Question

A rectangle has a length of \sqrt[3]{81} and a width of 3^{\frac{2}{3}} inches. Find the area of the rectangle.

Answer 1

`\sqrt[3]{81} * 3^{ (2)/(3 ) } = \sqrt[3]{ {9}^(2) } * {3}^{ (2)/(3) } = {9}^{ (2)/(3) } * {3}^{ (2)/(3) } = \\ 3^{ (4)/(3) } * {3}^{ (2)/(3) } = 3^{ (6)/(3) } = 9`

Answer 2

The area of the rectangle is 9.

Explanation:

The first thing to take into account is that the area of a rectangle is obtained by multiplying its length and width.

`area=length* width`

The problem says that the length is
`\sqrt[3]{81}` and the width is
`3^{(2)/(3)}`. So, to find the area we must replace the values of length and width into the previous expression.

`area=length* width`

`area=(\sqrt[3]{81})* (3^{(2)/(3)})`

The previous seems to be difficult but with a little of manipulation we can get a result without a calculator.

First step: the term
`\sqrt[3]{81}` could be written as exponential expression.

`area=(81^{(1)/(3)})* (3^{(2)/(3)})`

Second step: the term
`3^{(2)/(3)}` could be written in other way.

`area=(81^{(1)/(3)})* ((3^2)^{(1)/(3)})`

Third step: in the same way, the term
`81^{(1)/(3)}` could be written in other way taking into account that
`81=9^2`.

`area=((9^2)^{(1)/(3)})* ((3^2)^{(1)/(3)})`

Forth step: the two terms must have the same base to apply other exponential rules; for example, the base could be 9.

`area=(9^{(2)/(3)})* (9^{(1)/(3)})`

Fifth step: as both terms have the same base, we must add the exponents and simplify the expression.

`area=9^{(2)/(3)}* 9^{(1)/(3)}`

`area=9^{(2)/(3)+(1)/(3)}`

`area=9^{(3)/(3)}`

`area=9^1`

`area=9`

Thus, the area of the rectangle is 9.