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.