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
  • C2√7
  • D8
  • E2√2

Q2:

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

  • A−11√565
  • B3765
  • C12−23√513
  • D60−23√565

Q3:

Simplify 9√13+√7+9√13−√7.

  • A3√7
  • B−3√13
  • C3√13
  • 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√43
  • B2√11+√43
  • C√43−2√11
  • D2√11−√43
  • E2√11

Q6:

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

  • A957+√29
  • B33√29
  • C957√29
  • D957−√29

Q7:

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

  • A11384
  • B521
  • C43+27√10
  • D43
  • E43−27√10

Q8:

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

  • A𝑦
  • B16𝑦
  • C𝑦4
  • D𝑦16
  • E4𝑦

Q9:

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

  • A8√95
  • B−78
  • C−8√95
  • D78

Q10:

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

  • A20√91
  • B40√91
  • C182√91
  • D2√91
  • E182√7+182√13

Q11:

Express 21√66√11 in its simplest form.

  • A√66
  • B21√6
  • C√11
  • D√6
  • E21√11

Q12:

Simplify −20√−7 by rationalizing the denominator.

  • A207
  • B−207√49
  • C207√49
  • D−207

Q13:

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

  • A19√333
  • B3733
  • C9+10√311
  • D27+10√333

Q14:

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

  • A20
  • B60
  • C2√5
  • D12
  • E2√3

Q15:

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

  • A𝑎=−1, 𝑏=7
  • B𝑎=1, 𝑏=7
  • C𝑎=7, 𝑏=1
  • D𝑎=1, 𝑏=−7

Q16:

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

  • A𝑎=−1, 𝑏=6
  • B𝑎=−6, 𝑏=−1
  • C𝑎=6, 𝑏=−1
  • D𝑎=6, 𝑏=1

Q17:

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

  • A𝑎=1, 𝑏=16
  • B𝑎=1, 𝑏=−16
  • C𝑎=−1, 𝑏=16
  • D𝑎=16, 𝑏=1

Q18:

Simplify −1118√2 by rationalizing the denominator.

  • A−11√236
  • B−11√218
  • C−18√211
  • D11√236

Q19:

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

  • A12943−12√7
  • B12943+12√7
  • C43−12√7
  • D43+12√7
  • E14329+12√7

Q20:

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

  • A9√2−√6
  • B−36√2+√6
  • C9√2+√6
  • D−36√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
  • D4√14

Q22:

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

  • A−21+4√7028
  • B40−3√7040
  • C40−3√7010
  • D160−12√70160
  • E160+12√70160

Q23:

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

  • A12√15+6√5
  • B2√15
  • C12√5
  • D12√3
  • E12√15

Q24:

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

  • A−√7
  • B−9√7
  • C9√7
  • D√7

Q25:

Given that 𝑥=1√6−√5 and 𝑦=√6−√5, find (𝑥+𝑦).

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