Multiplying and Dividing Square Roots

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The rules for doing arithmetic with square roots are quite simple:

(i) multiplication:

“the product of square roots is the square root of the product”

(ii) division:

“the quotient of square roots is the square root of the quotient” }

Thus, for example:

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(iii) addition and subtraction: There is NO simple relationship between the square roots of a sum or a difference and the square roots of its terms. In particular, in symbols,

You need to use algebraic methods to demonstrate why these prohibitions apply. However, it is possible to see the problem using a simple numerical example.

Note that

Thus does not give the same value as . That is:

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Since the two forms are unequal in this specific example, we have demonstrated that you cannot rely on to be equivalent to . (In fact, these two forms are only equivalent in the relatively uninteresting cases where a or b or both are equal to zero!)

Similarly,

which demonstrates that

NOTE: The rules described above for doing arithmetic with square roots also apply to any other order of roots (cube roots, fourth roots, etc.) as long as all of the roots have the same order. So, for example,

and so on.