Rationalizing Denominators.

Remember that an irrational number is any real number that can't be expressed with whole numbers or fractions. These are numbers whose decimal expressions have infinite digits without forming repeating sequences.

We can say that 0,5 is the same as .

is the same as 0,75.

All of these numbers are rational; we can write them as whole numbers or as fractions.

There are some numbers we can't express in that way; for example

These numbers are called irrational because if we wanted to write their value, we would never finish writing decimal quantities. There is no number multiplied by itself which will give you a result of 2, 3, 11, 13, etc. The square roots of these numbers is infinite.

It is convenient for fractions with an irrational denominator to be turned into rational. In other words, the process of getting fractions with no radicals in their denominator is called 'rationalizing radical denominators':

For example: . The denominator is an irrational number. You can try to calculate its value but the operations will never end.

We know that if we multiply or divide the numerator and the denominator of a fraction by the same number, its value will remain the same.

To make the denominator a rational number, the simplest way is to multiply it by itself: . However, for the fraction to be equal, we need to multiply the numerator by .

We could say that: are equal, butdoesn't have an irrational number as denominator.

10.76 Calculate:

Answer: .

10.77 Calculate:

Answer: .

Solution:

10.78 Calculate:

Answer: .

Solution:

Here are the operations step-by-step:

10.79 Calculate:

Answer:

Solution:

The process of calculating with letter variables is the same we use with numbers. Here is the solution step-by-step:

10.80 Calculate:

Answer: .

Solution:

Now, we are dealing with a cube root. If we have and you want to remove the cube root, we need to make the radicant exponent (which is 1) equal to the index of the radical (which is 3). To be equal to ,

we need to multiply it by . In the denominator, when we multiply times , you will have to add the exponents keeping the same base: .

To keep the same value in the fraction, we need to multiply the numerator by :

10.81 Calculate:

Answer: .

Solution:

To be able to remove the radical from , 5 needs to have 7 as an exponent. We see that we would need to multiply it by . When we add the exponents, the result will be equal to the index (7) of the radical. Now, we can simplify. In order for the value of the fraction not to vary, we will have to multiply the numerator by also:

10.82 Calculate:

Answer: .