# How do you find the exact square root of 39?

Oct 6, 2015

$39 = 3 \cdot 13$ has no square factors, so its square root cannot be simplified. It is an irrational number, so cannot be represented by a fraction or by a terminating or repeating decimal expansion.

#### Explanation:

If you would like to prove that $\sqrt{39}$ is irrational, you can prove it in a similar way to $\sqrt{15}$ as shown here:

You can find a succession of rational approximations for $\sqrt{39}$ using a Newton Raphson type method.

Typically you would start with an approximation ${a}_{0}$ and iterate using a formula like:

${a}_{i + 1} = \frac{{a}_{i}^{2} + n}{2 {a}_{i}}$

where $n = 39$ is the number you are trying to approximate the square root of.

I prefer to split ${a}_{i}$ into a fraction ${p}_{i} / {q}_{i}$ and iterate using the formulas:

${p}_{i + 1} = {p}_{i}^{2} + n {q}_{i}^{2}$

${q}_{i + 1} = 2 {p}_{i} {q}_{i}$

If the resulting ${p}_{i + 1}$ and ${q}_{i + 1}$ have a common factor, then divide both by that factor before the next iteration...

So in our case, let $n = 39$, ${p}_{0} = 6$ and ${q}_{0} = 1$. We use an initial approximation $\sqrt{39} \approx \frac{6}{1}$ since ${6}^{2} = 36$.

Then:

${p}_{1} = {p}_{0}^{2} + n {q}_{0}^{2} = {6}^{2} + 39 \cdot {1}^{2} = 36 + 39 = 75$

${q}_{1} = 2 {p}_{0} {q}_{0} = 2 \cdot 6 \cdot 1 = 12$

Now both of these are divisible by $3$, so divide both by $3$ to get:

${p}_{1 a} = 25$

${q}_{1 a} = 4$

If we stopped here we would get $\sqrt{39} \approx \frac{25}{4} = 6.25$

Next iteration:

${p}_{2} = {p}_{1 a}^{2} + n {q}_{1 a}^{2} = {25}^{2} + 39 \cdot {4}^{2} = 625 + 624 = 1249$

${q}_{2} = 2 {p}_{1 a} {q}_{1 a} = 2 \cdot 25 \cdot 4 = 200$

If we stopped here we would get $\sqrt{39} \approx \frac{1249}{200} = 6.245$

Next iteration:

${p}_{3} = {p}_{2}^{2} + n {q}_{2}^{2} = {1249}^{2} + 39 \cdot {200}^{2} = 1560001 + 1560000 = 3120001$

${q}_{3} = 2 \cdot 1249 \cdot 200 = 499600$

If we stop here, we get the approximation:

$\sqrt{39} \approx \frac{3120001}{499600} \approx 6.2449979983987$

Actually:

$\sqrt{39} \approx 6.24499799839839820584$