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Simplifying Radicals
. . .
Rationalize the Denominator



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Simplifying Radicals & Roots - Rationalize the
Denominator . . .
radicals in fractions . . . Introduction to Radicals & Roots:

The "Radical Symbol" Math Symbol - Radical is a shorthand notation which stands for the roots of the number (or expression) in the radicand.


For example, the cube root of 8 stands for the number that can be multiplied by itself three times (cubed) to equal 8:

radical - cube root of 8 - Math

Since, 2 * 2 * 2 = 8, the cube root of 8 is 2.

radical symbol, radicand, and root index - Math

The 3 in the expression is called the root index, and the 8 is called the radicand.



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Rationalize the Denominator









What does "Rationalizing the Denominator" mean?

Rationalizing the Denominator simply means to remove all radicals from the denominator of a fraction without changing the value of the fraction.

When the denominator is rationalized, the original fraction is converted to the simplest equivalent fraction which does not have radicals in the denominator.

By removing all radicals from the denominator, all numbers in the denominator will be converted rational numbers (hence the term, "Rationalizing the Denominator").

For example, the following fraction has a square root in the denominator:

Math - fraction - 1 divided by the square root of 2

(Note that the square root of 2 is an
irrational number - a non-terminating
decimal without a repeating pattern.)

Rationalizing the Denominator means the fraction will be rewritten as the simplest equivalent fraction which does not have radicals in the denominator:

Math - fraction - the square root of 2 divided by 2

(Note that the denominator is now the
rational number 2.)




Why Rationalize the Denominator of a fraction?
Why not leave the denominator alone? What difference does it make?
What is the point?

Rationalizing the Denominator is the standard way of simplifying fractions so that they (fractions) can be readily understood and easily compared with other fractions.

The keys to understanding why Rationalizing the Denominator is important are: simplification and standardization:

Example of how the use of integers is standardized

Suppose you were telling a friend you had 5 apples yesterday, but today you only have 4 apples because you ate 1 apple.

How would you present the numbers 5 and 4?

Statement 1: "I had 5 apples yesterday. However, I ate 1 apple and now I have only 4 apples."

This statement is straightforward and makes it easy to compare how many apples you had yesterday with the number you have today (the numbers 5 and 4).

But that is not the only way you could communicate the numbers 5 and 4.

Statement 2: "I had 6 minus 1 apples yesterday; but, only 100 minus 96 apples today."

This statement makes it difficult to compare how many apples you had yesterday with the number you have today (the numbers 5 and 4) without completing a subtraction problem.

Statements 1 & 2 are mathematically equivalent. Each shows exactly how many apples you have on both days.

However, Statement 1 presents the numbers 5 and 4 as single integers . . . the standard for clarity and ease of comparison.

Example of how the use of fractions is standardized

Reducing a fraction to its simplest form makes it easier to understand and compare with other fractions in the majority of cases.

For example, the following four fractions have the exactly the same numerical value:

Math - equivalent fractions

However, if you were telling a friend you have half an apple, you would probably use the fractionMath - Fraction - one-half because it is easy to understand (it is the most reduced form of the fraction).

But you could use any of the four fractions listed. You could say you had Math - Fraction - 500 thousands of an apple.

Fractions are generally reduced to the simplest form because it makes them easier to understand.

Rationalizing the denominator of a fraction simplifies and standardizes fractions in the same way

Removing all radicals from the denominator of every fraction is a convention which allows fractions to be compared more easily.

The value of the fraction is not changed, but it is easier to understand and compare with other fractions.


Is it ever necessary to rationalize the numerator
of a fraction instead of a denominator?

Yes. When studying calculus, it will be necessary to find limits. In this case, rationalizing the numerator of a fraction rather than the denominator is part of the process. However, the basic technique for rationalization either the denominator or numerator is the same.


How to rationalize the denominator of a fraction


Video demonstrating How to Rationalize the Denominator - click here



The denominator is a monomial (1 term).

To rationalize the denominator, (1) multiply the denominator by a number (or expression) which will remove the radical from the denominator. (2) Multiply the numerator by the same number (or expression).



Math - Rationalize the Denominator - Monomial- example 1 Rationalize the denominator of: Math - Rationalize the Denominator - Monomial - Initial Problem - example 1
- simplifying radicals -

Math - Rationalize the Denominator - Monomial- example 1

The final answer is: Math - Rationalize the Denominator - Monomial - Final Answer - example 1



Math - Rationalize the Denominator - Monomial - Example 2 Rationalize the denominator of: Math - Rationalize the Denominator - Monomial - Initial Problem - example 2
- simplifying radicals -

Math - Rationalize the Denominator - Monomial- example 2

The final answer is: Math - Rationalize the Denominator - Monomial - Final Answer - example 2



Math - Rationalize the Denominator - Monomial - Example 3 Rationalize the denominator of: Math - Rationalize the Denominator - Monomial - Initial Problem - example 3
- simplifying radicals -

Math - Rationalize the Denominator - Monomial- example 3

The final answer is: Math - Rationalize the Denominator - Monomial - Final Answer - example 3



Math - Rationalize the Denominator - Monomial - Example 4 Rationalize the denominator of: Math - Rationalize the Denominator - Monomial - Initial Problem - example 4
- simplifying radicals -

Math - Rationalize the Denominator - Monomial- example 4

The final answer is: Math - Rationalize the Denominator - Monomial - Final Answer - example 4


The denominator is a binomial (2 terms).

To rationalize the denominator, (1) multiply the denominator by an expression which is the conjugate of the denominator. (2) Multiply the numerator by the same expression.

The conjugate of "A+B" is "A-B". The conjugate of "A-B" is "A+B".


Math - Rationalize the Denominator - Binomial - Example 5 Rationalize the denominator of: Math - Rationalize the Denominator - Binomial - Initial Problem - example 5
- simplifying radicals -

Math - Rationalize the Denominator - Binomial - example 5

The final answer is: Math - Rationalize the Denominator - Binomial - Final Answer - example 5



Math - Rationalize the Denominator - Binomial - Example 6 Rationalize the denominator of: Math - Rationalize the Denominator - Binomial - Initial Problem - example 6

Math - Rationalize the Denominator - Binomial - example 6

The final answer is: Math - Rationalize the Denominator - Binomial - Final Answer - example 6









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