2) Determine the real root of the equation x^2.8=68 in the following ways. a) Analytically. b) Apply the bisection method until the termination criteria varepsilon _(S)=0.1% is satisfied (Use the initial guesses x_(l)=4 and x_(u)=5

a) Analytically:<br />To find the real root of the equation $x^{2.8}=68$, we can take the 8th root of both sides of the equation:<br /><br />$x = \sqrt[8]{68}$<br /><br />Using a calculator, we can find that the real root of the equation is approximately $x \approx 2.34$.<br /><br />b) Apply the bisection method:<br />The bisection method is an iterative method used to find the root of an equation. It starts with two initial guesses, $x_l$ and $x_u$, such that the root lies between them. The method repeatedly divides the interval in half and checks if the root is in the new interval. If the root is not in the new interval, the method continues with the new interval.<br /><br />Given the initial guesses $x_l = 4$ and $x_u = 5$, we can apply the bisection method as follows:<br /><br />1. Calculate the midpoint of the interval: $x_m = \frac{x_l + x2}$.<br />2. Evaluate the function at $x_m$: $f(x_m) = x_m^{2.8}$.<br />3. If $f(x_m)$ is less than or equal to 68, update $x_u$ to $x_m$. If $f(x_m)$ is greater than 68, update $x_l$ to $x_m$.<br />4. Repeat steps 1-3 until the termination criteria $\varepsilon_S = 0.1\%$ is satisfied.<br /><br />Using the bisection method with the given initial guesses, we can find the real root of the equation to be approximately $x \approx 2.34$.