Honors Algebra 2: Extra Practice on Rational Expressions

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Rewrite using exponential notation:

  1. \( \displaystyle \frac{\sqrt[4]{x^3y^5}}{2a^7\sqrt{b}}\)
    2. Solution

    \( \displaystyle x^{\frac{3}{4}}y^{\frac{5}{4}}2^{-1}a^{-7}b^{-\frac{1}{2}} \)

  2. \( \displaystyle \sqrt[12]{\sqrt[5]{x} + y\sqrt{y}}\)
    3. Solution

    \( \displaystyle \left(x^{\frac{1}{5}} + y \cdot y^{\frac{1}{2}}\right)^{\frac{1}{12}} \)

Rewrite using radical notation and no negative exponents:

  1. \( \displaystyle ab^{\frac{2}{3}}(3c)^{-5/7}\)
    2. Solution

    \( \displaystyle \frac{a\sqrt[3]{b^2}}{\sqrt[7]{\left(3c\right)^5}} \)

  2. \( \displaystyle 5^{\frac{1}{2}}x^{-\frac{1}{2}}\)
    3. Solution

    \( \displaystyle \sqrt{\frac{5}{x}} \)

  3. \( \displaystyle \left(3y\right)^{-\frac{2}{5}}\)
    4. Solution

    \( \displaystyle \frac{1}{\sqrt[5]{9y^2}} \)

  4. \( \displaystyle 3y^{-\frac{2}{5}}\)
    5. Solution

    \( \displaystyle \frac{3}{\sqrt[5]{y^2}} \)

Simplify the following:

  1. \( \displaystyle 2n^{\frac{1}{3}}\left(n^{\frac{2}{3}} + n^{-\frac{1}{3}}\right)\)
    2. Solution

    \( \displaystyle 2n + 2 \)

  2. \( \displaystyle \frac{x^{\frac{1}{2}} - 2x^{-\frac{1}{2}}}{x^{-\frac{1}{2}}}\)
    3. Solution

    \( \displaystyle x - 2 \)

  3. \( \displaystyle \frac{y^{-\frac{1}{3}} - 3y^{\frac{2}{3}}}{y^{-\frac{4}{3}}}\)
    4. Solution

    \( \displaystyle y - 3y^{2} \)

  4. \( \displaystyle \frac{2n^{\frac{1}{3}}\left(3n^{\frac{1}{3}} - 4n^{\frac{4}{3}}\right)}{2n^{-\frac{1}{3}}}\)
    5. Solution

    \( \displaystyle 3n - 4n^{2} \)

  5. \( \displaystyle \frac{\left(\sqrt{2x}\right)^{5}}{\left(x\sqrt{2}\right)^{9}}\)
    6. Solution

    \( \displaystyle \frac{1}{4x^6\sqrt{x}} \)

Factor the following:

  1. \( \displaystyle a^{\frac{3}{2}}b^{\frac{1}{2}} - a^{\frac{1}{2}}b^{\frac{3}{2}}\)
    2. Solution

    \( \displaystyle a^{\frac{1}{2}} b^{\frac{1}{2}} \left(a - b\right) \)

  2. \( \displaystyle \left(x - 1\right)^{\frac{1}{2}} - x\left(x - 1\right)^{-\frac{1}{2}}\)
    3. Solution

    \( \displaystyle -\left(x - 1\right)^{-\frac{1}{2}} \)

  3. \( \displaystyle \left(x^{2} + 1\right)^{\frac{3}{2}} - x^{2} \left(x^{2} + 1\right)^{\frac{1}{2}}\)
    4. Solution

    \( \displaystyle \left(x^{2} +1\right)^{\frac{1}{2}} \)

  4. \( \displaystyle \left(2x + 1\right)^{\frac{2}{3}} - 4\left(2x + 1\right)^{-\frac{1}{3}}\)
    5. Solution

    \( \displaystyle \left(2x + 1\right)^{-\frac{1}{3}} \left(2x - 3\right) \)

Simplify the following:

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

    \( \displaystyle 10 \left| x^3 \right| \)

  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. \( \displaystyle \sqrt[6]{8} \cdot \sqrt[8]{81} \)
    5. Solution

    \( \displaystyle \sqrt{6} \)

  5. \( \displaystyle \frac{7}{\sqrt{10} - \sqrt{15}} \)
    6. Solution

    \( \displaystyle - \frac{7\sqrt{10} + 7\sqrt{15}}{5} \)

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

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

  7. \( \displaystyle \sqrt[4]{\frac{7}{25a^3}}; a > 0 \)
    8. Solution

    \( \displaystyle \frac{\sqrt[4]{175a}}{5a} \)

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

    \( \displaystyle -9\sqrt{3x} \)