Question

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

Short answer

The monomial expression for the area of the square with sides of length $3x{y}^2$is $9x^2{y}^4$.

Step-by-step solution

Step 1. State the concept of ‘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.

Step 2. State the concept of Area of square.

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,

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

Step 3. State the law of indices for multiplying powers with the same exponents.

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

Step 4. State the law of indices for power of index numbers.

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

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

Step 5. Calculate the area of the square.

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.

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

Step 6. State the conclusion.

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 $9x^2{y}^4$.