The square root of a fraction

By definition we cannot derive the square root of a fraction. Thus we must find a way to get a fraction out of the radical.

Consider the square root of 1/2

sqrt(1/2)

how can we get the fraction out? We have to multiply and make the denominator a perfect square. But what we multiply to the bottom we must also multiply to the top

sqrt((1/2)*(2/2))

=

1/2*sqrt(2)

Now we have got the fraction out of the radical and created a square root we can rationalize.

Another example

sqrt(3/5)

To get the denominator out we multiply by (5/5) (which is equal to 1)

sqrt((3/5)*(5/5))

=

(1/5)*sqrt(3*5)

=

(1/5)*sqrt(15)

Simplifying Radicals

Like so much of algebra, it is good to know how to simplify radicals for purposes of canceling out or combining like terms.

Consider the square root of 27 or sqrt(27) there is no whole number that equals sqrt(27) but we can write the radical as sqrt(9*3) and that is equal to 3*sqrt(3)

Thus we have simplified the radical for purposes of mathematical calculation or canceling.

Square roots are radical

It is true, taking the square root of an expression can also be called "The radical"

We will write square root as sqrt on this website, thus

sqrt(4) = 2

and

sqrt(9x²y¹⁰) ?

To solve this, it is good to factor under the radical ( or factor the square root)

So we get

sqrt(9x²y¹⁰)

=

(3xy⁵)