Practice 1 Solutions

Simplify. Write answers in simplified radical form.

$\pm \sqrt{12 a^{10} b^{6}}$

$\pm \sqrt{2^{2} \cdot 3 a^{10} b^{6}} \pm (2^{\frac{2}{2}} \cdot 3^{\frac{1}{2}} a^{\frac{10}{2}} b^{\frac{6}{2}}) \pm (2^{1} \cdot 3^{\frac{1}{2}} a^{5} b^{3})$

$\pm 2 |a^{5} b^{3}| \sqrt{3}$

Note

Q: What happens when the index does not divide into the exponent evenly?

A: This is a term that will be the radicand.

${(11^{3} q^{4} c^{10})}^{\frac{1}{2}}$

$11^{\frac{3}{2}} q^{\frac{4}{2}} c^{\frac{10}{2}} 11^{1 \frac{1}{2}} q^{2} c^{5}$

$11 q^{2} |c^{5}| \sqrt{11}$

Note

Q: What is the index of the problem? Explain.

A: 2, because this is the denominator of the fractional exponent.

$\sqrt[5]{32 n^{15}}$

$\sqrt[5]{2^{5} n^{15}} 2^{\frac{5}{5}} n^{\frac{15}{5}}$

$2 n^{3}$

$- {(8^{5} v^{13} w^{33})}^{\frac{1}{3}}$

$- {({(2^{3})}^{5} v^{13} w^{33})}^{\frac{1}{3}} - {(2^{15} v^{13} w^{33})}^{\frac{1}{3}} - (2^{\frac{15}{3}} v^{\frac{13}{3}} w^{\frac{33}{3}}) - (2^{5} v^{4 \frac{1}{3}} w^{11})$

$- 32 v^{4} w^{11} \sqrt[3]{v}$

$\sqrt[4]{81 k^{44} u^{13} r^{51}}$

$\sqrt[4]{3^{4} k^{44} u^{13} r^{51}} 3^{\frac{4}{4}} k^{\frac{44}{4}} u^{\frac{13}{4}} r^{\frac{51}{4}} 3^{1} k^{11} u^{3 \frac{1}{4}} r^{12 \frac{3}{4}}$

$3 |k^{11} u^{3}| r^{12} \sqrt[4]{u r^{3}}$

${(7^{6} g^{7} h^{16})}^{\frac{1}{4}}$

$7^{\frac{6}{4}} g^{\frac{7}{4}} h^{\frac{16}{4}} 7^{1 \frac{2}{4}} g^{1 \frac{3}{4}} h^{4} 7 | g | h^{4} \sqrt[4]{7^{2} g^{3}}$

$7 | g | h^{4} \sqrt[4]{49 g^{3}}$

Note

You should maintain all denominators using the index so the answer can be written in simplified form. This is why $1 \frac{2}{4}$ is used rather than $1 \frac{1}{2}$ .

$\sqrt[3]{125 x^{33} y^{14}}$

$\sqrt[3]{5^{3} x^{33} y^{14}} 5^{\frac{3}{3}} x^{\frac{33}{3}} y^{\frac{14}{3}} 5^{1} x^{11} y^{4 \frac{2}{3}}$

$5 x^{11} y^{4} \sqrt[3]{y^{2}}$

$\sqrt[5]{16 b^{19} c^{13} d^{34}}$

$\sqrt[5]{2^{4} b^{19} c^{13} d^{34}} 2^{\frac{4}{5}} b^{\frac{19}{5}} c^{\frac{13}{5}} d^{\frac{34}{5}} 2^{\frac{4}{5}} b^{3 \frac{4}{5}} c^{2 \frac{3}{5}} d^{6 \frac{4}{5}} b^{3} c^{2} d^{6} \sqrt[5]{2^{4} b^{4} c^{3} d^{4}}$

$b^{3} c^{2} d^{6} \sqrt[5]{16 b^{4} c^{3} d^{4}}$

Note

Q: Why is there no coefficient outside of the radical?

A: Because 16 is $2^{4}$ and the exponent is smaller than the index.

$3 \sqrt{12 x y^{3}} \cdot 6 \sqrt{3 x^{5} y^{3}}$

$18 \sqrt{12 \cdot 3 x^{6} y^{6}} 18 \sqrt{2^{2} \cdot 3^{2} x^{6} y^{6}} 18 (2^{\frac{2}{2}} \cdot 3^{\frac{2}{2}} x^{\frac{6}{2}} y^{\frac{6}{2}}) 18 \cdot 2 \cdot 3 |x^{3} y^{3}|$

$108 |x^{3} y^{3}|$

$- 4 x \sqrt{7 g^{8} h^{2}} \cdot 5 \sqrt{21 g^{5} h^{11}}$

$- 20 x \sqrt{7 \cdot 21 g^{13} h^{13}} - 20 x \sqrt{7^{2} \cdot 3 g^{13} h^{13}} - 20 x (7^{\frac{2}{2}} \cdot 3^{\frac{1}{2}} g^{\frac{13}{2}} h^{\frac{13}{2}}) - 20 x (7^{1} \cdot 3^{\frac{1}{2}} g^{6 \frac{1}{2}} h^{6 \frac{1}{2}})$

$- 140 x g^{6} h^{6} \sqrt{3 g h}$

Note

Q: Why does x not belong inside absolute value bars?

