# Mental Math: Square root of 9.8596 =___this is mental math question. Is there any trick to do this without using calculator? Thanks

lfryerda | High School Teacher | (Level 2) Educator

Posted on

There's another way to easily estimate square roots using Newton's method, which is from calculus.  This won't get you a perfect result like the method given above, but is also easy to give a quick (30 second) answer.  Also, the method above is only going to work with special exact square roots, whereas this method gives quick estimates for all square roots.

Newton's method is to find the zeros of a function and takes an initial guess at the zero and iterates it until you are happy with the result.

The formula itself is `x_{n+1}=x_n-{f(x_n)}/{f'(x_n)}`

In this case, finding the square root of a number is the same as finding the zero of the function `f(x)=x^2-a`

where a is the number you want for a square root (in this case 9.8596).

Its derivative is `f'(x)=2x` , so Newton's method simplifies to

`x_{n+1}=x_n-{x_n^2-a}/{2x_n}`

`={2x_n^2-x_n^2+a}/{2x_n}`

`={x_n^2+a}/{2x_n}`

`=1/2x_n+1/2 a/x_n`

Now at first this looks very complicated, but it's easy to get a quick result.

In this example, a=9.8596, and the square root of a is going to be a little bigger than 3.  So let's guess `x_1=3.1`.

The next guess uses the formula for Newton's method.  This means that we have

`x_2=1/2 (3.1)+ 1/2 (9.8596) 1/3`

Notice that I cheated a bit on the second term.  That's because it's just an estimate and I only want to be fast with the division, not super-accurate.  The first part is 1.55, and then we just do some estimating on the second part.  One half of 9.8596 is about 4.93 and now we can divide that by 3 to get about 1.64.  This means that our final guess for the square root is 1.55 and 1.64 added together which is about 3.19.

We can take a third guess, and calculate, but this starts to get a little tedious using mental math.

For very little calculation, we now have the square root of 9.8596 is 3.19, which is very close to the actual answer of 3.14

Sources:

embizze | High School Teacher | (Level 1) Educator Emeritus

Posted on

Find the square root of 9.8596:

96 is not a perfect square, but 196 is. 9 is a perfect square.

So you try `(3+14/100)^2` (Since `14^2=196)`

Expanding the binomial you get `3^2+2(42/100)+196/10000`

which is `9+84/100+1/100+96/10000=9.8596` as required.