What is the square root of 67?

Question

What is the square root of 67?

2 Answers

Impact of this question

23291 views around the world

You can reuse this answer
Creative Commons License

Answer 1 · 2016-10-12T17:25:34

$67$ is a prime, and cannot be factored......

Explanation:

.........and thus $67^(1/2)$ $=$ $+-sqrt67$ .

Answer 2 · 2017-02-19T00:27:38

$sqrt(67) ~~ 34313/4192 ~~ 8.185353$

Explanation:

$67$ is a prime number, so in particular has no square factors. So its square root is irrational and not simplifiable.

There are several methods you can use to find rational approximations.

Here's a method based on the Babylonian method...

To find the square root of a number $n$ , choose an initial approximation $p_0/q_0$ where $p_0, q_0$ are integers.

Then apply the following formulas repeatedly to get better approximations:

${ (p_(i+1) = p_i^2+n q_i^2), (q_(i+1) = 2 p_i q_i) :}$

In our example, let $n = 67$ , $p_0 = 8$ and $q_0 = 1$ , since $8^2 = 64$ is quite close to $67$ . Then:

${ (p_1 = p_0^2+n q_0^2 = 8^2+67*1^2 = 64+67 = 131), (q_1 = 2 p_0 q_0 = 2*8*1 = 16) :}$

${ (p_2 = p_1^2 + n q_1^2 = 131^2+67*16^2 = 17161+17152 = 34313), (q_2 = 2 p_1 q_1 = 2*131*16 = 4192) :}$

If we stop here, we get:

$sqrt(67) ~~ 34313/4192 ~~ 8.185353$

which is accurate to $6$ decimal places.

Science

Math

Humanities

... and beyond

What is the square root of 67?

2 Answers

Explanation:

Explanation:

Related questions

Impact of this question