Honors Algebra 2: Assignment 43: Unit Review

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Simplify Completely. Make sure the domain and range of the simplified version match the domain and range of the original:

  1. \( \displaystyle \sqrt[5]{4x^6} \)
    2. Solution

    \( \displaystyle x \sqrt[5]{4x} \)

  2. \( \displaystyle \sqrt{189x^5} \)
    3. Solution

    \( \displaystyle 3x^2 \sqrt{21x} \)

  3. \( \displaystyle \sqrt[4]{\left(-81 \right) ^{-2}} \)
    4. Solution

    \( \displaystyle \frac{1}{9} \)

  4. \( \displaystyle \sqrt[6]{8} \cdot \sqrt[6]{27} \)
    5. Solution

    \( \displaystyle \sqrt{6} \)

  5. \( \displaystyle \frac{5}{\displaystyle \sqrt{10} - \sqrt{15}} \)
    6. Solution

    \( \displaystyle -\sqrt{10} - \sqrt{15} \)

  6. \( \displaystyle \frac{\sqrt{x + y}}{5 - \sqrt{x + y}} \) given \( x > 0\) and \(y > 0\)
    7. Solution

    \( \displaystyle \frac{5\sqrt{x + y} + x + y}{25 - x - y} \)

  7. \( \displaystyle \sqrt[4]{\displaystyle \frac{7}{250a^3}} \) given \( a > 0\)
    8. Solution

    \( \displaystyle \frac{\sqrt[4]{280a}}{10a} \)

  8. \( \displaystyle 5\sqrt{12x} - 6\sqrt{48x} + 75\sqrt{x} \)
    9. Solution

    \( \displaystyle -14\sqrt{3x} + 75 \sqrt{x} \)

For each of the following, evaluate or identify as undefined:

  1. \( \displaystyle \sqrt[3]{-54} \)
    2. Solution

    \( \displaystyle -3\sqrt[3]{2} \)

  2. \( \displaystyle \sqrt[5]{(-32)^4} \)
    3. Solution

    16

  3. \( \displaystyle \sqrt{-\frac{1}{8}} \)
    4. Solution

    undefined

  4. \( \displaystyle \sqrt{\frac{1}{8}} \)
    5. Solution

    \( \displaystyle \frac{\sqrt{2}}{4} \)

Simplify completely. Where appropriate, make sure the domain and range of the simplified version is the same as the original.

  1. \( \displaystyle \sqrt{50} + 3\sqrt{18x^2} + x\sqrt{8} \)
    2. Solution

    \( \displaystyle 5\sqrt{2}+ 2x\sqrt{2} + 9|x|\sqrt{2} \)

  2. \( \displaystyle \sqrt[3]{16x^3y^5} \)
    3. Solution

    \( \displaystyle 2xy\sqrt[3]{2y^2} \)

  3. \( \displaystyle \sqrt[4]{\frac{16x^4y^9}{z}} \)
    4. Solution

    \( \displaystyle \frac{2 \left| x \right| y^2 \sqrt[4]{yz^3}}{\left| z \right|} \) where \( y\) and \(z\) have the same sign.

  4. \( \displaystyle \sqrt{48} \cdot \sqrt[6]{216} \)
    5. Solution

    \( \displaystyle 12\sqrt{2} \)

  5. \( \displaystyle \sqrt{27a^2b^4} \cdot \sqrt{3ab^3} \)
    6. Solution

    \( \displaystyle 9ab^3\sqrt{ab} \) where \( a\) and \(b \) have the same sign.

  6. \( \displaystyle \sqrt[3]{\frac{128x^6}{2y^5}} \)
    7. Solution

    \( \displaystyle \frac{4x^2\sqrt[3]{y}}{y^2} \)

  7. \( \displaystyle \frac{4}{2 + 2\sqrt{3}} \)
    8. Solution

    \( \displaystyle \sqrt{3} - 1 \)

  8. \( \displaystyle \left( 2 - 3\sqrt{5} \right)^2 \)
    9. Solution

    \( \displaystyle 49-12\sqrt{5} \)

  9. \( \displaystyle \sqrt{\frac{3}{5}} \left( \frac{\sqrt{5}}{2} + \sqrt{6} \right) \)
    10. Solution

    \( \displaystyle \frac{\left( 5 \sqrt{3} + 6\sqrt{10}\right)}{10} \)

  10. \( \displaystyle \sqrt{4x^{-1} + 4y^{-1}} \cdot \sqrt{ \left( x + y \right) ^{-1}} \), given \( x > 0 \) and \(y > 0\)
    11. Solution

    \( \displaystyle \frac{2 \sqrt{xy}}{xy} \)

  11. \( \displaystyle \sqrt[3]{16x^4y^7} - xy\sqrt[3]{2xy^4} \)
    12. Solution

    \( \displaystyle xy^2 \sqrt[3]{2xy} \)

  12. \( \displaystyle \sqrt[10]{\frac{1}{32}} \)
    13. Solution

    \( \displaystyle \frac{\sqrt{2}}{2} \)

Solve the following:

  1. \( \displaystyle m + \sqrt{m - 2} - 4 = 0 \)
    2. Solution

    \( \displaystyle \{ 3 \} \)

  2. \( \displaystyle \sqrt{m + 10} + \sqrt{2m + 4} - 8 = 0 \)
    3. Solution

    \( \displaystyle \{ 6 \} \)