How do we find our approximation for ${{2.9}^{5}}$ ?

Hint: To solve this question, you can use the differentiation method. First you have to take x = 3 which is approximately near to 2.9. And then you have to differentiate and then find the tangent line around x = 3. Then you just have to substitute the value 2.9 and then you can get your answer.

Complete step by step solution:
According to the problem, we are asked to find our approximation for ${{2.9}^{5}}$.
For this, we use the differentiation method. First you have to take x = 3 which is approximately near to 2.9. And then you have to differentiate and then find the tangent line around x = 3. Then we will just have to substitute the value 2.9 and then we can get our answer.
Here, we first use $y={{x}^{5}}$. We take this as equation 1.
$\Rightarrow y={{x}^{5}}$------- (1)
Now we differentiate equation 1. Then we get:
$\Rightarrow \dfrac{dy}{dx}=5{{x}^{4}}$ ------- (2)
Now we write the tangent line. The slope can be derived as ${{5.3}^{4}}=405$ and the point of intersection can be derived as $\left( 3,{{3}^{5}} \right)=\left( 3,243 \right)$. Therefore, we get the tangent equation as:
$\Rightarrow \left( y-243 \right)=405\left( x-3 \right)$-
Now, we just have to substitute the value of 2.9 in that equation in x. Therefore after substituting, we get the answer as
$\Rightarrow y=202.5$
Therefore, after all the solving, we get the final answer of the question as y = 202.5 which is nearly equal to its original value 205.11.

Note: You need to be careful while calculating the values and while substituting the values. In order to solve this question you have to know how to write the tangent equation for a given equation. You can also solve the given question using binomial expansion.