Question

A rectangle has a length of the fifth root of 16 inches and a width of 2 to the 1 over 5 power inches. Find the area of the rectangle. A. 2 to the 3 over 5 power inches squared B. 2 to the 4 over 5 power inches squared C. 2 inches squared D. 2 to the 2 over 5 power inches squared

Answer 1

C.
`2\text{ Inches}^2`

Explanation:

We have been given that length of rectangle is
`\sqrt[5]{16}` inches and width is
`2^{(1)/(5)}` inches. We are asked to find the area of the given rectangle.

`\text{Area of rectangle}=\text{Length}* \text{Width}`

Substitute given values:

`\text{Area of rectangle}=\sqrt[5]{16}\text{ Inches}* 2^{(1)/(5)}\text{ Inches}`

Writing 16 as
`2^4`.

`\text{Area of rectangle}=\sqrt[5]{2^4}\text{ Inches}* 2^{(1)/(5)}\text{ Inches}`

Using property
`\sqrt[n]{a^m}=a^(m)/(n)}`, we will get:

`\text{Area of rectangle}=2^{(4)/(5)}\text{ Inches}* 2^{(1)/(5)}\text{ Inches}`

`\text{Area of rectangle}=2^{(4)/(5)}* 2^{(1)/(5)}\text{ Inches}^2`

Using property
`a^m* a^n=a^(m+n)`, we will get:

`\text{Area of rectangle}=2^{(4)/(5)+(1)/(5)}\text{ Inches}^2`

`\text{Area of rectangle}=2^{(4+1)/(5)}\text{ Inches}^2`

`\text{Area of rectangle}=2^{(5)/(5)}\text{ Inches}^2`

`\text{Area of rectangle}=2^(1)\text{ Inches}^2`

`\text{Area of rectangle}=2\text{ Inches}^2`

Therefore, the area of the given rectangle is 2 square inches and option C is the correct choices.

Answer 2

The area (A) of the rectangle is the product of its width (W) and length (L). Mathematically,
A = L x W
Substituting the given values,
A = (16^1/5) x (2^1/5)
Since they have the same exponents, the simplified expression for the area is,
A = 32^1/5
The value of area is 2. Thus, the answer is letter C.