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
and

Answer 1

To approximate the square root of 46, we recognize that it falls between the whole numbers 6 and 7. We can check the squares of these two numbers:

• \( 6^2 = 36 \)
• \( 7^2 = 49 \)

Since \( 36 < 46 < 49 \), we confirm that:

\[ 6 < \sqrt{46} < 7 \]

Now, let's narrow it down further by testing some numbers between 6 and 7:

1. Check \( 6.5 \): \[ 6.5^2 = 42.25 \] \( 42.25 < 46 \)

2. Check \( 6.7 \): \[ 6.7^2 = 44.89 \] \( 44.89 < 46 \)

3. Check \( 6.8 \): \[ 6.8^2 = 46.24 \] \( 46.24 > 46 \)

So now we have:

\[ 6.7 < \sqrt{46} < 6.8 \]

Thus, \( \sqrt{46} \) must lie between 6.7 and 6.8.

To one decimal place, we estimate that:

\[ \sqrt{46} \approx 6.8 \]

So, the final answer would be that \( \sqrt{46} \) must lie between 6.7 and 6.8.