(see attachment)

To approximate the square root of 13:

Working from the top down...

Enter the number you are trying to approximate in the top box: [tex]\boxed{\sf \sqrt{13}}[/tex]

Find the perfect squares directly below and above 13.

Perfect squares: 1, 4, 9, 16, 25, 36, ...

Therefore, the perfect squares below and above 13 are: 9 and 16

Enter these with square root signs in the next two boxes: [tex]\boxed{\sf \sqrt{9}}[/tex] and [tex]\boxed{\sf \sqrt{16}}[/tex]

Carry out the operation and enter [tex]\boxed{\sf 3}[/tex] and [tex]\boxed{\sf 4}[/tex] in the next two boxes.

Enter the number you are trying to square root (13) in the top left box, the perfect square above it (16) in the box below, then the perfect square below it (9) in the two boxes to the right of these. Carry out the subtractions and place the numbers in the boxes to the right.

[tex]\dfrac{\boxed{\sf 13}-\boxed{\sf 9}}{\boxed{\sf 16}-\boxed{\sf 9}}=\dfrac{\boxed{\sf 4}}{\boxed{\sf 7}}[/tex]

Now enter the number you are trying to square root (13) under the square root sign. Place the square root of the perfect square below it (3) in the box to the right. Copy the fraction from above (4/7). Finally, enter this mixed number into a calculator and round to the nearest hundredth.

[tex]\sf \sqrt{13}=\boxed{\sf3}\dfrac{\boxed{\sf 4}}{\boxed{\sf 7}}=\boxed{\sf3.57}[/tex]