Question

Please help me figure out how to solve the problem X squared to the Y square root 18 X to the fifth Y to the second

Answer 1

To solve this problem, we will use the following property of exponents:

`x^nx^m=x^(n+m).`

Now, notice that:

`x^5=x^(2+3)=x^2x^3\text{.}`

Therefore, we can rewrite the given expression as follows:

`x^2y\sqrt[]{18x^2y^2x^3}.`

Recall that:

`\sqrt[]{ab}=\sqrt[]{a}\sqrt[]{b}.`

Therefore, we can split the root as follows:

`x^2y\sqrt[]{18x^2y^2x^3}=x^2y\sqrt[]{18}\sqrt[]{x^2y^2}\sqrt[]{x^3}\text{.}`

Simplifying we get:

`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}.`

Multiplying like terms, we get:

`x^2yxy\sqrt[]{18x^3}=x^4y^2\sqrt[]{18x}=x^4y^2\sqrt[]{(9\cdot2)x^2}=3x^4y^2\sqrt[]{2x}.`

Answer:

`3x^4y^2\sqrt[]{2x^3}\text{.}`

Example:

Simplify

`2x^2y^3\sqrt[]{9x^3y^2}.`

First, we notice that:

`\begin{gathered} x^3=x^2x, \\ 9=3^2. \end{gathered}`

Therefore, we can rewrite the expression as:

`2x^2y^3\sqrt[]{3^2x^2xy^2}=2x^2y^3\sqrt[]{3^2x^2y^2}\sqrt[]{x}=2x^2y^3(3xy)\sqrt[]{x}.`

Simplifying we get:

`6x^3y^4\sqrt[]{x}.`