Practice 2 Solutions

Simplify. Write answers in simplified radical form.

$\sqrt[3]{24 x^{6} y^{10} z^{12}}$

$\sqrt[3]{2^{3} \cdot 3 x^{6} y^{10} z^{12}} 2^{\frac{3}{3}} \cdot 3^{\frac{1}{3}} x^{\frac{6}{3}} y^{\frac{10}{3}} z^{\frac{12}{3}} 2^{1} \cdot 3^{\frac{1}{3}} x^{2} y^{3 \frac{1}{3}} z^{4}$

$2 x^{2} y^{3} z^{4} \sqrt[3]{3 y}$

$- \sqrt{120 g h^{8} j^{19}}$

$- \sqrt{2^{3} \cdot 3 \cdot 5 g h^{8} j^{19}} - (2^{\frac{3}{2}} \cdot 3^{\frac{1}{2}} \cdot 5^{\frac{1}{2}} g^{\frac{1}{2}} h^{\frac{8}{2}} j^{\frac{19}{2}}) - (2^{1 \frac{1}{2}} \cdot 3^{\frac{1}{2}} \cdot 5^{\frac{1}{2}} g^{\frac{1}{2}} h^{4} j^{9 \frac{1}{2}}) - 2 h^{4} |j^{9}| \sqrt{2 \cdot 3 \cdot 5 g j}$

$- 2 h^{4} |j^{9}| \sqrt{30 g j}$

$\sqrt[4]{625 f^{28} g^{48}}$

$\sqrt[4]{5^{4} f^{28} g^{48}} 5^{\frac{4}{4}} f^{\frac{28}{4}} g^{\frac{48}{4}}$

$5 |f^{7}| g^{12}$

${(3^{8} x^{3} y^{13} z^{15})}^{\frac{1}{5}}$

$3^{\frac{8}{5}} x^{\frac{3}{5}} y^{\frac{13}{5}} z^{\frac{15}{5}} 3^{1 \frac{3}{5}} x^{\frac{3}{5}} y^{2 \frac{3}{5}} z^{3} 3 y^{2} z^{3} \sqrt[5]{3^{3} x^{3} y^{3}}$

$3 y^{2} z^{3} \sqrt[5]{27 x^{3} y^{3}}$

$\pm {(6^{4} a^{7} b^{3} c^{10})}^{\frac{1}{2}}$

$\pm (6^{\frac{4}{2}} a^{\frac{7}{2}} b^{\frac{3}{2}} c^{\frac{10}{2}}) \pm (6^{2} a^{3 \frac{1}{2}} b^{1 \frac{1}{2}} c^{5})$

$\pm 36 |a^{3} b c^{5}| \sqrt{a b}$

$\sqrt[3]{54 x^{4} y^{3}}$

$\sqrt[3]{2 \cdot 3^{3} x^{4} y^{3}} 2^{\frac{1}{3}} \cdot 3^{\frac{3}{3}} x^{\frac{4}{3}} y^{\frac{3}{3}} 2^{\frac{1}{3}} \cdot 3^{1} x^{1 \frac{1}{3}} y^{1}$

$3 x y \sqrt[3]{2 x}$

$\sqrt[5]{243 q^{5} v^{9}}$

$\sqrt[5]{3^{5} q^{5} v^{9}} 3^{\frac{5}{5}} q^{\frac{5}{5}} v^{\frac{9}{5}} 3^{1} q^{1} v^{1 \frac{4}{5}}$

$3 q v \sqrt[5]{v^{4}}$

${(48 x^{10} y^{6} z^{2})}^{\frac{1}{4}}$

${(2^{4} \cdot 3 x^{10} y^{6} z^{2})}^{\frac{1}{4}} 2^{\frac{4}{4}} \cdot 3^{\frac{1}{4}} x^{\frac{10}{4}} y^{\frac{6}{4}} z^{\frac{2}{4}} 2^{1} \cdot 3^{\frac{1}{4}} x^{2 \frac{2}{4}} y^{1 \frac{2}{4}} z^{\frac{2}{4}}$

$2 x^{2} |y| \sqrt[4]{3 x^{2} y^{2} z^{2}}$

$- 5 x^{2} \sqrt{x y} \cdot 2 x \sqrt{x^{3} y}$

$- 10 x^{3} \sqrt{x^{4} y^{2}} - 10 x^{3} (x^{\frac{4}{2}} y^{\frac{2}{2}}) - 10 x^{3} (x^{2} y^{1})$

$- 10 x^{5} |y|$

$\sqrt[3]{7 c^{3} n^{4}} \cdot \sqrt[3]{42 c^{17} n^{13}}$

$\sqrt[3]{7 \cdot 42 c^{20} n^{17}} \sqrt[3]{2 \cdot 3 \cdot 7^{2} c^{20} n^{17}} 2^{\frac{1}{3}} \cdot 3^{\frac{1}{3}} \cdot 7^{\frac{2}{3}} c^{\frac{20}{3}} n^{\frac{17}{3}} 2^{\frac{1}{3}} \cdot 3^{\frac{1}{3}} \cdot 7^{\frac{2}{3}} c^{6 \frac{2}{3}} n^{5 \frac{2}{3}} c^{6} n^{5} \sqrt[3]{2 \cdot 3 \cdot 7^{2} c^{2} n^{2}}$

