Radicals Practice Problems

Here you will find a concise collection of radicals practice problems. Visit this site directly at hunkim.com/radicals

  1. Evaluate
    1. \sqrt{9}
    2. \sqrt{1}
    3. \sqrt{0}
    4. \sqrt{\frac{4}{9}}
    5. \sqrt{4,000,000}
    6. \sqrt{0.25}
    7. \sqrt[3]{-8}
    8. \sqrt{-1}
  2. True or False: \sqrt{4}=\pm2
  3. Write \sqrt{8} as a mixed radical.
  4. Write 3\sqrt{3} as an entire radical.
  5. Write \sqrt[3]{5400} as a mixed radical.
  6. Write -2\sqrt[3]{3} as an entire radical.
  7. Give an example of a radical that is not an irrational number.
  8. Arrange from least to greatest:
    • I: \sqrt{9}
    • II: 2\sqrt{3}
    • III: -\sqrt{100}
    • IV: \pi
  9. Arrange from least to greatest (assume x>1):
    • I: \sqrt{x^2}
    • II: \sqrt[4]{x}
    • III: \sqrt[3]{x}
  10. \left(\sqrt{x}\right)\left(\sqrt[3]{x}\right)=x^k. Find k.
  11. Simplify \sqrt{8}+3\sqrt{2}
  12. Simplify \sqrt{8}-\sqrt[3]{32}+3\sqrt{2}+\sqrt[3]{4}
  13. Simplify \frac{-2+\sqrt{12}}{-2}
  14. Expand and simplify:
    1. \left(2-\sqrt{2}\right)^2
    2. \left(\sqrt{3}-\sqrt{2}\right)\left(\sqrt{3}+\sqrt{2}\right)
    3. 3\left(\sqrt{8}-\sqrt{2}\right)\left(1-\sqrt{8}\right)
  15. A rectangle has a base of 4\sqrt{2}-2\sqrt{3} and a height \sqrt{8}-\sqrt{3}
    1. Area in simplified form?
    2. Perimeter in simplified form?
  16. Solve the radical equation:
    1. \sqrt{x}=3
    2. 2\sqrt{x}=4
    3. \sqrt{x-2}=3
    4. \sqrt{x+4}=2-x
    5. \sqrt{2x+2}+3=x
    6. \sqrt{x+2}-1=\sqrt{x-3}
  17. Rationalize:
    1. \frac{1}{\sqrt{2}}
    2. \frac{3}{\sqrt{3}}
    3. \frac{2}{\sqrt[3]{2}}
  18. Rationalize the denominator by multiplying by the conjugate: \frac{4\sqrt{2}-5\sqrt{3}}{\sqrt{3}-4}
  19. y=\sqrt{x}
    1. Sketch
    2. Domain?
    3. Range?
  20. Use Desmos to investigate the effects of parameters a and b: y=a\sqrt{x-b}+c.
  21. f(x)=\sqrt{x-2}
    1. Sketch
    2. Domain?
    3. Range?
    4. Evalaute f(11)
  22. What is the domain to y=\sqrt{1-2x}
  23. y=\sqrt[3]{x}
    1. Sketch
    2. Domain?
    3. Range?
  24. Define y=|x|
  25. Simplify:
    1. \sqrt{x^2}
    2. \sqrt{x^4}
    3. \sqrt{x^5}
    4. \sqrt{x^8}
    5. \sqrt{x^{10}}

Answers

    1. 3
    2. 1
    3. 0
    4. 2/3
    5. 2000
    6. 0.5
    7. -2
    8. undefined or i
  2. False. \sqrt{4}=2 but if x^2=4 then x=\pm2
  3. 2\sqrt2
  4. \sqrt{27}
  5. 6\sqrt[3]{25}
  6. -\sqrt[5]{24} or \sqrt[3]{-24}
  7. ex. \sqrt{4}
  8. III, I, IV, II
  9. II, III, I
  10. 5/6
  11. 5\sqrt{2}
  12. 5\sqrt{2}-\sqrt[3]{4}
  13. 1-2\sqrt{3}
    1. 6-4\sqrt{2}
    2. 1
    3. 3\sqrt{2}-12
    1. 22-8\sqrt{6}
    2. 12\sqrt{2}-6\sqrt{3}
    1. 9
    2. 4
    3. 11
    4. 0
    5. 7
    6. 7
    1. \frac{1}{\sqrt{2}}
    2. \frac{3}{\sqrt{3}}
    3. \frac{2}{\sqrt[3]{2}}
  18. \frac{20\sqrt{3}+15-16\sqrt{2}-4\sqrt{6}}{13}
    1. See graph below:
    2. x\geq0
    3. y\geq0
  20. a affects the vertical stretch.
    b shifts the graph left or right.
    c shifts the graph up or down.
    1. See graph below:
    2. x\geq2
    3. y\geq0
  22. x\leq\frac{1}{2}
    1. See graph below:
    2. x\in\reals
    3. y\in\reals
  24. y=\begin{cases} x & x\geq0 \\ -x & x<0 \end{cases}
    1. |x|
    2. x^2
    3. x^2\sqrt{x}
    4. x^4
    5. |x^5|
  • Click here for more math practice problems.