Answer: The correct option is the second one, counting from the top.
Step-by-step explanation:
For a rectangle of length L, and width W, the area can be calculated as:
A = L*W
In this case, we know that:
[tex]L = \sqrt{x}[/tex]
and
[tex]W = \sqrt[3]{x^2}[/tex]
First, we need to remember the relations;
[tex]\sqrt[n]{x^m} = x^{m/n}[/tex]
[tex]x^m*x^n = x^{m + n}[/tex]
[tex]\frac{x^n}{x^m} = x^{n - m}[/tex]
a) Now we can calculate the area of the rectangle as:
[tex]A = L*W = \sqrt{x} *\sqrt[3]{x^2} = (x^{1/2})*(x^{2/3}) = x^{1/2 + 2/3} = x^{3/6 + 4/6} = x^{1/6 + 1}[/tex]
And we can write that last part as:
[tex]x^{1/6 + 1} = x*x^{1/6} = x*\sqrt[6]{x}[/tex]
b) Now we want to find the ratio between the width and the length:
[tex]\frac{W}{L} = \frac{\sqrt[3]{x^2} }{\sqrt{x} } = \frac{x^{2/3}}{x^{1/2}} = x^{2/3 - 1/2} = x^{4/6 - 3/6} = x^{1/6} = \sqrt[6]{x}[/tex]
Now, if x = 1, the ratio will be equal to 1.
if x > 1, the ratio will be larger than 1.
if 0 < x < 1, the ratio will be smaller than 1.
The correct option is the second one, counting from the top.
