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

Mar 12, 2018

$2 \setminus {\text{m}}^{2}$

#### Explanation:

The width of the rectangle is ${2}^{\frac{1}{3}} \setminus \text{m}$ long, while the length of the rectangle is ${4}^{\frac{1}{3}} \setminus \text{m}$ long.

We have:

${4}^{\frac{1}{3}} = {\left({2}^{2}\right)}^{\frac{1}{3}}$

$= {2}^{\frac{2}{3}}$

So, the length of the rectangle can be written as ${2}^{\frac{2}{3}} \setminus \text{m}$ long.

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

$A = l \cdot w$

$= {2}^{2} / 3 \setminus \text{m"*2^1/3 \ "m}$

Recall that ${a}^{b} \cdot {a}^{c} = {a}^{b + c}$. Therefore,

$= {2}^{\frac{2}{3} + \frac{1}{3}} \setminus {\text{m}}^{2}$

$= {2}^{\frac{3}{3}} \setminus {\text{m}}^{2}$

But, $\frac{3}{3} = 1$, and so we got,

$= {2}^{1} \setminus {\text{m}}^{2}$

Another important fact is that ${a}^{1} = a$, and so we have,

$= 2 \setminus {\text{m}}^{2}$

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

Mar 12, 2018

The area of the rectangle is $2 {m}^{2}$.

#### 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: ${\left({a}^{x}\right)}^{y} = {a}^{x y}$
4. Power of a product rule: ${\left(a b\right)}^{x} = {a}^{x} \times {b}^{x}$
5. Power of a quotient rule: ${\left(\frac{a}{b}\right)}^{x} = \frac{{a}^{x}}{{b}^{x}}$
6. Zero exponent: ${a}^{0} = 1$
7. Negative exponent: ${a}^{-} x = \frac{1}{a} ^ x$
8. Fractional exponent: ${a}^{\frac{x}{y}} = \sqrt[y]{{a}^{x}}$

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

$A = {2}^{\frac{1}{3}} \times {4}^{\frac{1}{3}}$

Using rule 4 - Power of a product rule,

$A = {\left(2 \times 4\right)}^{\frac{1}{3}}$
$\textcolor{w h i t e}{A} = {8}^{\frac{1}{3}}$
$\textcolor{w h i t e}{A} = 2$

Therefore, the area of the rectangle is $2 {m}^{2}$.