Rationalize Denominators - Questions with Solutions

Grade 10 questions on how to rationalize radical expressions with solutions are presented.

To rationalize radical expressions with denominators is to express the denominator without radicals

Examples with Solutions

The following identities may be used to rationalize denominators of rational expressions.

equation 1

Examples

Rationalize the denominators of the following expressions and simplify if possible.

equation 2

solution

Because of \( \sqrt 2 \) in the denominator, multiply numerator and denominator by \( \sqrt 2 \) and simplify

equation 3

equation 4

solution

Because of \( \sqrt[3]{x} \) in the denominator, multiply the numerator and denominator by \( \left( \sqrt[3]{x} \right)^2 \) and simplify.

equation 5

equation 6

solution

Because of the expression \( \sqrt{3} - \sqrt{2} \) in the denominator, multiply the numerator and denominator by the conjugate of \( \sqrt{3} - \sqrt{2} \) which is \( \sqrt{3} + \sqrt{2} \) to obtain:

equation 7

equation 8

solution

Because of the expression \( \sqrt[3]{x^2} \) in the denominator, multiply the numerator and denominator by \( \left( \sqrt[3]{x^2} \right)^2 \) to obtain

equation 9

Simplify and cancel terms

equation 10

equation 11

solution

Because of the expression \( y + \sqrt{x^2 + y^2} \) in the denominator, multiply numerator and denominator by its conjugate \( y - \sqrt{x^2 + y^2} \) to obtain

equation 12

Questions

Rationalize the denominators of the following expressions and simplify if possible.

equation 13

Solutions to the Above Problems

  1. Multiply numerator and denominator by \( \sqrt 5 \)

    equation 14

    and simplify

    equation 15

  2. Multiply numerator and denominator by \( \sqrt 2 - \sqrt 3 \)

    equation 16

  3. Multiply numerator and denominator by \( \left( \sqrt[3]{x^4} \right)^2 \)

    equation 17

    and simplify

    equation 18

  4. Multiply numerator and denominator by \( y - \sqrt{x^2 + y^2} \)

    equation 19

    and simplify

    equation 20

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