To solve the problem of finding the approximate value of $80^{\frac{3}{4}}$ using linear approximation, we will follow these steps:

Step 1: Define the function

The given expression is $80^{\frac{3}{4}}$. Let: $f(x) = x^{\frac{3}{4}}$

We will approximate $f(80)$ using the linear approximation formula.

Step 2: Linear Approximation Formula

Linear approximation is given by: $f(x) \approx f(a) + f'(a)(x - a)$ where $a$ is a value near $x$ for which the calculation of $f(a)$ is easy.

Step 3: Choose $a$ and calculate $f(a)$

We choose $a = 81$ because $81$ is a perfect fourth power, and: $f(81) = 81^{\frac{3}{4}} = (81^{\frac{1}{4}})^3 = (3)^3 = 27$

Step 4: Compute the derivative $f'(x)$

The derivative of $f(x) = x^{\frac{3}{4}}$ is: $f'(x) = \frac{3}{4}x^{-\frac{1}{4}} = \frac{3}{4\sqrt[4]{x}}$

At $x = 81$: $f'(81) = \frac{3}{4\sqrt[4]{81}} = \frac{3}{4 \cdot 3} = \frac{1}{4}$

Step 5: Use the linear approximation formula

Now substitute into the formula: $f(x) \approx f(a) + f'(a)(x - a)$ Substitute $a = 81$, $f(81) = 27$, $f'(81) = \frac{1}{4}$, and $x = 80$: $f(80) \approx 27 + \frac{1}{4}(80 - 81)$ $f(80) \approx 27 + \frac{1}{4}(-1)$ $f(80) \approx 27 - \frac{1}{4} = 27 - 0.25 = 26.75$

Final Answer:

The approximate value of $80^{\frac{3}{4}}$ using linear approximation is: $\boxed{26.75}$

Would you like additional details or have any follow-up questions?

Here are some related questions you might find helpful:

1. How does linear approximation compare to the actual value of $80^{3/4}$?
2. Can we use a different value of $a$ for the approximation? What happens if $a = 64$?
3. How do we calculate higher-order approximations for this problem?
4. Why is $a = 81$ a good choice for this problem?
5. How does the derivative influence the linear approximation's accuracy?

Tip: Always choose a nearby value of $a$ that simplifies calculations when using linear approximation.