Question

simplify the following expression.
\sqrt{x^{7}y^{7}} \cdot 3\sqrt{5x^{11}y^{2}}
\sqrt{x^{7}y^{7}} \cdot 3\sqrt{5x^{11}y^{2}} = \square
(simplify your answer. type an exact answer, using radicals as needed.)

Explanation:

Step1: Simplify each radical

First, simplify \(\sqrt{x^{7}y^{7}}\). We can rewrite the exponents as \(x^{6 + 1}y^{6+1}\), so \(\sqrt{x^{7}y^{7}}=\sqrt{x^{6}x y^{6}y}\). Using the property \(\sqrt{ab}=\sqrt{a}\cdot\sqrt{b}\) and \(\sqrt{a^{n}} = a^{\frac{n}{2}}\) (for even \(n\)), we get \(\sqrt{x^{6}x y^{6}y}=x^{3}y^{3}\sqrt{xy}\).

Next, simplify \(3\sqrt{5x^{11}y^{2}}\). Rewrite \(x^{11}\) as \(x^{10+1}\), so \(3\sqrt{5x^{11}y^{2}} = 3\sqrt{5x^{10}x y^{2}}\). Using the same radical properties, we have \(3\sqrt{5x^{10}x y^{2}}=3x^{5}y\sqrt{5x}\).

Step2: Multiply the simplified radicals

Now, multiply \(x^{3}y^{3}\sqrt{xy}\) and \(3x^{5}y\sqrt{5x}\). First, multiply the coefficients and the variables with the same base:

• Coefficients: \(1\times3 = 3\)
• Variables: \(x^{3}\cdot x^{5}=x^{3 + 5}=x^{8}\), \(y^{3}\cdot y=y^{3+1}=y^{4}\)
• Radicals: \(\sqrt{xy}\cdot\sqrt{5x}=\sqrt{xy\cdot5x}=\sqrt{5x^{2}y}\)

Step3: Simplify the resulting radical

Simplify \(\sqrt{5x^{2}y}\). Using the radical property, \(\sqrt{5x^{2}y}=x\sqrt{5y}\) (since \(x^{2}\) is a perfect square).

Now, combine all the parts:

• Coefficient: \(3\)
• Variables: \(x^{8}\cdot x=x^{9}\), \(y^{4}\)
• Radical: \(\sqrt{5y}\)

Wait, let's re - check the multiplication of the radicals:

Wait, when we multiply \(x^{3}y^{3}\sqrt{xy}\times3x^{5}y\sqrt{5x}\), the product of the radicals is \(\sqrt{xy}\times\sqrt{5x}=\sqrt{5x^{2}y}=x\sqrt{5y}\) (assuming \(x\geq0\)). The product of the variables: \(x^{3}\times x^{5}=x^{8}\), \(y^{3}\times y = y^{4}\), and then multiply by \(x\) from the radical simplification: \(x^{8}\times x=x^{9}\), and the coefficient is \(3\), and the \(y^{4}\) and \(\sqrt{5y}\). Wait, no, let's do it step by step again.

Wait, \(\sqrt{x^{7}y^{7}}\cdot3\sqrt{5x^{11}y^{2}}\) can also be combined into a single radical first (since \(a\sqrt{b}\cdot c\sqrt{d}=ac\sqrt{bd}\)). Let's try this approach for accuracy.

Using the property \(a\sqrt{b}\cdot c\sqrt{d}=ac\sqrt{bd}\), we have:

\(\sqrt{x^{7}y^{7}}\cdot3\sqrt{5x^{11}y^{2}}=3\sqrt{x^{7}y^{7}\cdot5x^{11}y^{2}}\)

Multiply the expressions inside the radical:

\(x^{7}y^{7}\cdot5x^{11}y^{2}=5x^{7 + 11}y^{7+2}=5x^{18}y^{9}\)

So now we have \(3\sqrt{5x^{18}y^{9}}\)

Now, simplify \(\sqrt{5x^{18}y^{9}}\). We can rewrite the exponents:

\(x^{18}\) is a perfect square (\(18\) is even), \(y^{9}=y^{8 + 1}\)

So \(\sqrt{5x^{18}y^{9}}=\sqrt{5x^{18}y^{8}y}=x^{9}y^{4}\sqrt{5y}\) (since \(\sqrt{x^{18}}=x^{9}\), \(\sqrt{y^{8}}=y^{4}\))

Then, multiply by the coefficient \(3\): \(3\times x^{9}y^{4}\sqrt{5y}=3x^{9}y^{4}\sqrt{5y}\)

Answer:

\(3x^{9}y^{4}\sqrt{5y}\)