Question

what the route of 67

Answer 1

Given:

The number 67.

To find the square root of 67, use the rational approximation,

$\begin{gathered} \frac{p_0}{q_0},Herep_0,q_0\text{ are integers.} \\ p_{\mleft\{i+1\mright\}}=p^2_i+nq^2_i \\ q_{\mleft\{i+1\mright\}}=2p_iq_i \end{gathered}$

Apply this formula to get better approximation.

$\begin{gathered} p_0=8,n=67,q_0=1 \\ p_1=p^2_0+nq^2_0 \\ p_1=(8)^2+67(1)^2=131 \\ q_1=2p_0q_0=2(8)(1)=16 \end{gathered}$

Again apply the formula,

$\begin{gathered} p_2=p^2_1+nq^2_1 \\ p_2=131^2+(67)(16)^2=34313 \\ q_2=2p_1q_1=2(131)(16)=4192 \end{gathered}$

Now thw approximation becomes,

$\sqrt[]{67}=\frac{p_2}{q_2}_{}=\frac{34313}{4192}=8.18535$

Answer:

$\sqrt[]{67}=8.1854\text{ ( up to 4 decimal)}$