\[1300\]
is approximately 36 - \[36^2=1296\]
, so the square root is just a little bit more.We can find a better estimate by writing
\[\sqrt{1300}=36+x\]
where \[x\]
is small.Then
\[1300=(36+x)^2=1296+72x+x^2\]
.Since
\[x\]
is small, \[x^2\]
is very small and we can ignore it, so\[1300 \simeq 1296+72x\]
.\[4 \simeq 72x\]
.\[x \simeq \frac{4}{72} =0.0555555...\]
.Then
\[\sqrt{1300} \simeq 36.055555...\]
.In fact
\[\sqrt{1300}=36.0555127...\]
.The error is
\[\frac{36.0555555-36.0555127}{36.0555127} \times 100% = 1.2 \times 10^{-4} %=\]
.