Use a linear approximation to estimate (8.2)^(2/3)

in general

f(x) = f(x0) + f ' (x0)*(x-x0),
where x0 is a value close to x and a value which we can calculate.

let f(x) = x^(2/3)
f ' (x) = (2/3)x^(-1/3) = 2/(3x^(1/3))

f(8) = 8^(2/3) = 4
f ' (8) = 2/(3(2)) = 1/3

f(8.2) = f(8) + f ' (8)*(8.2 - 8)
= 4 + (1/3)(.2)
= 4 + .2/3
= 4 + 2/30
= 122/30 or 61/15

using my calculator:
8.2^(2/3) = 4.066391931
my appr of 61/15 = 4.066666..
for an error of .000274.. , not bad

To estimate (8.2)^(2/3) using linear approximation, we can use the formula:

f(x) ≈ f(a) + f'(a) * (x - a)

where f(x) is the function we want to approximate, a is a known point, and f'(a) is the derivative of the function at point a.

Let's consider the function f(x) = x^(2/3), and choose a = 8 (since it is close to 8.2). The derivative of f(x) = x^(2/3) is given by:

f'(x) = (2/3) * x^(-1/3)

Now, we can plug in the values into the formula:

f(x) ≈ f(a) + f'(a) * (x - a)
≈ f(8) + f'(8) * (8.2 - 8)

First, let's find f(8):

f(8) = 8^(2/3)
= 4

Next, let's find f'(8):

f'(8) = (2/3) * 8^(-1/3)
= (2/3) * (1/2)
= 1/3

Now, we can substitute the calculated values into the formula:

f(x) ≈ 4 + (1/3) * (8.2 - 8)

Simplifying further:

f(x) ≈ 4 + (1/3) * 0.2
≈ 4 + 0.0667
≈ 4.0667

Therefore, using linear approximation, we estimate that (8.2)^(2/3) is approximately 4.0667.