Question

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

Short answer

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

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 triangle.

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

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

State the multiplication rule of ‘laws of indices’.

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

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

Step 4. Calculate the area of the triangle.

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

Step 5. State the conclusion.

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 $10a^2{b}^2$.