Express the area of a square with sides of length$3x{y}^2$as a monomial.

A monomial is a type of polynomial, which is an algebraic expression having only a non-zero single term. Monomial consists of only a single term which makes it easy to do the operation of addition, subtraction and multiplication. It consists of either only one variable or one coefficient or product of a variable and a coefficient with exponents as whole numbers, which represent only one term, unlike binomial and trinomial, which consist of two and three terms respectively. It cannot have a variable in the denominator.

A square is a 2D figure in which all the sides are of equal measure. Since all the sides are equal, the area would be length times width, which is equal to side × side. Hence, the area of a square is side square. That is,

$\text{Area\hspace{0.17em}\hspace{0.17em}of\hspace{0.17em}\hspace{0.17em}square}\left(A\right)={\left(\text{lenght\hspace{0.17em}\hspace{0.17em}of\hspace{0.17em}\hspace{0.17em}side}\right)}^2$.

When two variables with different bases, but same indices are multiplied together, we have to multiply its base and raise the same index to multiplied variables.

In general:$x^m{y}^m={\left(xy\right)}^m$

If a term with a power is itself raised to a power then the powers are multiplied together.

In general:${\left(x^m\right)}^n=x^{m\times n}$

The length of each sides of the given square is$3x{y}^2$.

$Area\left(A\right)={\left(3x{y}^2\right)}^2$

Apply the laws of indices to simplify further.

$$\begin{gathered} Area\left(A\right)={\left(3x{y}^2\right)}^2 \\ ={\left(3\right)}^2{\left(x\right)}^2{\left(y^2\right)}^2 \\ =9{\left(x\right)}^2\left(y^{2\times 2}\right) \\ =9\left(x^2\right)\left(y^4\right) \\ =9x^2{y}^4 \end{gathered}$$

From the definition a monomial expression must contain only one non-zero term.

Note: Even though it contains two variables$x,y$and a constant 9, all those are considered as one single term by the definition of monomial.

Therefore the area of the square with sides of length $3x{y}^2$as a monomial is $9x^2{y}^4$.