Question

what is the following product? assume x≥0 and y≥0
\sqrt{5x^8y^2} \cdot \sqrt{10x^3} \cdot \sqrt{12y}
\bigcirc \\ 3x^5y\sqrt{3xy}
\bigcirc \\ 10x^5y\sqrt{6xy}
\bigcirc \\ 3x^3y\sqrt{3x^2y^2}
\bigcirc \\ 10x^3y\sqrt{6x^2y^2}

Explanation:

Step1: Multiply the radicands

Using the property \(\sqrt{a}\cdot\sqrt{b}=\sqrt{ab}\), we combine the three square roots into one:
$$\sqrt{5x^{8}y^{2}\cdot10x^{3}\cdot12y}$$
Multiply the coefficients and the variables separately. For coefficients: \(5\times10\times12 = 600\). For \(x\)-terms: \(x^{8}\cdot x^{3}=x^{8 + 3}=x^{11}\). For \(y\)-terms: \(y^{2}\cdot y=y^{2+1}=y^{3}\). So the radicand becomes \(600x^{11}y^{3}\).

Step2: Simplify the radicand

Factor the radicand into perfect squares and remaining factors.

• Factor \(600\): \(600 = 100\times6=10^{2}\times6\)
• Factor \(x^{11}\): \(x^{11}=x^{10}\cdot x=(x^{5})^{2}\cdot x\)
• Factor \(y^{3}\): \(y^{3}=y^{2}\cdot y=(y)^{2}\cdot y\)

So, \(600x^{11}y^{3}=10^{2}\times6\times(x^{5})^{2}\times x\times y^{2}\times y\)

Step3: Take square roots of perfect squares

Using \(\sqrt{a^{2}} = a\) (for \(a\geq0\)):
$$\sqrt{10^{2}\times(x^{5})^{2}\times y^{2}\times6xy}=\sqrt{10^{2}}\cdot\sqrt{(x^{5})^{2}}\cdot\sqrt{y^{2}}\cdot\sqrt{6xy}$$
Simplify each perfect square root: \(\sqrt{10^{2}} = 10\), \(\sqrt{(x^{5})^{2}}=x^{5}\), \(\sqrt{y^{2}} = y\) (since \(y\geq0\)).

Step4: Combine the simplified terms

Multiply the terms outside the square root: \(10\times x^{5}\times y=10x^{5}y\)
The remaining radicand is \(6xy\), so the expression becomes \(10x^{5}y\sqrt{6xy}\)

Answer:

B. \(10x^{5}y\sqrt{6xy}\)