Use the following approximation
\[
\left(1+x^{2}\right)^{-3} \approx 1-3 x^{2}+6 x^{4}-10 x^{6}+15 x^{8}-21 x^{10}
\]
to estimate $\left(\frac{16}{17}\right)^{3}$. Round the answer to three decimal places.

Answer

Expert–verified

Hide Steps

Answer

Final Answer: The approximation of $\left(\frac{16}{17}\right)^{3}$ is $\boxed{1.199}$.

Steps

Step 1 ：Rewrite $\left(\frac{16}{17}\right)^{3}$ as $\left(1-\frac{1}{17}\right)^{3}$, which is in the form of $\left(1+x^{2}\right)^{-3}$ where $x^2 = -\frac{1}{17}$.

Step 2 ：Substitute $x^2$ into the approximation and calculate the result.

Step 3 ：$x^2 = -0.058823529411764705$

Step 4 ：approximation = 1.199461635925308

Step 5 ：Final Answer: The approximation of $\left(\frac{16}{17}\right)^{3}$ is $\boxed{1.199}$.

Use the following approximation\[\left(1+x^{2}\right)^{-3} \approx 1-3 x^{2}+6 x^{4}-10 x^{6}+15 x^{8}-21 x^{10}\]to estimate $\left(\frac{16}{17}\right)^{3}$. Round the answer to three decimal places.

Answer

Popular Problems

Use the following approximation
\[
\left(1+x^{2}\right)^{-3} \approx 1-3 x^{2}+6 x^{4}-10 x^{6}+15 x^{8}-21 x^{10}
\]
to estimate $\left(\frac{16}{17}\right)^{3}$. Round the answer to three decimal places.