Question

How to find irrational numbers in roots

Answer 1

Answer:

Step-by-step explanation:

Find a Starting Value

Find the perfect squares that would be to either side of √8 on the numberline. In this case, √4 = 2 and √9 = 3. Choose the one that's closest to your target number. Since 8 is much closer to 9 than to 4, choose √9 = 3.

Divide by Your Estimate

Next, divide the number whose root you want – 8 – by your estimate. Continuing the example, you have:

8 ÷ 3 = 2.67

Compute the Average

Now, find the average of the result from Step 2 with the divisor from Step 2. Here, that means averaging 3 and 2.67. First add the two numbers together, and then divide by two:

3 + 2.67 = 5.6667 (This is actually the repeating decimal 5.6666666666, but it has been rounded to four decimal places for the sake of brevity.)

5.6667 ÷ 2 = 2.83335

Repeat Steps 2 and 3 as Needed

The result from Step 3 still isn't exact, but it's getting closer. Repeat Steps 2 and 3 as needed, using the result from Step 3 as the new divisor in Step 2 every time.

To continue the example, you would divide 8 by the result from Step 3 (2.83335), which gives you:

8 ÷ 2.83335 = 2.8235 (Again, rounding to four decimal places for the sake of brevity.)

You would then average the result of your division with the divisor, which gives you:

2.83335 + 2.8235 = 5.65685

5.65685 ÷ 2 = 2.828425

You can continue this process, repeating Steps 2 and 3 as needed, until the answer is as exact as you need it to be.