6.3 Homework Problems: Product, Power and Quotient Rule
1. Multiply the following using the product rule and simplify as much as possible
a. [latex]{\mathrm{\ x}}^\frac{\mathrm{1}}{\mathrm{2}}\mathrm{\ \cdot\ }\mathrm{x}^\frac{\mathrm{1}}{\mathrm{3}}[/latex]
b. [latex]{\ y}^\frac{2}{3}\ \cdot\ y^\frac{3}{5}[/latex]
c. [latex]{\ 2x}^\frac{3}{2}\ \cdot\ 3x^\frac{2}{3}[/latex]
d. [latex]{\ 5y}^\frac{1}{7}\ \cdot\ 4y^\frac{3}{2}[/latex]
e. [latex]{\ x}^\frac{3}{2}\ y^\frac{1}{2}\ \cdot\ x^\frac{2}{3}\ y^\frac{2}{3}[/latex]
f. [latex]{\mathrm{\ 5x}}^\frac{\mathrm{5}}{\mathrm{3}}\mathrm{\ }\mathrm{y}^\frac{\mathrm{1}}{\mathrm{7}}\mathrm{\ \cdot\ 4}\mathrm{x}^\frac{\mathrm{2}}{\mathrm{9}}\mathrm{\ }\mathrm{y}^\frac{\mathrm{3}}{\mathrm{2}}[/latex]
g. [latex]{\ x}^\frac{3}{2}\ \cdot\ x^\frac{2}{-5}[/latex]
h. [latex]{\ 5y}^\frac{1}{7}\ \cdot\ {4y}^{-\ \frac{3}{2}}[/latex]
i. [latex]{\ 5x}^\frac{-4}{9}\ \cdot\ {-3x}^{\ \frac{2}{3}}[/latex]
j. [latex]{\ -5y}^\frac{3}{8}\ \cdot\ {-4y}^{\ \frac{-3}{2}}[/latex]
2. Divide the following using the quotient rule and simplify as much as possible
a. [latex]\frac{x^\frac{1}{2}}{x^\frac{3}{2}}[/latex]
b. [latex]\frac{y^\frac{2}{5}}{y^\frac{6}{7}}[/latex]
c. [latex]\frac{3x^\frac{1}{5}}{{6x}^\frac{3}{8}}[/latex]
d. [latex]\frac{12y^\frac{2}{5}}{4y^\frac{6}{7}}[/latex]
e. [latex]\frac{2x^\frac{1}{3}{\ y}^\frac{2}{3}}{8x^\frac{5}{8}{\ y}^\frac{1}{4}}[/latex]
f. [latex]\frac{12x^\frac{2}{5}\ y^\frac{2}{5}}{4x^\frac{1}{3}\ y^\frac{6}{7}}[/latex]
g. [latex]\frac{x^\frac{1}{5}}{x^{-\frac{3}{2}}}[/latex]
h. [latex]\frac{{4y}^\frac{-2}{5}}{{12y}^{-\frac{3}{2}}}[/latex]
3. Simplify the following using the power rule and simplify as much as possible
a. [latex]\left(x^\frac{2}{3}\right)^\frac{1}{2}[/latex]
b. [latex]\ \left({4x}^\frac{2}{3}\right)^\frac{3}{2}[/latex]
c. [latex]\left(x^\frac{2}{3}\right)^\frac{-3}{5}[/latex]
d. [latex]\left({3x}^\frac{-2}{3}\right)^\frac{5}{2}[/latex]