Find the approximate value of (127)¹/³

The correct option is 5.0267.

To find the approximate value of ${\left(127\right)}^{1/3}$, we can use the method of linear approximation (or differentials) from calculus.

Let us define a function:
$f\left(x\right)=x^{1/3}$

We want to find the value of $f\left(127\right)$. We choose a value $a$ close to $127$ whose cube root is easy to calculate.
Let $a=125$, since ${125}^{1/3}=5$.
The difference between the target value and $a$ is:
$\Delta x=127-125=2$

Using the differential approximation formula:
$f\left(a+\Delta x\right)\approx f\left(a\right)+f^{\text{'}}\left(a\right)·\Delta x$

First, we find the derivative of $f\left(x\right)$:
$f^{\text{'}}\left(x\right)=\frac{1}{3}{x}^{-2/3}=\frac{1}{3x^{2/3}}$

Now, evaluate the function and its derivative at $a=125$:
$f\left(125\right)={125}^{1/3}=5$
$f^{\text{'}}\left(125\right)=\frac{1}{3·{\left({125}^{1/3}\right)}^2}=\frac{1}{3·5^2}=\frac{1}{3·25}=\frac{1}{75}$

Substitute these values back into the approximation formula:
${\left(127\right)}^{1/3}\approx 5+\left(\frac{1}{75}\right)·2$
${\left(127\right)}^{1/3}\approx 5+\frac{2}{75}$

Now, convert the fraction to a decimal:
$\frac{2}{75}\approx 0.02667$

Adding this to $5$ gives:
${\left(127\right)}^{1/3}\approx 5+0.02667=5.02667$

Rounding to four decimal places, we get approximately 5.0267.