Arithmetic of Irrational Expressions

The arithmetic of irrational expressions is just like the arithmetic of algebraic expressions. Like terms can be added to and subtracted from simpler expressions, but unlike terms cannot. For example, 3x + 5x = 8x, but 3x + 5y cannot be simplified. Like and unlike terms can be multiplied and divided. For example, (3x)(5x) = 15x2, and (3x)(5y) = 15xy.

, and


cannot be simplified because the radicands—the terms under the radical—are different.


At first look you might think that these terms cannot be combined because the radicands are different. However, if you simplify

to be
, the problem now becomes to simplify to
, which equals


. The third term in the problem is different in that there is a fraction within the square root. Simplifying the fraction makes
. Although this is correct, it does not give a form that can be simplified with the other terms. If you multiply both the numerator and the denominator by
, you get an equivalent fraction with a rational denominator (which is why this process is called rationalizing the denominator), and
. Simplifying the terms yields


These are cube roots, so you need to be thinking about perfect cubes when simplifying terms.

. Rewriting 1/4 as the equivalent fraction 2/8 makes
. Therefore,

To rationalize the denominator of a term in which the denominator is a binomial, such as

, multiply the numerator and denominator by the conjugate of the given denominator, thus using the formula for the difference of two squares.

Rationalize the denominator of

, and simplify the fraction.

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