Simplify Radicals Questions with Solutions

Questions on simplifying expressions with square root radicals are presented. The answers to the questions are at the bottom of the page and the solutions with full explanations to these questions are also included.

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Rules for Square Root Radicals

  1. \( \sqrt{x^2} = | x | \)
    Example: \( \sqrt{3^2} = | 3 | = 3 \)

  2. \( \sqrt{x \times y} = \sqrt{x } \sqrt{y} \)         for \( x \ge 0 \) and \( y \ge 0 \)
    Example: \( \sqrt{9 \times 100} = \sqrt{ 9 } \sqrt{ 100 } = 3 \times 10 = 30 \) \)

  3. \( \sqrt{\dfrac{x} {y}} = \dfrac{\sqrt x}{\sqrt y} \)         for \( x \ge 0 \) and \( y \ge 0 \)
    Example: \( \sqrt{\dfrac{1} {36}} = \dfrac{\sqrt 1}{\sqrt 36} = \dfrac{1}{6} \)

  4. We can group expressions with the same radicand (the term under the radical)
    Example: \( 5 \sqrt 3 + 8 \sqrt 3 = (5 + 8) \sqrt 3 = 13 \sqrt 3 \)
    Example: The expression \( 2 \sqrt 5 + 8 \sqrt 3 \) cannot be simplified because the radicands \( 5 \) and \( 3 \) are different.

Common Mistakes to Avoid when Working with Radicals
1) \( \sqrt{x + y} \)  
IS NOT EQUAL TO   \( \sqrt x + \sqrt y \)
Example: \( \sqrt{16 + 9} = \sqrt{25} = 5 \)  
IS NOT EQUAL TO   \( \sqrt {16} + \sqrt 9 = 4 + 3 = 7 \)

2) \( \sqrt{x - y} \)  
IS NOT EQUAL TO   \( \sqrt x - \sqrt y \)
Example: \( \sqrt{25 - 16} = \sqrt{9} = 3 \)  
IS NOT EQUAL TO   \( \sqrt {25} - \sqrt {16} = 5 - 4 = 1 \)


Questions to Simplify Square Root Radicals

  1. Simplify the expression \( 2 \sqrt{50} + 12 \sqrt{8} \).
    A) \( 14 \sqrt{58} \)
    B) 240
    C) 280
    D) 98
    E) \( 34 \sqrt 2 \)

  2. Simplify the expression \( \sqrt{27} - \sqrt{300} \).
    A) \( \sqrt{-273} \)
    B) \( -7 \sqrt 3 \)
    C) -147
    D) \( 3 - 10 \sqrt3 \)
    E) \( 7\sqrt3 \)

  3. Simplify the expression \( - 2 \sqrt{16y} + 10 \sqrt y \).
    A) \( 8 \sqrt y \)
    B) \( 8 \sqrt{-15y} \)
    C) \( 8 \sqrt{17y} \)
    D) \( 2 \sqrt{16y} \)
    E) \( 2 \sqrt y \)



  4. Simplify the expression \( 2 \sqrt{x + 1} + 3 \sqrt{16x + 16} \).
    A) \( 14 \sqrt{x + 1} \)
    B) \( 5 \sqrt{17x + 17} \)
    C) 5
    D) \( \sqrt{17x + 17} \)
    E) \( 6 \sqrt{17x + 17} \)

  5. \( 2 \sqrt 3 + 4 \sqrt{12} + 3 \sqrt{48} = \)
    A) \( 9 \sqrt3 \)
    B) \( 9 \sqrt{63} \)
    C) \( 22 \)
    D) \( 22 \sqrt 3 \)
    E) \( 9 \sqrt{48} \)

  6. Rewrite the expression \( \dfrac {\sqrt3 + \sqrt{12}} {\sqrt 3 - \sqrt{12}} \) without radicals.
    A) 1
    B) 0
    C) -3
    D) - 1
    E) 3

  7. Simplify the expression \( 5 \sqrt x + 6 \sqrt {9x} - 10 \sqrt {16x} \).

    A) \( \sqrt x \)
    B) \( -17 \sqrt x \)
    C) \( \sqrt{46 x} \)
    D) \( -2 \sqrt x \)
    E) \( \sqrt {-25 x} \)

  8. \( 2 \sqrt {27} + 2 \sqrt{75} = \)
    A) \( 16 \sqrt 3 \)
    B) \( 4 \sqrt 3 \)
    C) \( 4 \sqrt{102} \)
    D) 16
    E) 204

  9. \( \sqrt {10^3} + \sqrt {10^5} = \)
    A) \( \sqrt {10,100} \)
    B) \( 110 \sqrt {10} \)
    C) 10,000
    D) \( 2 \sqrt { 1000} \)
    E) \( 2 \sqrt { 100,000} \)

  10. Simplify and rewrite the expression \( \sqrt 8 \sqrt 3 \sqrt 6 \) without radicals.
    A) 144
    B) 3
    C) 17
    D) 12
    E) 4

Answers to the Above Questions
  1. E
  2. B
  3. E
  4. A
  5. D

  6. C
  7. B
  8. A
  9. B
  10. D



More References and Links

Simplify Radical Expressions with variables
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