Question

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

Answer 1

`2 \ "m"^2`

Explanation:

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.

Answer 2

The area of the rectangle is `2m^2`.

Explanation:

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`.