Respuesta :
[tex] \sqrt[3]{81} \times 3^{ \frac{2}{3 } } = \sqrt[3]{ {9}^{2} } \times {3}^{ \frac{2}{3} } = {9}^{ \frac{2}{3} } \times {3}^{ \frac{2}{3} } = \\ 3^{ \frac{4}{3} } \times {3}^{ \frac{2}{3} } = 3^{ \frac{6}{3} } = 9[/tex]
Answer:
The area of the rectangle is 9.
Step-by-step explanation:
The first thing to take into account is that the area of a rectangle is obtained by multiplying its length and width.
[tex]area=length\times width[/tex]
The problem says that the length is [tex]\sqrt[3]{81}[/tex] and the width is [tex]3^{\frac{2}{3}}[/tex]. So, to find the area we must replace the values of length and width into the previous expression.
[tex]area=length\times width[/tex]
[tex]area=(\sqrt[3]{81})\times (3^{\frac{2}{3}})[/tex]
The previous seems to be difficult but with a little of manipulation we can get a result without a calculator.
First step: the term [tex]\sqrt[3]{81}[/tex] could be written as exponential expression.
[tex]area=(81^{\frac{1}{3}})\times (3^{\frac{2}{3}})[/tex]
Second step: the term [tex]3^{\frac{2}{3}}[/tex] could be written in other way.
[tex]area=(81^{\frac{1}{3}})\times ((3^2)^{\frac{1}{3}})[/tex]
Third step: in the same way, the term [tex]81^{\frac{1}{3}}[/tex] could be written in other way taking into account that [tex]81=9^2[/tex].
[tex]area=((9^2)^{\frac{1}{3}})\times ((3^2)^{\frac{1}{3}})[/tex]
Forth step: the two terms must have the same base to apply other exponential rules; for example, the base could be 9.
[tex]area=(9^{\frac{2}{3}})\times (9^{\frac{1}{3}})[/tex]
Fifth step: as both terms have the same base, we must add the exponents and simplify the expression.
[tex]area=9^{\frac{2}{3}}\times 9^{\frac{1}{3}}[/tex]
[tex]area=9^{\frac{2}{3}+\frac{1}{3}}[/tex]
[tex]area=9^{\frac{3}{3}}[/tex]
[tex]area=9^1[/tex]
[tex]area=9[/tex]
Thus, the area of the rectangle is 9.
