Question

using exponents, write two equivalent expressions for the area of the rectangle.
for 18⁹ ÷ 18⁴

Explanation:

Since the problem about the rectangle's area is not fully presented (we need the length and width expressions with exponents or variables), but assuming a common case where length and width are, for example, \( a^m \) and \( a^n \) (or other base - exponent forms). Let's assume the length is \( x^3 \) and the width is \( x^2 \) (just an example to show the method).

Step1: Recall the area formula of a rectangle

The area of a rectangle \( A = \text{length} \times \text{width} \). If length \( l=a^{m}\) and width \( w = a^{n}\), by the product of powers rule \( a^{m}\times a^{n}=a^{m + n}\)

Step2: Write the first expression

Using the formula directly, if length is \( a^{m}\) and width is \( a^{n}\), the first expression for area is \( A=a^{m}\times a^{n}\)

Step3: Write the second expression

Using the product of powers property \( a^{m}\times a^{n}=a^{m + n}\), the second expression is \( A = a^{m + n}\)

For example, if the length is \( x^{2}\) and the width is \( x^{3}\):

Step1: Apply the area formula

\( A=x^{2}\times x^{3}\)

Step2: Simplify using exponent rule

By \( a^{m}\times a^{n}=a^{m + n}\), \( x^{2}\times x^{3}=x^{2 + 3}=x^{5}\)

So two equivalent expressions could be \( x^{2}\times x^{3}\) and \( x^{5}\) (depending on the actual length and width expressions of the rectangle which were not fully provided in the problem, but this is the general method)

Answer:

(Assuming length \( = x^{2}\), width \( = x^{3}\)) Two equivalent expressions are \( \boldsymbol{x^{2}\times x^{3}} \) and \( \boldsymbol{x^{5}} \) (The answer will vary depending on the actual length and width of the rectangle given in the complete problem)