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How to Get a Square Root Out of a Numerator

In algebra, getting a square root out of a numerator is not as common as getting it out of a denominator. However, you may need to do so occasionally to reduce fractions. We call this process "rationalizing the numerator," which means rewriting the fraction with a rational number in the numerator. Remember that you never change the value of the fraction when you rationalize a quantity. You are only changing the way the expression looks. The trick is to multiply the quantity by 1.

Instructions

    • 1

      Identify the number of terms in the numerator. If there is only one term inside the square root, proceed to Step 2. If there are two terms, go to Step 3.

    • 2

      Multiply both the numerator and the denominator by the same square root as the original numerator, if there is only one term inside the square root. For example, to rationalize sqrt(5)/2, multiply sqrt(5)/sqrt(5) to sqrt(5)/2. Then sqrt(5) times sqrt (5) = 5. The final answer is 5/(2sqrt(5)).

    • 3

      Multiply both the numerator and the denominator by the conjugate of the numerator, if the numerator contains two terms. For example, if numerator is 2+sqrt(3), its conjugate is 2 - sqrt(3). Observe that when we multiply 2+sqrt(3) by its conjugate 2-sqrt(3), the square root disappears and the product becomes 4- 3, which is 1.

      If the numerator contains two terms where at least one term contains a square root, you can rationalize the numerator by multiplying both the numerator and the denominator by its conjugate. For example, [3-sqrt(5)]/7 = [3-sqrt(5)][3+sqrt(5)]/ [7(3+sqrt(5)] = (9-5)/[7(3+sqrt(5)] = 4/[7(3+sqrt(5)].

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