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.
  1. xx=(x)2=x
  2. x3(x3)2=(x3)3=x
  3. (xy)(x+y)=(x)2(y)2=xy
  4. (xy)(x+y)=x2(y)2=x2y

Examples

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

  1. 12

    solution

    Because of 2 in the denominator, multiply numerator and denominator by 2 and simplify 12=1222=2(2)2=22

  2. 1x3

    solution

    Because of x3 in the denominator, multiply the numerator and denominator by (x3)2 and simplify. 1x3=1x3(x3)2(x3)2=x23x

  3. 432

    solution

    Because of the expression 32 in the denominator, multiply the numerator and denominator by the conjugate of 32 which is 3+2 to obtain: 432=4323+23+2 =4(3+2)(32)(3+2) =4(3+2)(3)2(2)2 =4(3+2)32=4(3+2)

  4. 5x2x23

    solution

    Because of the expression x23 in the denominator, multiply the numerator and denominator by (x23)2 to obtain 5x2x23=5x2x23(x23)2(x23)2 =5x2x43(x23)3 Simplify and cancel terms: =5x2x43x2=5x43=5xx3

  5. 5)x2y+x2+y2

    solution

    Because of the expression y+x2+y2 in the denominator, multiply numerator and denominator by its conjugate yx2+y2 to obtain x2y+x2+y2=x2y+x2+y2yx2+y2yx2+y2 =x2(yx2+y2)(y)2(x2+y2)2 =x2(yx2+y2)y2(x2+y2) =x2(yx2+y2)x2 =y+x2+y2

Questions

Rationalize the denominators of the following expressions and simplify if possible.
  1. 105
  2. 223232+3
  3. 7x4x43
  4. x2y+x2+y2

Solutions to the Above Problems


  1. Multiply numerator and denominator by 5 105=10555=105(5)2=1055=25

  2. Multiply numerator and denominator by 23 223232+3=223232+32323 and simplify =223(23)2(2)2(3)2 =223(2)2+(3)222323 =2232+32231 =223+5223 =5

  3. Multiply numerator and denominator by (x43)2 7x4x43=7x4x43(x43)2(x43)2 and simplify =7x4x83(x43)3=7x4x83x4=7x83=7x2x23

  4. Multiply numerator and denominator by yx2+y2 x2y+x2+y2=x2y+x2+y2yx2+y2yx2+y2 and simplify =x2(yx2+y2)y2(x2+y2) =x2(yx2+y2)x2=yx2+y2

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