Question

simplify. 9\sqrt{56x^{7}y^{12}} assume x and y are nonnegative. 18x^{3}y^{6}\sqrt{7x} 18x^{3}y^{6}\sqrt{14x} 18x^{3}y^{6}\sqrt{14} 2x^{3}y^{6}\sqrt{14x}

Explanation:

Step1: Factorize the number inside the square - root

Factorize 56 as \(56 = 2^3\times7\), so \(9\sqrt{56x^{7}y^{12}}=9\sqrt{2^3\times7\times x^{7}y^{12}}\).

Step2: Use the property of square - roots \(\sqrt{ab}=\sqrt{a}\cdot\sqrt{b}\)

\(9\sqrt{2^3\times7\times x^{7}y^{12}} = 9\sqrt{2^2\times2\times7\times x^{6}\times x\times y^{12}}\).

Step3: Take out the perfect - square factors from the square - root

We know that \(\sqrt{2^2}=2\), \(\sqrt{x^{6}} = x^{3}\), \(\sqrt{y^{12}}=y^{6}\). So \(9\sqrt{2^2\times2\times7\times x^{6}\times x\times y^{12}}=9\times2\times x^{3}\times y^{6}\sqrt{2\times7\times x}\).

Step4: Simplify the coefficient

\(9\times2 = 18\), and \(9\times2\times x^{3}\times y^{6}\sqrt{2\times7\times x}=18x^{3}y^{6}\sqrt{14x}\).

Answer:

\(18x^{3}y^{6}\sqrt{14x}\) (corresponding to the second option)