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      FIGURE 2-1: Radicals on the number line.

      Notice that radicals of square numbers are always equivalent to integers. In contrast, radicals like and are irrational numbers that fit between these integers on the number line.

      When you see this ordered relationship, you can estimate the value of a radical by finding the two integer values where it must fit on the number line. For example:

       Where on the number line does the value of occur?

      (A) Between 4 and 5

      (B) Between 5 and 6

      (C) Between 6 and 7

      (D) Between 7 and 8

      The number 39 falls on the number line between the square numbers 36 and 49. Therefore, falls between and , so Answer C is correct.

      Multiplying and dividing radicals

      You can multiply any pair of radicals simply by multiplying the values inside the radicals. For example:

      Similarly, you can divide one radical by another by dividing their contents:

      If x is equivalent to the expression shown here, what is the value of ?

      To find the value of x, first simplify . Begin by multiplying the two radicals in the numerator:

      Now, divide the numerator by the denominator and simplify:

      Thus, , so plug this value into and evaluate:

Therefore, the answer is 11.

      Simplifying radicals

      In some cases, you can simplify a radical by factoring it and then evaluating one of the factors as a whole number. This is always the case when the value inside the radical is divisible by a square number, such as 4, 9, 16, 25, and so forth.

      For example, here are three radicals that can be simplified by factoring out and then simplifying this value to 2:

      Similarly, you can simplify the following three radicals by factoring out and then simplifying this value to 3:

      These three radicals can be simplified by factoring out :

      And these three can be simplified by factoring out :

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