Worksheet: Rationalizing Radical Expressions

Practice

In this worksheet, we will practice rationalizing radical expressions.

Q1:

Given that 𝑥 = 5 √ 7 − √ 2 and 𝑦 = √ 7 − √ 2 , find 𝑥 + 𝑦 expressing your answer in simplest form.

  • A 56
  • B 8
  • C 2 √ 2
  • D 2 √ 7
  • E 28

Q2:

Simplify 1 2 √ 5 − 2 3 1 3 √ 5 by rationalising the denominator.

  • A 3 7 6 5
  • B 1 2 − 2 3 √ 5 1 3
  • C − 1 1 √ 5 6 5
  • D 6 0 − 2 3 √ 5 6 5

Q3:

Simplify 9 √ 1 3 + √ 7 + 9 √ 1 3 − √ 7 .

  • A − 3 √ 1 3
  • B − 3 √ 7
  • C 3 √ 7
  • D 3 √ 1 3

Q4:

Given that 𝑥 = 3 0 √ 6 , 𝑦 = √ 2 1 6 + √ 3 , and 𝑧 = √ 3 0 0 + √ 6 , find the value of ( 𝑥 − 𝑦 + 𝑧 ) expressing your answer in simplest form.

Q5:

What is the conjugate of 1 2 √ 1 1 + √ 4 3 ? Express your answer in simplest form.

  • A √ 4 3 − 2 √ 1 1
  • B 2 √ 1 1 − √ 4 3
  • C 2 √ 1 1
  • D 2 √ 1 1 + √ 4 3
  • E √ 4 3

Q6:

Given that 𝑎 = 9 5 7 𝑥 and 𝑎 = √ 2 9 − 𝑦 , find the value of 𝑎 𝑥 + 𝑦 .

  • A 9 5 7 + √ 2 9
  • B 9 5 7 √ 2 9
  • C 9 5 7 − √ 2 9
  • D 3 3 √ 2 9

Q7:

Given that 𝑥 = 2 √ 5 + √ 2 2 √ 7 and 𝑦 = √ 5 − 3 √ 2 √ 4 2 , find the value of 𝑥 + 𝑦 2 2 .

  • A 4 3 + 2 7 √ 1 0
  • B 5 2 1
  • C 4 3 − 2 7 √ 1 0
  • D 4 3
  • E 1 1 3 8 4

Q8:

Given that 𝑥 = √ 1 2 + √ 7 and 𝑦 = √ 1 9 2 + √ 1 1 2 , express 𝑥 in terms of 𝑦 .

  • A 𝑦 1 6
  • B 4 𝑦
  • C 1 6 𝑦
  • D 𝑦 4
  • E 𝑦 4

Q9:

Simplify √ 2 0 − √ 1 9 √ 2 0 + √ 1 9 − √ 2 0 + √ 1 9 √ 2 0 − √ 1 9 .

  • A78
  • B 8 √ 9 5
  • C − 7 8
  • D − 8 √ 9 5

Q10:

Given that 𝑥 = √ 7 √ 1 3 and 𝑦 = √ 1 3 √ 7 , find 1 8 2 ( 𝑥 + 𝑦 ) expressing your answer in simplest form.

  • A 1 8 2 √ 9 1
  • B 1 8 2 √ 7 + 1 8 2 √ 1 3
  • C 2 0 √ 9 1
  • D 4 0 √ 9 1
  • E 2 √ 9 1

Q11:

Express 2 1 √ 6 6 √ 1 1 in its simplest form.

  • A √ 6
  • B 2 1 √ 1 1
  • C √ 1 1
  • D 2 1 √ 6
  • E √ 6 6

Q12:

Simplify − 2 0 √ − 7 3 by rationalizing the denominator.

  • A 2 0 7
  • B − 2 0 7 √ 4 9 3
  • C − 2 0 7
  • D 2 0 7 √ 4 9 3

Q13:

Simplify 9 √ 3 + 1 0 1 1 √ 3 by rationalising the denominator.

  • A 3 7 3 3
  • B 9 + 1 0 √ 3 1 1
  • C 1 9 √ 3 3 3
  • D 2 7 + 1 0 √ 3 3 3

Q14:

Given that 𝑥 = 2 √ 5 − √ 3 and 𝑦 = √ 5 − √ 3 , find 𝑥 + 𝑦 expressing your answer in simplest form.

  • A 60
  • B 12
  • C 2 √ 3
  • D 2 √ 5
  • E 20

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