Question

simplify the following expression.\\(\sqrt{x^{7}y^{7}}\cdot 4\sqrt{3x^{3}y^{10}}\\)\\(\sqrt{x^{7}y^{7}}\cdot 4\sqrt{3x^{3}y^{10}}=\square\\)\\((\text{simplify your answer. type an exact answer, using radicals as needed.})\\)

Explanation:

Step1: Rewrite radicals as exponents

Recall that \(\sqrt[n]{a^m}=a^{\frac{m}{n}}\). So, \(\sqrt{x^7y^7}=(x^7y^7)^{\frac{1}{2}} = x^{\frac{7}{2}}y^{\frac{7}{2}}\) and \(\sqrt{3x^3y^{10}}=(3x^3y^{10})^{\frac{1}{2}}=3^{\frac{1}{2}}x^{\frac{3}{2}}y^{5}\).

Step2: Multiply the coefficients and use exponent rules for variables

Multiply the coefficients: \(1\times4\times3^{\frac{1}{2}} = 4\sqrt{3}\) (since \(3^{\frac{1}{2}}=\sqrt{3}\)). For the \(x\)-terms: \(x^{\frac{7}{2}}\times x^{\frac{3}{2}}=x^{\frac{7 + 3}{2}}=x^{5}\) (using \(a^m\times a^n=a^{m + n}\)). For the \(y\)-terms: \(y^{\frac{7}{2}}\times y^{5}=y^{\frac{7}{2}+\frac{10}{2}}=y^{\frac{17}{2}}\) (converting \(5\) to \(\frac{10}{2}\) to add the exponents).

Step3: Rewrite \(y^{\frac{17}{2}}\) as a radical and simplify

\(y^{\frac{17}{2}}=y^{8+\frac{1}{2}}=y^{8}\times y^{\frac{1}{2}}=y^{8}\sqrt{y}\) (since \(y^{\frac{1}{2}}=\sqrt{y}\)). Now combine all parts: \(4\sqrt{3}\times x^{5}\times y^{8}\sqrt{y}=4x^{5}y^{8}\sqrt{3y}\) (using \(\sqrt{a}\times\sqrt{b}=\sqrt{ab}\) for \(\sqrt{3}\times\sqrt{y}=\sqrt{3y}\)). Wait, let's check the exponent addition again. Wait, in step 2, when we multiply \(\sqrt{x^7y^7}\) and \(4\sqrt{3x^3y^{10}}\), we can also use the property \(\sqrt{a}\times\sqrt{b}=\sqrt{ab}\) directly. Let's redo that way for accuracy. \(\sqrt{x^7y^7}\times4\sqrt{3x^3y^{10}} = 4\sqrt{x^7y^7\times3x^3y^{10}}=4\sqrt{3x^{7 + 3}y^{7+10}}=4\sqrt{3x^{10}y^{17}}\). Now, simplify \(\sqrt{3x^{10}y^{17}}\): \(x^{10}\) is a perfect square (\((x^{5})^2\)), \(y^{17}=y^{16}\times y=(y^{8})^2\times y\). So, \(\sqrt{3x^{10}y^{17}}=\sqrt{3}\times x^{5}\times y^{8}\sqrt{y}=x^{5}y^{8}\sqrt{3y}\). Then multiply by 4: \(4x^{5}y^{8}\sqrt{3y}\). Wait, but let's check \(y^{17}\): \(17\) divided by 2 is 8 with a remainder of 1, so \(y^{17}=y^{2\times8 + 1}=(y^{8})^2\times y\), so \(\sqrt{y^{17}}=y^{8}\sqrt{y}\), and \(\sqrt{x^{10}}=x^{5}\), \(\sqrt{3}\) remains. So combining, \(\sqrt{3x^{10}y^{17}}=x^{5}y^{8}\sqrt{3y}\), then multiply by 4 gives \(4x^{5}y^{8}\sqrt{3y}\). Wait, but let's check the first method's mistake. In the first method, when we multiplied the exponents, we had \(y^{\frac{7}{2}}\times y^{5}=y^{\frac{7}{2}+\frac{10}{2}}=y^{\frac{17}{2}}\), which is correct, and \(x^{\frac{7}{2}}\times x^{\frac{3}{2}}=x^{5}\), correct. Then \(y^{\frac{17}{2}}=y^{8.5}=y^{8}\sqrt{y}\), and \(x^{5}\), and \(4\sqrt{3}\), so \(4x^{5}y^{8}\sqrt{3}\times\sqrt{y}=4x^{5}y^{8}\sqrt{3y}\), which matches the second method. So that's the simplified form.

Answer:

\(4x^{5}y^{8}\sqrt{3y}\)