# What is the square root of 41.7 and 0.6781, and 0.8?

Mar 17, 2018

$\sqrt{41.7} \approx 6.4576$
$\sqrt{0.6781} \approx 0.8196$
$\sqrt{0.8} \approx 0.89443$

#### Explanation:

Given: Find the square root of 41.7, 0.6781 and 0.8

If you use a calculator:
$\sqrt{41.7} \approx 6.4576$
$\sqrt{0.6781} \approx 0.8196$
$\sqrt{0.8} \approx 0.89443$

To find a square root without a calculator takes some time.
For example, hopefully you know that the $\sqrt{36} = 6$ and the $\sqrt{49} = 7$.

Since $36 < 41.7 < 49$, you would know that the $\sqrt{41.7}$ is between $6$ and $7$.

If you take the difference between 41.7 and 36 & 49 and 41.7, you would see that 41.7 is closer to 36. This would mean the #sqrt(41.7) is less than 6.5.

${6.5}^{2} = 42.25$
${6.4}^{2} = 40.96$

This means $\sqrt{41.7} \approx 6.4 \ldots$

${6.45}^{2} = 41.6025$

As you can see we are getting closer to $41.7$. We need a number slightly larger.

Try ${6.455}^{2} = 41.667025$

As you can see we are getting closer to $41.7$

Try ${6.456}^{2} = 41.679936$

Try ${6.457}^{2} = 41.692849$

Try ${6.458}^{2} = 41.705764$

Because ${6.457}^{2} < 41.7 < {6.458}^{2}$ We would have to add another decimal to $6.457$ to approach $41.7$

Try ${6.4575}^{2} = 41.69930625$

This process can be very tedious, but works.