sqrt{4} \cdot \sqrt{14} \cdot \sqrt{x^{6} \cdot x} \cdot \sqrt{y^{12}} = 2 \cdot \sqrt{14} \cdot x^{3} \cdot y^{6} \cdot \sqrt{x} \]

Now we can combine everything back together: \[ 9 \cdot 2 \cdot \sqrt{14} \cdot x^{3} \cdot y^{6} \cdot \sqrt{x} = 18 \sqrt{14} x^{3} y^{6} \sqrt{x} \]

Final Answer