Chapters

What Is Rationalization of Radicals?
Case 1
Case 2
Case 3
Examples of Radical Rationalization Exercises

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What Is Rationalization of Radicals?

Rationalization of radicals consists of removing radicals from the denominator, which facilitates the calculation of operations such as adding fractions.

We can distinguish three cases:

Case 1

Rationalization of the type $\text{[math]}$

We multiply the numerator and denominator by $\text{[math]}$ .

$\text{[math]}$

Examples

1 Rationalize the expression $\text{[math]}$

We multiply numerator and denominator by the square root of 2, perform the calculations, and simplify the fraction:

$\text{[math]}$

2 Rationalize the expression $\text{[math]}$

To perform the addition, we rationalize the 2nd term by multiplying and dividing by the square root of 2, and perform the addition:

$\text{[math]}$ $\text{[math]}$

Case 2

Rationalization of the type $\text{[math]}$

We multiply numerator and denominator by $\text{[math]}$ .

$\text{[math]}$

Example

Rationalize the expression $\text{[math]}$

We express the radicand $\text{[math]}$ in exponential form: $\text{[math]}$

We must multiply the numerator and denominator by the fifth root of $\text{[math]}$

We multiply the radicals in the denominator, extract factors from the radical, and simplify the fraction:

$\text{[math]}$

Case 3

Rationalization of the type $\text{[math]}$

And in general when the denominator is a binomial with at least one radical.

We multiply the numerator and denominator by the conjugate of the denominator.

The conjugate of a binomial is equal to the binomial with the middle sign changed:

$\text{[math]}$

We also need to keep in mind that: "sum times difference equals difference of squares."

$\text{[math]}$

Examples

1 Rationalize the expression $\text{[math]}$

We multiply numerator and denominator by the conjugate of the denominator, remove parentheses in the numerator, and perform the sum times difference in the denominator, which gives us a difference of squares:

$\text{[math]}$

In the denominator, we extract the radicands and divide by $\text{[math]}$ , that is, we change the sign of the numerator:

$\text{[math]}$

2 Rationalize the expression $\text{[math]}$

We multiply and divide the fraction by the conjugate of the denominator:

$\text{[math]}$

We perform the sum times difference in the denominator, carry out the operations, and simplify the fraction by dividing by $\text{[math]}$ :

$\text{[math]}$

3 Rationalize the expression $\text{[math]}$

$\text{[math]}$

In the numerator, we factor 12 and extract factors; we finish by performing the operations in the denominator:

$\text{[math]}$

Examples of Radical Rationalization Exercises

Rationalize the following expressions:

Score:

$\text{[math]}$

Solution

$\text{[math]}$

Solution

$\text{[math]}$

Solution

$\text{[math]}$

Solution

$\text{[math]}$

Solution

$\text{[math]}$

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Rationalization of Radicals

What Is Rationalization of Radicals?

Case 1

Examples

Case 2

Example

Case 3

Examples

Examples of Radical Rationalization Exercises

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