Simplifying Products and Quotients Involving Square Roots
The basic strategy here is to simplify any square roots that remain after the two expressions involving square roots are multiplied, or after one is divided by another. The simplification is assisted by removing perfect squares from the square roots, as illustrated previously in these notes.
We could start by simply multiplying the coefficients and multiplying the square roots to get
Now check to see if the square root can be simplified. Checking for perfect square factors in 900, we see immediately that
900 = 9 Ã— 100 = (3 2 ) Ã— (10 2 )
as the simplest form (with no square root at all remaining).
We could also have attempted to simplify the original square roots first before multiplying. This would have first given:
since . We got the same final result, of course using both methods.
In the case of division (or a quotient), it is usually more efficient to simplify the two square roots first. So here, we have
So, now we can write
We were able to cancel common factors in the numerator and denominator to reduce the final result to the simplest form indicated. This method of simplification of fractions is described and illustrated at greater length in the section of these notes dealing with fractions.