Worksheet: Rationalizing Denominators of Radicals

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In this worksheet, we will practice rationalizing denominators of radicals.

Q1:

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

  • A28
  • B56
  • C 2 √ 7
  • D8
  • E 2 √ 2

Q2:

Simplify 12√5−2313√5 by rationalizing the denominator.

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

Q3:

Simplify 9√13+√7+9√13−√7.

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

Q4:

Given that 𝑥=30√6, 𝑦=√216+√3, and 𝑧=√300+√6, find the value of (𝑥−𝑦+𝑧) expressing your answer in simplest form.

Q5:

What is the conjugate of 12√11+√43? Express your answer in simplest form.

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

Q6:

Given that 𝑎=957 and 𝑎=√29, find the value of 𝑎.

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

Q7:

Given that 𝑥=2√5+√22√7 and 𝑦=√5−3√2√42, find the value of 𝑥+𝑦.

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

Q8:

Given that 𝑥=√12+√7 and 𝑦=√192+√112, express 𝑥 in terms of 𝑦.

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

Q9:

Simplify √20−√19√20+√19−√20+√19√20−√19.

  • A 8 √ 9 5
  • B − 7 8
  • C − 8 √ 9 5
  • D78

Q10:

Given that 𝑥=√7√13 and 𝑦=√13√7, find 182(𝑥+𝑦) expressing your answer in simplest form.

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

Q11:

Express 21√66√11 in its simplest form.

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

Q12:

Simplify −20√−7 by rationalizing the denominator.

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

Q13:

Simplify 9√3+1011√3 by rationalizing the denominator.

  • A 1 9 √ 3 3 3
  • B 3 7 3 3
  • C 9 + 1 0 √ 3 1 1
  • 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.

  • A20
  • B60
  • C 2 √ 5
  • D12
  • E 2 √ 3

Q15:

Given that −47√2−7=𝑎√2+𝑏, find the values of 𝑎 and 𝑏.

  • A 𝑎 = − 1 , 𝑏 = 7
  • B 𝑎 = 1 , 𝑏 = 7
  • C 𝑎 = 7 , 𝑏 = 1
  • D 𝑎 = 1 , 𝑏 = − 7

Q16:

Given that 676√2+√5=𝑎√2+𝑏√5, find the values of 𝑎 and 𝑏.

  • A 𝑎 = − 1 , 𝑏 = 6
  • B 𝑎 = − 6 , 𝑏 = − 1
  • C 𝑎 = 6 , 𝑏 = − 1
  • D 𝑎 = 6 , 𝑏 = 1

Q17:

Given that 7674√48−1=𝑎+𝑏√3, find the values of 𝑎 and 𝑏.

  • A 𝑎 = 1 , 𝑏 = 1 6
  • B 𝑎 = 1 , 𝑏 = − 1 6
  • C 𝑎 = − 1 , 𝑏 = 1 6
  • D 𝑎 = 1 6 , 𝑏 = 1

Q18:

Simplify −1118√2 by rationalizing the denominator.

  • A − 1 1 √ 2 3 6
  • B − 1 1 √ 2 1 8
  • C − 1 8 √ 2 1 1
  • D 1 1 √ 2 3 6

Q19:

Simplify 6+√76−√7 by rationalizing the denominator.

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

Q20:

Simplify −36√2−√6 by rationalizing the denominator.

  • A 9 √ 2 − √ 6
  • B − 3 6 √ 2 + √ 6
  • C 9 √ 2 + √ 6
  • D − 3 6 √ 2 − √ 6
  • E √ 2 + √ 6

Q21:

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

  • A28
  • B56
  • C8
  • D 4 √ 1 4

Q22:

Simplify 4√10−3√74√10 by rationalizing the denominator.

  • A − 2 1 + 4 √ 7 0 2 8
  • B 4 0 − 3 √ 7 0 4 0
  • C 4 0 − 3 √ 7 0 1 0
  • D 1 6 0 − 1 2 √ 7 0 1 6 0
  • E 1 6 0 + 1 2 √ 7 0 1 6 0

Q23:

Express √45+105√15+√55√3−3 in its simplest form.

  • A 1 2 √ 1 5 + 6 √ 5
  • B 2 √ 1 5
  • C 1 2 √ 5
  • D 1 2 √ 3
  • E 1 2 √ 1 5

Q24:

Express √875−64√8+√512+√448 in its simplest form.

  • A − √ 7
  • B − 9 √ 7
  • C 9 √ 7
  • D √ 7

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