In Exercises simpradfirst - simpradlast , perform the indicated operations and simplify.
\(\sqrt [3]{8t^3} = \answer {2t}\)
\(\sqrt {4t^2 + 4t + 1}\)
\(\sqrt {w^2 - 16w + 64} = \answer {|w-8|}\)
\(\sqrt {(\sqrt {12x} - \sqrt {3x})^2+1}\)
\(\sqrt {\frac {c^2 - v^2}{c^2}}\)
\(\frac {\sqrt {c^2-v^2}}{|c|}\)
\(\sqrt [3]{\frac {24 \pi r^5}{L^3}}\)
\(\frac {2r \sqrt [3]{3 \pi r^2}}{L}\)
\(\sqrt [4]{\frac {32 \pi \varepsilon ^8}{\rho ^{12}}}\)
\(\frac {2 \varepsilon ^2 \sqrt [4]{2\pi }}{|\rho ^3|}\)
\(\sqrt {x} - \frac {x+1}{\sqrt {x}}\)
\(-\frac {1}{\sqrt {x}}\)
\(3 \sqrt {1-t^2} + 3t\left (\frac {1}{2 \sqrt {1-t^2}}\right )(-2t)\)
\(\frac {3-6t^2}{\sqrt {1-t^2}}\)
\(2 \sqrt [3]{1-z} + 2z \left (\frac {1}{3 \left (\sqrt [3]{1-z}\right )^2}\right )(-1)\)
\(\frac {6-8z}{3 (\sqrt [3]{1-z})^2}\)
\(\frac {3}{\sqrt [3]{2x-1}} + (3x)\left (-\frac {1}{3 \left (\sqrt [3]{2x-1} \right )^4}\right )(2)\)
\(\frac {4x-3}{(2x-1)\sqrt [3]{2x-1}}\)
In Exercises
algineqexfirst -
algineqexlast , find all real solutions.
\(\frac {(1-2y)^{4}}{3} = 27\)
\(y = \answer {-1}\) (smaller solution)
\(y = \answer {2}\) (larger solution)
\(\frac {1}{1 + 2t^3} = 4\)
\(t = -\frac {\sqrt [3]{3}}{2}\)
\(\sqrt {3x+1} = 4\)
\(x = \answer {5}\)
\(5 - \sqrt [3]{t^2+1} = 1\)
\(x+1 = \sqrt {3x+7}\)
\(x=\answer {3}\)
\(y + \sqrt {3y+10} = -2\)
\(3t+\sqrt {6-9t}=2\)
\(t = \answer {-\frac {1}{3}}\) (smaller solution)
\(t = \answer {\frac {2}{3}}\) (larger solution)
\(2x - 1 = \sqrt {x + 3}\)
\(x = \frac {5 + \sqrt {57}}{8}\)
\(w = \sqrt [4]{12-w^2}\)
\(w = \answer {\sqrt {3}}\)
\(\sqrt {x - 2} + \sqrt {x - 5} = 3\)
\(\sqrt {2x+1} = 3 + \sqrt {4-x}\)
\(x = \answer {4}\)
In Exercises radliteqnfirst - radliteqnlast , solve each equation for the indicated variable. Assume all quantities represent positive real
numbers.
Solve for
\(h\) :
\(I = \frac {bh^3}{12}\) .
\(h = \sqrt [3]{\frac {12I}{b}}\)
Solve for
\(a\) :
\(I_{0} = \frac {5\sqrt {3} a^4}{16}\)
\(a = \frac {2 \sqrt [4]{I_{0}}}{\sqrt [4]{5\sqrt {3}}}\)
Solve for
\(g\) :
\(T = 2\pi \sqrt {\frac {L}{g}}\)
\(g = \frac {4 \pi ^2 L}{T^2}\)
Solve for
\(v\) :
\(L = L_{0} \sqrt {1 - \frac {v^2}{c^2}}\) .
\(v = \frac {c \sqrt {L_{0}^2 - L^2}}{L_{0}}\)
In Exercises rationalizefirst - rationalizelast , rationalize the numerator or denominator, and simplify.
\(\frac {4}{3 - \sqrt {2}}\)
\(\frac {12 + 4\sqrt {2}}{7}\)
\(\frac {7}{\sqrt [3]{12x^7}}\)
\(\frac {7 \sqrt [3]{18x^2}}{6x^3}\)
\(\frac {\sqrt {x} - \sqrt {c}}{x - c}\)
\(\frac {1}{\sqrt {x}+ \sqrt {c}}\)
\(\frac {\sqrt {2x+2h+1} - \sqrt {2x+1}}{h}\)
\(\frac {2}{\sqrt {2x+2h+1} + \sqrt {2x+1}}\)
\(\frac {\sqrt [3]{x+1} - 2}{x- 7}\)
\(\frac {1}{(\sqrt [3]{x+1})^2 + 2\sqrt [3]{x+1} + 4}\)
\(\frac {\sqrt [3]{x+h} - \sqrt [3]{x}}{h}\)
\(\frac {1}{(\sqrt [3]{x+h})^2 + \sqrt [3]{x+h}\sqrt [3]{x} + (\sqrt [3]{x})^2}\)
