To solve this problem, we will use the following property of exponents:
[tex]x^nx^m=x^{n+m}.[/tex]
Now, notice that:
[tex]x^5=x^{2+3}=x^2x^3\text{.}[/tex]
Therefore, we can rewrite the given expression as follows:
[tex]x^2y\sqrt[]{18x^2y^2x^3}.[/tex]
Recall that:
[tex]\sqrt[]{ab}=\sqrt[]{a}\sqrt[]{b}.[/tex]
Therefore, we can split the root as follows:
[tex]x^2y\sqrt[]{18x^2y^2x^3}=x^2y\sqrt[]{18}\sqrt[]{x^2y^2}\sqrt[]{x^3}\text{.}[/tex]
Simplifying we get:
[tex]x^2y\sqrt[]{18}\sqrt[]{x^2y^2}\sqrt[]{x^3}=x^2y\sqrt[]{9\cdot2}\sqrt[]{x^2}\sqrt[]{y^2}\sqrt[]{x^3}=x^2y3\sqrt[]{2}xy\sqrt[]{x^2x}.[/tex]
Multiplying like terms, we get:
[tex]x^2yxy\sqrt[]{18x^3}=x^4y^2\sqrt[]{18x}=x^4y^2\sqrt[]{(9\cdot2)x^2}=3x^4y^2\sqrt[]{2x}.[/tex]
Answer:
[tex]3x^4y^2\sqrt[]{2x^3}\text{.}[/tex]
Example:
Simplify
[tex]2x^2y^3\sqrt[]{9x^3y^2}.[/tex]
First, we notice that:
[tex]\begin{gathered} x^3=x^2x, \\ 9=3^2. \end{gathered}[/tex]
Therefore, we can rewrite the expression as:
[tex]2x^2y^3\sqrt[]{3^2x^2xy^2}=2x^2y^3\sqrt[]{3^2x^2y^2}\sqrt[]{x}=2x^2y^3(3xy)\sqrt[]{x}.[/tex]
Simplifying we get:
[tex]6x^3y^4\sqrt[]{x}.[/tex]