$c^{6} n^{5} \sqrt[3]{294 c^{2} n^{2}}$

$\sqrt[4]{32 b^{47} d^{29}} \cdot \sqrt[4]{13 b^{50} d^{8}}$

$\sqrt[4]{2^{5} \cdot 13 b^{97} d^{37}} 2^{\frac{5}{4}} \cdot 13^{\frac{1}{4}} b^{\frac{97}{4}} d^{\frac{37}{4}} 2^{1 \frac{1}{4}} \cdot 13^{\frac{1}{4}} b^{24 \frac{1}{4}} d^{9 \frac{1}{4}} 2 b^{24} |d^{9}| \sqrt[4]{2 \cdot 13 b d}$

$2 b^{24} |d^{9}| \sqrt[4]{26 b d}$

${(12 m^{7} n^{14} p^{7})}^{\frac{1}{4}} \cdot {(4 m^{5} n^{2} p^{3})}^{\frac{1}{4}}$

${(12 \cdot 4 m^{12} n^{16} p^{10})}^{\frac{1}{4}} {(2^{4} \cdot 3 m^{12} n^{16} p^{10})}^{\frac{1}{4}} 2^{\frac{4}{4}} \cdot 3^{\frac{1}{4}} m^{\frac{12}{4}} n^{\frac{16}{4}} p^{\frac{10}{4}} 2^{1} \cdot 3^{\frac{1}{4}} m^{3} n^{4} p^{2 \frac{2}{4}}$

$2 |m^{3}| n^{4} p^{2} \sqrt[4]{3 p^{2}}$

$11 x^{2} \sqrt{24 x^{2} y^{2}} \cdot 7 x^{8} \sqrt{2 y^{8}}$

$77 x^{10} \sqrt{24 \cdot 2 x^{2} y^{10}} 77 x^{10} \sqrt{2^{4} \cdot 3 x^{2} y^{10}} 77 x^{10} (2^{\frac{4}{2}} \cdot 3^{\frac{1}{2}} x^{\frac{2}{2}} y^{\frac{10}{2}}) 77 x^{10} (2^{2} \cdot 3^{\frac{1}{2}} x^{1} y^{5}) 77 \cdot 2^{2} |x^{11} y^{5}| \sqrt{3}$

$308 |x^{11} y^{5}| \sqrt{3}$

$5 \sqrt[3]{7^{6} n^{4} u^{18}} \cdot 13 n u \sqrt[3]{7^{3} n u}$

$65 n u \sqrt[3]{7^{9} n^{5} u^{19}} 65 n u (7^{\frac{9}{3}} n^{\frac{5}{3}} u^{\frac{19}{3}}) 65 n u (7^{3} n^{1 \frac{2}{3}} u^{6 \frac{1}{3}}) 65 \cdot 7^{3} n^{2} u^{7} \sqrt[3]{n^{2} u}$

$22, 295 n^{2} u^{7} \sqrt[3]{n^{2} u}$

Determine the area of a triangle with a base of ${(12 a^{3} b^{4})}^{\frac{1}{2}}$ units and a height of $6 a^{2} \sqrt{3 a b}$ units. Write your answer in simplified radical form.

$\begin{array}{rcl} A & = & \frac{1}{2} b h \\ A & = & \frac{1}{2} {(12 a^{3} b^{4})}^{\frac{1}{2}} \cdot 6 a^{2} \sqrt{3 a b} \\ = & 3 a^{2} {(12 \cdot 3 a^{4} b^{5})}^{\frac{1}{2}} \\ = & 3 a^{2} {(2^{2} \cdot 3^{2} a^{4} b^{5})}^{\frac{1}{2}} \\ = & 3 a^{2} (2^{\frac{2}{2}} \cdot 3^{\frac{2}{2}} a^{\frac{4}{2}} b^{\frac{5}{2}}) \\ = & 3 a^{2} (2^{1} \cdot 3^{1} a^{2} b^{2 \frac{1}{2}}) \\ = & 3 \cdot 2 \cdot 3 a^{4} b^{2} \sqrt{b} \end{array}$

$A = 18 a^{4} b^{2} \sqrt{b} {units}^{2}$

Determine the area of the rectangle with the dimensions ${(8 x y^{3})}^{\frac{1}{3}}$ and ${(216 x^{8})}^{\frac{1}{3}}$ units.

$\begin{array}{rcl} A & = & b h \\ A & = & {(8 x y^{3})}^{\frac{1}{3}} \cdot {(216 x^{8})}^{\frac{1}{3}} \\ = & {(2^{3} x y^{3})}^{\frac{1}{3}} \cdot {(6^{3} x^{8})}^{\frac{1}{3}} \\ = & {(2^{3} \cdot 6^{3} x^{9} y^{3})}^{\frac{1}{3}} \\ = & 2^{\frac{3}{3}} \cdot 6^{\frac{3}{3}} x^{\frac{9}{3}} y^{\frac{3}{3}} \\ = & 2^{1} \cdot 6^{1} x^{3} y^{1} \end{array}$

$A = 12 x^{3} y {units}^{2}$