Question

A rectangle has a length of ∛81 and a width of 3 2/3 inches. Find the area of the rectangle.

A.3 2/3 power inches squared
B.3 8/3 power inches squared
C.9 inches squared
D. 9 2/3 power inches squared

Answer 1

We have
length = ∛81
width =
`3^{ (2)/(3) }`

Area =
`\sqrt[3]{81}` ×
`3^{ (2)/(3) }`
Area =
`81^{ (1)/(3) }` ×
`3^{ (2)/(3) }`
Area =
`( 3^(4)) ^{ (1)/(3) }` ×
`3^{ (2)/(3) }`
Area =
`3^{ (4)/(3) }` ×
`3^{ (2)/(3) }`
Area =
`3^{ (6)/(3) }`
Area =
`3^(2)`
Area = 9 inches squared

Answer 2

First express ∛81 as an exponential expression, using the properties of exponents and roots:


`\sqrt[3]{81}= 81^{ (1)/(3) }= (3^(4)) ^{ (1)/(3) }= 3^{ (4)/(3) }`

The area of the rectangle = length*width

=
`3^{ (4)/(3) } * 3^{ (2)/(3) } =3^{((4)/(3) +(2)/(3)) }=3^{ (6)/(3) }= 3^(2)=9` (inches squared).


Answer: 9 inches squared