Question

What is the side length of a cube with a volume of $128m^3.$ Is the side of the cube a surd?

• A. ${128}^{\frac{1}{3}}m$, yes
• B. $\frac{128}{3}m$, no
• C. ${128}^{3}m$, yes
• D. ${128}^{\frac{1}{2}}m$, no

Answer 1

The correct option is A ${128}^{\frac{1}{3}}m$, yes

Let the side of the cube be $a$ metres
Volume of cube $={\text{side}}^3$
$=a^3$
Volume of the cube given$=128m^3$
Therefore,
$a^3=128$
Taking $3^{rd}$ root of $128$
$a=(128{)}^{\frac{1}{3}}m$

We know: $128=2^7\Rightarrow (128{)}^{\frac{1}{3}}=(2{)}^{\frac{7}{3}}$

Answer: Side length of the cube is $(2{)}^{\frac{7}{3}}$ which does not yield a rational number. Therefore it is a surd.