Let's approximate the value of

Question

26
​
.
26
​
is the same as the side length of a square whose area is 26 units
2
. So, to approximate the value of
26
​
, you can find which two perfect squares a square with an area of 26 units
2
is between. Start by finding the areas of squares with the following whole number side lengths.
26
​
units
A = 26 units
2


Area (square units)
1
2
=
2
2
=
3
2
=
4
2
=
5
2
=
6
2
=

Answer 1

To approximate \(\sqrt{26}\), we first find the areas of squares with integer side lengths, as follows:

• For \(1\): \[ 1^2 = 1 \text{ square unit} \]

• For \(2\): \[ 2^2 = 4 \text{ square units} \]

• For \(3\): \[ 3^2 = 9 \text{ square units} \]

• For \(4\): \[ 4^2 = 16 \text{ square units} \]

• For \(5\): \[ 5^2 = 25 \text{ square units} \]

• For \(6\): \[ 6^2 = 36 \text{ square units} \]

Now we can see that the area of the square with an area of 26 square units lies between the areas of squares with the following side lengths:

• The area of a square with a side length of \(5\) (which is \(25\) square units) is less than \(26\).
• The area of a square with a side length of \(6\) (which is \(36\) square units) is greater than \(26\).

Thus, we can conclude that: \[ 5 < \sqrt{26} < 6 \]

This gives us the approximation of \(\sqrt{26}\) lying between the integer values \(5\) and \(6\). To further narrow it down, you might want to estimate between \(5.0\) and \(6.0\) based on additional calculations or a calculator. The approximate value of \(\sqrt{26}\) is about \(5.1\) to \(5.2\).