Find the approximate value of (82)¹/⁴

The correct option is 3.00925.

To find the approximate value of ${82}^{\frac{1}{4}}$, we can use the concept of differentials or linear approximation from calculus.
Let us define the function:
$f\left(x\right)=x^{\frac{1}{4}}$

We choose a value of $x$ close to 82 that is a perfect fourth power. The nearest perfect fourth power to 82 is 81, since:
$3^4=3\times 3\times 3\times 3=81$

Thus, we set:
$x=81$
$\Delta x=dx=82-81=1$

First, we calculate the value of the function at $x=81$:
$f\left(81\right)={81}^{\frac{1}{4}}=3$

Next, we find the derivative of $f\left(x\right)$:
$f^{\prime }\left(x\right)=\frac{d}{dx}\left(x^{\frac{1}{4}}\right)=\frac{1}{4}{x}^{-\frac{3}{4}}=\frac{1}{4x^{\frac{3}{4}}}$

Now, we evaluate the derivative at $x=81$:
$f^{\prime }\left(81\right)=\frac{1}{4\times {81}^{\frac{3}{4}}}$
Since ${81}^{\frac{1}{4}}=3$, we have:
${81}^{\frac{3}{4}}={\left({81}^{\frac{1}{4}}\right)}^3$
Therefore:
$f^{\prime }\left(81\right)=\frac{1}{4\times 27}$

Using the linear approximation formula:
$f(x+\Delta x)\approx f\left(x\right)+f^{\prime }\left(x\right)\Delta x$
We substitute the values we computed:
${82}^{\frac{1}{4}}\approx 3+\frac{1}{108}\times 1$
${82}^{\frac{1}{4}}\approx 3+\mathrm{0.009259...}$
${82}^{\frac{1}{4}}\approx 3.00925$

Thus, the approximate value of ${82}^{\frac{1}{4}}$ matches option 3.00925.