Question

Express the area of each triangle as a monomial.

Short answer

The monomial expression for the area of the triangle with height$8c^2{d}^4$ and base$5c^{3}d$ is $20c^5{d}^5$.

Step-by-step solution

Step 1. State the concept of ‘monomial’.

A polynomial which contains only one non-zero single term is called a monomial. A monomial consists either one variable or a constant or products of more than one variable with a coefficient along with the exponents as a whole numbers.

Note: A monomial cannot have a variable in the denominator. The variable in the denominator is considered as second term.

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$\text{Area\hspace{0.17em}\hspace{0.17em}of\hspace{0.17em}\hspace{0.17em}triangle}\left(A\right)=\frac{1}{2}\times \text{height×base\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}\dots \left(1\right)$localid="1647932944313" $\text{Area\hspace{0.17em}\hspace{0.17em}of\hspace{0.17em}\hspace{0.17em}triangle}\left(A\right)=\frac{1}{2}\times \text{height×base\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}\dots \left(1\right)$

 Step 3 State the multiplication rule of ‘laws of indices’.

If the two terms having the same base are multiplied together, then their indices are added.

In general: $x^m\times x^n=x^{m+n}$localid="1647932950985" $x^m\times x^n=x^{m+n}$

Step 4. Calculate the area of the triangle.

Observe the figure given below.

From the figure, height of the triangle is $8c^2{d}^4$and base of the triangle is $5c^{3}d$.

Substitute $8c^2{d}^4$as the height and $5c^{3}d$as the base in the equation (1) to get the area of the given triangle.

$Area\left(A\right)=\frac{1}{2}\times \left(8c^2{d}^4\right)\times \left(5c^{3}d\right)$

Collect the like terms together.

$\begin{array}{c}Area\left(A\right)=\frac{1}{2}\times (8\times 5)\times (c^2\times c^3)\times (d^4\times d)\\ =\frac{1}{2}\left(40\right)\left(c^{2+3}\right)\left(d^{4+1}\right)\\ =\left(\frac{40}{2}\right)\left(c^5\right)\left(d^5\right)\\ =20c^5{d}^5\end{array}$

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 $c,d$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 $8c^2{d}^4$and a base of $5c^{3}d$is $20c^5{d}^5$.