Question

Use the Babylonian method to estimate √80 to the nearest hundredth.

Answer 1

The answer is 8.94.

Most people guess that a square root is half, so we will make that our first guess; 80/2 = 40, so 40 is our guess.
80/40 = 2
Average 40 & 2: (40+2)/2 = 21; this is our new guess.

80/21 = 3.80952381
Average 21 & 3.80952381: 12.40476191 is our new guess

80/12.40476191 = 6.449136274
Average 12.40476191 & 6.449136274: 9.426949092 is our new guess

80/9.426949092 = 8.48630869
Average 9.426949092 & 8.48630869 = 8.956628891 is our new guess

80/8.956628891 = 8.931931977
Average 8.956628891 & 8.931931977: 8.944280434 is our new guess

80/8.944280434 = 8.944280434
Average 8.944280434 & 8.944280434 = 8.944263386

By this point, we have the same number to the hundredths place, so we stop. Our estimate is 8.94.

Answer 2

Using a calculator we get that
`√(80)` to the nearest hundredth is 8.94, so we are going to use the Babylonian method to get that number:

Step 1. We are going to make a guess. Our guess is 9 because
`9^2=81`.

Step 2. We are going to dive the original radicand (80) by our guest:

`(80)/(9) =8.88`

Step 3. We are going to find the average between our initial guest and the result of our previous calculation:

`(8.88+9)/(2) = (17.88)/(2) =8.94`

We can conclude that
`√(80)` to the nearest hundredth using the Babylonian method is: 8.94