A rectangle has width 2^(1/3)m and length 4^(1/3)m. What is the area of the rectangle?

The width of the rectangle is #2^(1/3) \ "m"# long, while the length of the rectangle is #4^(1/3) \ "m"# long.

We have:

#4^(1/3)=(2^2)^(1/3)#

#=2^(2/3)#

So, the length of the rectangle can be written as #2^(2/3) \ "m"# long.

Area of a rectangle is given by the length multiplied by the width. So we have,

#A=l*w#

#=2^2/3 \ "m"*2^1/3 \ "m"#

Recall that #a^b*a^c=a^(b+c)#. Therefore,

#=2^(2/3+1/3) \ "m"^2#

#=2^(3/3) \ "m"^2#

But, #3/3=1#, and so we got,

#=2^1 \ "m"^2#

Another important fact is that #a^1=a#, and so we have,

#=2 \ "m"^2#

So, the area of this rectangle is #2# meters squared.

Before we start, let's revise the exponent rules,

1. Product rule: #a^x xxa^y=a^(x+y#
2. Quotient rule: #a^x -:a^y=a^(x-y#
3. Power rule: #(a^x)^y=a^(xy)#
4. Power of a product rule: #(ab)^x=a^x xx b^x#
5. Power of a quotient rule: #(a/b)^x=(a^x)/(b^x)#
6. Zero exponent: #a^0=1#
7. Negative exponent: #a^-x=1/a^x#
8. Fractional exponent: #a^(x/y)=root(y)(a^x)#

Now let's begin, let the area of the rectangle be #A#,

#A=2^(1/3)xx4^(1/3)#

Using rule 4 - Power of a product rule,

#A=(2xx4)^(1/3)#
#color(white)(A)=8^(1/3)#
#color(white)(A)=2#

Therefore, the area of the rectangle is #2m^2#.