How to Find the Square Root of Any Number Faster Than a Calculator

[00:00:00]Hey everyone, welcome back.

[00:00:03]Quick question.

[00:00:04]If I asked you to find the square root of 87 right now without using a calculator, could you do it?

[00:00:14]Most people would probably reach for their phone immediately.

[00:00:18]But today, I'm going to show you a mental math shortcut that feels almost like a magic trick.

[00:00:26]Here's the shortcut.

[00:00:28]The square root of x squared plus a is approximately equal to x plus a divided by 2x.

[00:00:45]Let's start with the square root of 87.

[00:00:49]First, find the nearest perfect square below 87.

[00:00:55]That's 81.

[00:00:57]And 81 is 9 squared.

[00:01:00]So, our value of x is 9.

[00:01:04]The difference between 87 and 81 is 6.

[00:01:08]So, a equals 6.

[00:01:11]Now, apply the shortcut.

[00:01:14]Take 9 and add 6 divided by 2 * 9.

[00:01:21]2 * 9 is 18.

[00:01:24]So, we get 9 + 6 over 18. But, wait a second. 6 over 18 simplifies to 1/3.

[00:01:36]And 1/3 is about 0.33.

[00:01:40]So, our estimate becomes 9.33.

[00:01:45]Let's compare that with the actual value.

[00:01:48]The square root of 87 is approximately 9.327.

[00:01:55]That's incredibly close.

[00:01:58]Let's try another one.

[00:02:00]This time, the square root of 50.

[00:02:04]The nearest perfect square below 50 is 49.

[00:02:09]And 49 is 7 squared. So, X = 7.

[00:02:14]The difference between 50 and 49 is 1.

[00:02:18]So, A = 1.

[00:02:20]Now, plug it into the formula.

[00:02:23]7 + 1 divided by 2 * 7.

[00:02:27]That's 7 + 1 over 14.

[00:02:31]1 over 14 is approximately 0.071.

[00:02:36]So, our estimate is 7.071.

[00:02:40]And the actual value?

[00:02:43]The square root of 50 is approximately 7.07106.

[00:02:49]That's amazingly accurate. We practically nailed it.

[00:02:54]Ready for a slightly bigger number?

[00:02:57]Let's try the square root of 105.

[00:03:01]The nearest perfect square below it is 100.

[00:03:05]And 100 is 10 squared.

[00:03:08]So, X = 10.

[00:03:11]The difference is 5. So, A = 5.

[00:03:15]Using the shortcut.

[00:03:17]10 + 5 / 20.

[00:03:21]5 / 20 is 1/4.

[00:03:24]And 1/4 is 0.25.

[00:03:28]So, our estimate is 10.25.

[00:03:32]Let's check.

[00:03:33]The actual square root of 105 is approximately 10.247.

[00:03:40]Once again, extremely close. And all of that was done mentally.

[00:03:46]Let's do one more, rapid-fire style.

[00:03:50]The square root of 18.

[00:03:53]>> [clears throat] >> Nearest perfect square? 16. So, X = 4.

[00:03:59]The difference between 18 and 16 is 2.

[00:04:04]That means A = 2.

[00:04:07]Now, use the shortcut.

[00:04:10]4 + 2 / 8.

[00:04:14]2 / 8 is 1/4, which is 0.25.

[00:04:20]So, our estimate is 4.25.

[00:04:24]And the actual value?

[00:04:27]Approximately 4.242.

[00:04:31]Once again, remarkably close.

[00:04:34]Now, is this formula always exact? No.

[00:04:39]It's an approximation. But for numbers that are reasonably close to a perfect square, the accuracy is often surprisingly good, and that's what makes it such a powerful mental math trick.

[00:04:54]Now, here's a challenge for you. Try estimating the square root of 27 or the square root of 39 using this method.

[00:05:03]Then, compare your answer with a calculator.

[00:05:06]Drop your estimates in the comments and let me know how close you got.

[00:05:11]And I'll see you in the next one.