A: Because x was already outside the radical.

$7 v^{2} \sqrt[3]{4 v} \cdot 2 \sqrt[3]{2 v}$

$14 v^{2} \sqrt[3]{4 \cdot 2 v^{2}} 14 v^{2} \sqrt[3]{2^{3} v^{2}} 14 v^{2} (2^{\frac{3}{3}} v^{\frac{2}{3}}) 14 v^{2} (2^{1} v^{\frac{2}{3}})$

$28 v^{2} \sqrt[3]{v^{2}}$

${(3^{4} a^{3} b^{5})}^{\frac{1}{5}} \cdot {(3^{2} a^{4} b^{10})}^{\frac{1}{5}}$

${(3^{6} a^{7} b^{15})}^{\frac{1}{5}} 3^{\frac{6}{5}} a^{\frac{7}{5}} b^{\frac{15}{5}} 3^{1 \frac{1}{5}} a^{1 \frac{2}{5}} b^{3}$

$3 a b^{3} \sqrt[5]{3 a^{2}}$

$\sqrt[4]{27 a^{10} b^{9} c} \cdot \sqrt[4]{18 a^{26} b^{21} c^{3}}$

$\sqrt[4]{27 \cdot 18 a^{36} b^{30} c^{4}} \sqrt[4]{3^{5} \cdot 2 a^{36} b^{30} c^{4}} 3^{\frac{5}{4}} \cdot 2^{\frac{1}{4}} a^{\frac{36}{4}} b^{\frac{30}{4}} c^{\frac{4}{4}} 3^{1 \frac{1}{4}} \cdot 2^{\frac{1}{4}} a^{9} b^{7 \frac{2}{4}} c^{1} 3 |a^{9} b^{7} c| \sqrt[4]{3 \cdot 2 b^{2}}$

$3 |a^{9} b^{7} c| \sqrt[4]{6 b^{2}}$

$- \sqrt{11 p^{4} q^{6}} \cdot 21 \sqrt{33 q^{16}}$

$- 21 \sqrt{11 \cdot 33 p^{4} q^{22}} - 21 \sqrt{11^{2} \cdot 3 p^{4} q^{22}} - 21 (11^{\frac{2}{2}} \cdot 3^{\frac{1}{2}} p^{\frac{4}{2}} q^{\frac{22}{2}}) - 21 (11^{1} \cdot 3^{\frac{1}{2}} p^{2} q^{11})$

$- 231 p^{2} |q^{11}| \sqrt{3}$

Note

Problems 15–16

Remember to use absolute value bars when an even index results in an odd power after simplifying.

Q: What is needed in your answer when the index is even and their result is an odd power?
A: Absolute value bars around the base with an odd power.

Find the area of a rectangle with the dimensions ${(5 x^{3} y^{6} z)}^{\frac{1}{2}}$ and $\sqrt{5 x y}$ units. Write your answer in simplified radical form.

$\begin{array}{rcl} A & = & l w \\ A & = & {(5 x^{3} y^{6} z)}^{\frac{1}{2}} \cdot {(5 x y)}^{\frac{1}{2}} \\ = & {(5^{2} x^{4} y^{7} z)}^{\frac{1}{2}} \\ = & 5^{\frac{2}{2}} x^{\frac{4}{2}} y^{\frac{7}{2}} z^{\frac{1}{2}} \\ = & 5^{1} x^{2} y^{3 \frac{1}{2}} z^{\frac{1}{2}} \end{array}$

$A = 5 x^{2} |y^{3}| \sqrt{y z} {units}^{2}$

A triangle has a height of ${(21 a^{4} b^{7})}^{\frac{1}{3}}$ centimeters, and base of $\sqrt[3]{49 a b^{2}}$ centimeters. Find the area of the triangle in simplified radical form.

$\begin{array}{rcl} A & = & \frac{1}{2} b h \\ A & = & \frac{1}{2} {(49 a b^{2})}^{\frac{1}{3}} \cdot {(21 a^{4} b^{7})}^{\frac{1}{3}} \\ = & \frac{1}{2} {(49 \cdot 21 a^{5} b^{9})}^{\frac{1}{3}} \\ = & \frac{1}{2} {(7^{3} \cdot 3 a^{5} b^{9})}^{\frac{1}{3}} \\ = & \frac{1}{2} (7^{\frac{3}{3}} \cdot 3^{\frac{1}{3}} a^{\frac{5}{3}} b^{\frac{9}{3}}) \\ = & \frac{1}{2} (7^{1} \cdot 3^{\frac{1}{3}} a^{1 \frac{2}{3}} b^{3}) \end{array}$

$A = \frac{7 a b^{3} \sqrt[3]{3 a^{2}}}{2} {cm}^{2}$