Roots: Different Base, Same Root

To sum up:

[[summary]]

We've seen the basic rules for arithmetic operations involving roots with the same base. Since roots are simply fractional exponents, the rules for roots were the same as the rules for powers - we just added the necessary step of converting root to power form.

With regards to roots with different bases, we treat them as we would powers with the same exponents, along with the same stipulation: the following is true for arithmetic operations (multiplication / division, addition / subtraction) involving roots with different bases, as long as the roots are the same: √a and √b.

Multiplication / Division:

As long as the root is equal, multiplying and dividing roots with different bases involves combining the two bases under the same root.

Example (multiplying roots with different base, same root): √2·√8 = √(2·8) = √16 = 4

Example (dividing roots with different base, same root): √75 / √3 = √(75 / 3) = √25 = 5

Note that the above examples show a neat way to avoid difficult roots - √75 is not something that can be worked out without a calculator, but combining and reducing the two bases under the same root allows you to reduce the problem to a manageable √25 = 5.

The reverse is also true - a complex base under a root can be split into smaller, more manageable building blocks under the same root. If, for example, a problem requires us to calculate √50, split 50 into (2·25) and put each individual block under a root:

√50 = √(25·2) = √25·√2 = 5·√2.

Notice that we chose to split 50 into 25 and 2, rather than, say, 5 and 10. This is because 25 is a Perfect square and has an integer square root - neither 5 nor 10 have easily calculated roots. When splitting complex bases, look for blocks that have easily calculated roots.

Remember that the above is only true for multiplication and division of same roots. Combining bases under different roots is an illogical concept. For example:

√25 / ∛5 ≠ √(25/5)

Addition / Subtraction

As you've seen in the Adding and Subtracting Powers lesson, this is a big no-no. The same goes for roots.  Adding √2 and √3 does NOT magically reach a result of √(2+3) = √5. 

The same goes for subtraction: √7 - √4 ≠ √3

In fact, there's not much you can do if faced with a situation involving adding or subtracting roots with different bases, except beware of the above traps, which will definitely be laid by our friends at GMAC.