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      This result is the most simplified form of , so Answer C is correct. (You can also find this answer directly as .)

       Other opportunities for simplifying radicals may arise when working with the quadratic formula, which includes a radical. In Chapter 3, I show you how to use this formula to solve quadratic equations. And then in Chapter 12, you use it again to find the roots of quadratic functions.

      Adding and subtracting radicals

      You can only add equivalent radicals. For example:

      You can add radical expressions that have the same radical parts by adding coefficients. For example:

      You can also subtract radicals in the same way:

      In some cases, you can add or subtract radical expressions with non-equivalent radical parts by simplifying them using the factoring method I describe in the previous section. For example, here’s an SAT question that depends upon your understanding this idea:

       Which of the following is equivalent to ?

      (A)

      (B)

      (C)

      (D)

      Begin by factoring and to simplify them:

      Now, rewrite and solve it:

      So Answer C is correct.

      Rationalizing radicals in the denominator

      In some cases, when a radical appears in the denominator of a fraction, an SAT question will require you to rationalize the denominator — that is, find an equivalent form of that fraction with an integer in the denominator.

      To rationalize the denominator of a fraction, multiply both the numerator and denominator by the radical that’s in the denominator. For example, here’s how you rationalize :

      In some cases when rationalizing, you may need to simplify the result. For example, here’s how you rationalize :

      This result can be simplified by dividing both the numerator and the denominator by 2:

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