Compute approximately: 1° sqrt(1.009); 2° sqrt(128); 3°

Compute approximately: 1° $\sqrt{1.009}$; 2° $\sqrt{128}$; 3° $\sqrt[7]{0.9979}$; 4° $\sqrt[10]{1027}$.

Step 1: The tool

From Problem 259, \[\sqrt[n]{a^{n}+x} \approx a+\frac{x}{na^{\,n-1}} \qquad (|x|\ll a^{n}), \tag{$*$}\]

equivalently $\bigl(1+\alpha\bigr)^{1/n}\approx1+\dfrac{\alpha}{n}$. The only decision in each part is which perfect power $a^{n}$ to take as the base point.

Step 2: Part 1°: $\sqrt{1.009}$

Base point $1$, so $\alpha = 0.009$ and $n = 2$: $\sqrt{1.009} \approx 1+\frac{0.009}{2} = 1.0045$.

Exact: $1.0044899$. The estimate is high by $10^{-5}$, and squaring back gives $1.0045^{2} = 1.00900025$.

Step 3: Part 2°: $\sqrt{128}$

The nearest perfect square below $128$ is $11^{2} = 121$, so $x = 7$: \[\sqrt{128} = \sqrt{11^{2}+7} \approx 11+\frac{7}{2\cdot11} = 11+\frac{7}{22} = 11.3182.\]

Exact: $8\sqrt2 = 11.3137085$, so the estimate is high by $0.00447$. This is the least accurate of the four, because $x/a^{n} = 7/121 \approx 5.8\%$ is much the largest relative perturbation here.

Step 4: Part 3°: $\sqrt[7]{0.9979}$

Base point $1$ again, with $\alpha = -0.0021$ and $n = 7$: $\sqrt[7]{0.9979} \approx 1-\frac{0.0021}{7} = 1-0.0003 = 0.9997$.

Exact: $0.9996997$ — agreement to seven decimal places.

Step 5: Part 4°: $\sqrt[10]{1027}$

Here the useful base point is $2^{10} = 1024$, giving $x = 3$: \[\sqrt[10]{1027} \approx 2+\frac{3}{10\cdot2^{9}} = 2+\frac{3}{5120} = 2+0.0005859 = 2.00059.\]

Exact: $2.0005852$. The error is under $10^{-6}$, since $x/a^{n} = 3/1024$ is tiny and $n = 10$ divides it further.

Step 6: Verification

\[\begin{array}{c|c|c|c|c} & a^{n}+x & x/a^{n} & \text{estimate} & \text{exact} \\ \hline \sqrt{1.009} & 1+0.009 & 0.9\% & 1.0045000 & 1.0044899 \\ \sqrt{128} & 121+7 & 5.8\% & 11.318182 & 11.313708 \\ \sqrt[7]{0.9979} & 1-0.0021 & 0.21\% & 0.9997000 & 0.9996997 \\ \sqrt[10]{1027} & 1024+3 & 0.29\% & 2.0005859 & 2.0005852 \end{array}\]

\[\text{Independent check of 4}^{\circ}\text{: } \ln1027 = 10\ln2+\ln\!\left(1+\tfrac{3}{1024}\right) \approx 6.93147+0.00293 = 6.93440\]

\[e^{0.693440} = 2\,e^{0.000293} \approx 2\times1.000293 = 2.000586 \;\;\checkmark\]