What is the square root of 3.4?

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What is the square root of 3.4?

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Answer 1 · 2015-09-18T06:37:48

$\sqrt{3.4} = \frac{\sqrt{85}}{5} \approx 1.8439$ $sqrt(3.4) = sqrt(85)/5 ~~ 1.8439$

Explanation:

If $a, b \geq 0$ $a, b >= 0$ then $\sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}}$ $sqrt(a/b) = sqrt(a)/sqrt(b)$

If $a \geq 0$ $a >= 0$ then $\sqrt{a^{2}} = a$ $sqrt(a^2) = a$

So:

$\sqrt{3.4} = \sqrt{\frac{85}{25}} = \frac{\sqrt{85}}{\sqrt{25}} = \frac{\sqrt{85}}{5}$ $sqrt(3.4) = sqrt(85/25) = sqrt(85)/sqrt(25) = sqrt(85)/5$

Here I have chosen to express $3.4$ $3.4$ as $\frac{85}{25}$ $85/25$ to get the smallest whole value for the denominator.

I could also have written $\sqrt{3.4} = \sqrt{\frac{17}{5}} = \frac{\sqrt{17}}{\sqrt{5}}$ $sqrt(3.4) = sqrt(17/5) = sqrt(17)/sqrt(5)$

If I wanted to calculate an approximation for $\sqrt{3.4}$ $sqrt(3.4)$ by hand then I would probably have used this instead:

$\sqrt{3.4} = \sqrt{\frac{340}{100}} = \frac{\sqrt{340}}{10}$ $sqrt(3.4) = sqrt(340/100) = sqrt(340)/10$

Then I would work out an approximation for $\sqrt{340}$ $sqrt(340)$ and divide by $10$ $10$ .

For example: $18^{2} = 324$ $18^2 = 324$ and $19^{2} = 361$ $19^2 = 361$ , so:

$18 = \sqrt{324} < \sqrt{340} < \sqrt{361} = 19$ $18 = sqrt(324) < sqrt(340) < sqrt(361) = 19$

Using a Newton Raphson type method to find $\sqrt{340}$ $sqrt(340)$ , I might chose $18.5 = \frac{37}{2}$ $18.5 = 37/2$ as my first approximation $a_{0}$ $a_0$ , then make better approximations using the formula:

$a_{i + 1} = \frac{a_{i}^{2} + n}{2 a_{i}}$ $a_(i+1) = (a_i^2 + n)/(2a_i)$

where $n = 340$ $n = 340$ .

Then

$a_{1} = \frac{37^{2} + 340 \cdot 2^{2}}{2 \cdot 37 \cdot 2} = \frac{1369 + 1360}{148}$ $a_1 = (37^2 + 340 * 2^2) / (2 * 37 * 2) = (1369+1360)/148$

$= \frac{2729}{148} \approx 18.439$ $= 2729 / 148 ~~ 18.439$

So $\sqrt{340} \approx 18.439$ $sqrt(340) ~~ 18.439$ and $\sqrt{3.4} \approx 1.8439$ $sqrt(3.4) ~~ 1.8439$

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What is the square root of 3.4?

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