Approximate square root of, 46, end square root

Question

46
​
by following the steps below.

square root of, 46, end square root
46
​
must lie between the whole numbers
6
6 and
7
7 because 6, squared, equals, start box, 36, end box6
2
=
36
​
and 7, squared, equals, start box, 49, end box7
2
=
49
​
, and 46 lies between these values.
Drag square root of, 46, end square root
46
​
based on your estimate above:
Real
√46
__
_
To one decimal place, square root of, 46, end square root
46
​
must lie between
6.7
6.7 and
6.8
6.8 because 6, point, 7, squared, equals, start box, 44, point, 8, 9, end box6.7
2
=
44.89
​
and 6, point, 8, squared, equals, start box, 46, point, 2, 4, end box6.8
2
=
46.24
​
, and 46 lies between these values.
Drag square root of, 46, end square root
46
​
based on your estimate above:
Real
√46
__
_
To two decimal places, square root of, 46, end square root
46
​
must lie between
and

Answer 1

To find the square root of $46$ to two decimal places, we first look for two decimal values between which the square root lies.

We have already established that:

• ${6.7}^2=44.89$
• ${6.8}^2=46.24$

Now let's check $6.7$ and $6.8$ further to narrow it down to two decimal places.

Checking $6.75$:

$${6.75}^2=45.5625$$

Checking $6.76$:

$${6.76}^2=45.6976$$

Checking $6.77$:

$${6.77}^2=45.8129$$

Checking $6.78$:

$${6.78}^2=45.8684$$

Checking $6.79$:

$${6.79}^2=46.0641$$

Since ${6.75}^2=45.5625<46$ and ${6.79}^2=46.0641>46$, we can narrow it down further.

Checking $6.76$:

Earlier we found ${6.76}^2=45.6976<46$

Checking $6.77$:

Earlier we found ${6.77}^2=45.8129<46$

Checking $6.78$:

Earlier we found ${6.78}^2=45.8684<46$

Checking $6.79$:

We found ${6.79}^2=46.0641>46$

Thus, we can conclude that:

$$\sqrt{46}\text{ lies between }6.78\text{ and }6.79.$$

So, the square root of $46$ lies between 6.78 and 6.79 to two decimal places.