Question

to rationalize the denominator we have to multiply by ________________​

Answer 1

Answer:

we have to change sign

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Answer 2

Answer:

Multiply numerator and denominator by a radical that will get rid of the radical in the denominator.

If the radical in the denominator is a square root, then you multiply by a square root that will give you a perfect square under the radical when multiplied by the denominator. If the radical in the denominator is a cube root, then you multiply by a cube root that will give you a perfect cube under the radical when multiplied by the denominator and so forth...

Note that the phrase "perfect square" means that you can take the square root of it. Just as "perfect cube" means we can take the cube root of the number, and so forth.

Keep in mind that as long as you multiply the numerator and denominator by the exact same thing, the fractions will be equivalent.

Step 2: Make sure all radicals are simplified.

Some radicals will already be in a simplified form, but make sure you simplify the ones that are not. If you need a review on this, go to Tutorial 39: Simplifying Radical Expressions.

Step 3: Simplify the fraction if needed.

Be careful. You cannot cancel out a factor that is on the outside of a radical with one that is on the inside of the radical. In order to cancel out common factors, they have to be both inside the same radical or be both outside the radical.

notebook Example 1: Rationalize the denominator example 1a.

Step 1: Multiply numerator and denominator by a radical that will get rid of the radical in the denominator.

Since we have a square root in the denominator, then we need to multiply by the square root of an expression that will give us a perfect square under the radical in the denominator.

Square roots are nice to work with in this type of problem because if the radicand is not a perfect square to begin with, we just have to multiply it by itself and then we have a perfect square.

So in this case we can accomplish this by multiplying top and bottom by the square root of 6:

example 1b

*Mult. num. and den. by sq. root of 6

*Den. now has a perfect square under sq. root

Step 2: Make sure all radicals are simplified

AND

Step 3: Simplify the fraction if needed.

example 1c

*Sq. root of 36 is 6

*Divide out the common factor of 2

Be careful when you reduce a fraction like this. It is real tempting to cancel the 3 which is on the outside of the radical with the 6 which is inside the radical on the last fraction. You cannot do that unless they are both inside the same radical or both outside the radical like the 4 in the numerator and the 6 in the denominator were in the second to the last fraction.

notebook Example 2: Rationalize the denominator example 2a.

Step 1: Multiply numerator and denominator by a radical that will get rid of the radical in the denominator.

Since we have a cube root in the denominator, we need to multiply by the cube root of an expression that will give us a perfect cube under the radical in the denominator.

So in this case, we can accomplish this by multiplying top and bottom by the cube root of example 2b:

example 2c

*Mult. num. and den. by cube root of example 2b

*Den. now has a perfect cube under cube root

Step 2: Make sure all radicals are simplified

AND

Step 3: Simplify the fraction if needed.