Question

How do you simplify `sqrt(75/4 )`?

Answer 1

`\frac{5sqrt(3)}{2}`

Explanation:

When you need to simplify numerical root, you should look for these two key facts:

• `sqrt(a^2) = a`
• `sqrt(ab)=sqrt(a)sqrt(b)`

So, anytime you have a number inside a root, you should try to write it as a product of other numbers, of which at least one is a perfect square. Let's analyze your case.

First of all, using the second property, we can write

`sqrt(75/4) = sqrt(75)/sqrt(4)`

In fact, every fraction can be read as multiplication using

`\frac{a}{b} = a * \frac{1}{b}`

Now let's deal with the two roots separately: I'd start with the denominator, since we already have a perfect square under a square root, so they simplify (see first rule above): we have

`sqrt(75) / sqrt(4) = sqrt(75)/2`

As for `sqrt(75)`, we can see that `75 = 25*3`, and `25=5^2` is a perfect square. So, by the second rule above, we have

`sqrt(75) = sqrt(25*3) =sqrt(25)*sqrt(3)=5sqrt(3)`

Which leads to the final answer

`sqrt(75/4) = \frac{5sqrt(3)}{2}`

But how do we find the most appropriate way to rewrite our number, in this case `75=25*3`? You can use the prime factorization:

`75 = 3*5^2`

and select only the primes with even exponent. In this case, `5^2`.

Answer 2

`+-(5sqrt(3))/2`

Explanation:

If you are stuck it is always worth have in a 'play' with numbers and see what comes up.

Consider the 75. This is exactly divisible by 5 so `75-:5=15`

Thus we have: `sqrt((5xx15)/4)`

But 15 is the same as `3xx5` giving:

`sqrt((5xx5xx3)/4)`

`sqrt(5^2xx3)/sqrt(4) = +-(5sqrt(3))/2`