Question

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

Answer 1

We have
length = ∛81
width = $3^{ \frac{2}{3} }$

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

Answer 2

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

$\sqrt[3]{81}= 81^{ \frac{1}{3} }= (3^{4}) ^{ \frac{1}{3} }= 3^{ \frac{4}{3} }$

The area of the rectangle = length*width

=$3^{ \frac{4}{3} } * 3^{ \frac{2}{3} } =3^{(\frac{4}{3} +\frac{2}{3}) }=3^{ \frac{6}{3} }= 3^{2}=9$ (inches squared).


Answer: 9 inches squared