Express the area of a triangle with height of$4a$ and a base of$5a{b}^2$ 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.

The area of a triangle (A) is half of the the product of its height and base. That is,

$\text{Area\hspace{0.17em}\hspace{0.17em}of\hspace{0.17em}\hspace{0.17em}triangle}\left(A\right)=\frac{1}{2}\times \text{height×base\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}\dots \left(1\right)$

If the two terms have the same base (in this case x) and are to be multiplied together their indices are added.

In general:$x^m\times x^n=x^{m+n}$

Substitute$4a$ as the height and $5a{b}^2$as the base in the equation (1) to get the area of the given triangle.

$Area\left(A\right)=\frac{1}{2}\times \left(4a\right)\times \left(5a{b}^2\right)$

Collect the like terms together.

$Area\left(A\right)=\frac{1}{2}\times \left(4\times 5\right)\times \left(a\times a\right)\times \left(b^2\right)$

Apply multiplication rule to simplify further.

width="195" height="113" role="math">AreaA=1220a1+1b2=1220a2b2=10a2b2

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

Note: Even though it contains two variables$a,b$and a constant 20, all those are considered as one single term by the definition of monomial.

Therefore the monomial expression for the area of the triangle with height $4a$and a base of$5a{b}^2$ is $10a^2{b}^2$.