Chapter 10

Assume that all variables represent positive real numbers. $$ \sqrt[3]{-27 x^{12} y^{9}} $$

Use rational exponents to simplify each radical. Assume that all variables represent positive real numbers. $$ \sqrt[8]{x^{4} y^{4}} $$

Use the quotient rule to divide. Then simplify if possible. Assume that all variables represent positive real numbers. \(\frac{8 \sqrt[3]{54 m^{7}}}{\sqrt[3]{2 m}}\)

Multiply. Then simplify if possible. Assume that all variables represent positive real numbers. $$ (\sqrt[3]{4}+2)(\sqrt[3]{2}-1) $$

Divide. Write your answers in the form \(a+b i\) $$ \frac{2-3 i}{2+i} $$

Rationalize each numerator. Assume that all variables represent positive real numbers. \(\frac{5+\sqrt{2}}{\sqrt{2 x}}\)

The period of a pendulum is the time it takes for the pendulum to make one full back-and-forth swing. The period of a pendulum depends on the length of the pendulum. The formula for the period \(P\), in seconds, is \(P=2 \pi \sqrt{\frac{l}{32}},\) where l is the length of the pendulum in feet. Klockit sells a 43 -inch lyre pendulum. Find the period of this pendulum. Round your answer to 2 decimal places. (Hint: First convert inches to feet.)

Assume that all variables represent positive real numbers. $$ \sqrt[3]{-8 a^{21} y^{6}} $$

Use rational exponents to simplify each radical. Assume that all variables represent positive real numbers. $$ \sqrt[9]{y^{6} z^{3}} $$

Use the quotient rule to divide. Then simplify if possible. Assume that all variables represent positive real numbers. \(\frac{\sqrt[3]{128 x^{3}}}{-3 \sqrt[3]{2 x}}\)

Multiply. Then simplify if possible. Assume that all variables represent positive real numbers. $$ (\sqrt[3]{3}+\sqrt[3]{2})(\sqrt[3]{9}-\sqrt[3]{4}) $$

Divide. Write your answers in the form \(a+b i\) $$ \frac{6+5 i}{6-5 i} $$

Rationalize each numerator. Assume that all variables represent positive real numbers. \(\frac{\sqrt{x}+1}{\sqrt{x}-1}\)

The period of a pendulum is the time it takes for the pendulum to make one full back-and-forth swing. The period of a pendulum depends on the length of the pendulum. The formula for the period \(P\), in seconds, is \(P=2 \pi \sqrt{\frac{l}{32}},\) where l is the length of the pendulum in feet. Find the length of a pendulum whose period is 4 seconds. Round your answer to 2 decimal places.

Assume that all variables represent positive real numbers. $$ \sqrt[4]{a^{16} b^{4}} $$

Use rational exponents to simplify each radical. Assume that all variables represent positive real numbers. $$ \sqrt[12]{a^{8} b^{4}} $$

Use the quotient rule to divide. Then simplify if possible. Assume that all variables represent positive real numbers. \(\frac{3 \sqrt{100 x^{2}}}{2 \sqrt{2 x^{-1}}}\)

Multiply. Then simplify if possible. Assume that all variables represent positive real numbers. $$ (\sqrt[3]{x}+1)(\sqrt[3]{x}-4 \sqrt{x}+7) $$

Find each power of \(i\). $$ i^{8} $$

Rationalize each numerator. Assume that all variables represent positive real numbers. \(\frac{\sqrt{x}+\sqrt{y}}{\sqrt{x}-\sqrt{y}}\)

The period of a pendulum is the time it takes for the pendulum to make one full back-and-forth swing. The period of a pendulum depends on the length of the pendulum. The formula for the period \(P\), in seconds, is \(P=2 \pi \sqrt{\frac{l}{32}},\) where l is the length of the pendulum in feet. Find the length of a pendulum whose period is 3 seconds. Round your answer to 3 decimal places.

Assume that all variables represent positive real numbers. $$ \sqrt[4]{x^{8} y^{12}} $$

Use rational exponents to simplify each radical. Assume that all variables represent positive real numbers. $$ \sqrt[10]{a^{5} b^{5}} $$

Use the quotient rule to divide. Then simplify if possible. Assume that all variables represent positive real numbers. \(\frac{\sqrt{270 y^{2}}}{5 \sqrt{3 y^{-4}}}\)

Multiply. Then simplify if possible. Assume that all variables represent positive real numbers. $$ (\sqrt[3]{3 x}+3)(\sqrt[3]{2 x}-3 x-1) $$

Find each power of \(i\). $$ i^{10} $$

Solve each equation. \(2 x-7=3(x-4)\)

The period of a pendulum is the time it takes for the pendulum to make one full back-and-forth swing. The period of a pendulum depends on the length of the pendulum. The formula for the period \(P\), in seconds, is \(P=2 \pi \sqrt{\frac{l}{32}},\) where l is the length of the pendulum in feet. Study the relationship between period and pendulum length in Exercises 69 through 72 and make a conjecture about this relationship.

Assume that all variables represent positive real numbers. $$ \sqrt[5]{-32 x^{10} y^{5}} $$

Use rational expressions to write as a single radical expression. $$ \sqrt[3]{y} \cdot \sqrt[5]{y^{2}} $$

Use the quotient rule to divide. Then simplify if possible. Assume that all variables represent positive real numbers. \(\frac{\sqrt[4]{96 a^{10} b^{3}}}{\sqrt[4]{3 a^{2} b^{3}}}\)

Multiply. Then simplify if possible. Assume that all variables represent positive real numbers. $$ (\sqrt{x-1}+5)^{2} $$

Find each power of \(i\). $$ i^{21} $$

Solve each equation. \(9 x-4=7(x-2)\)

Use rational expressions to write as a single radical expression. $$ \sqrt[3]{y^{2}} \cdot \sqrt[6]{y} $$

Use the quotient rule to divide. Then simplify if possible. Assume that all variables represent positive real numbers. \(\frac{\sqrt[4]{160 x^{10} y^{5}}}{\sqrt[4]{2 x^{2} y^{2}}}\)

Multiply. Then simplify if possible. Assume that all variables represent positive real numbers. $$ (\sqrt{3 x+1}+2)^{2} $$

The period of a pendulum is the time it takes for the pendulum to make one full back-and-forth swing. The period of a pendulum depends on the length of the pendulum. The formula for the period \(P\), in seconds, is \(P=2 \pi \sqrt{\frac{l}{32}},\) where l is the length of the pendulum in feet. Galileo experimented with pendulums. He supposedly made conjectures about pendulums of equal length with different bob weights. Try this experiment. Make two pendulums 3 feet long. Attach a heavy weight (lead) to one and a light weight (a cork) to the other. Pull both pendulums back the same angle measure and release. Make a conjecture from your observations.

Find each power of \(i\). $$ i^{15} $$

Solve each equation. \((x-6)(2 x+1)=0\)

Assume that all variables represent positive real numbers. $$ \sqrt{\frac{25}{49}} $$

Use rational expressions to write as a single radical expression. $$ \frac{\sqrt[3]{b^{2}}}{\sqrt[4]{b}} $$

Use the quotient rule to divide. Then simplify if possible. Assume that all variables represent positive real numbers. \(\frac{\sqrt[5]{64 x^{10} y^{3}}}{\sqrt[5]{2 x^{3} y^{-7}}}\)

Multiply. Then simplify if possible. Assume that all variables represent positive real numbers. $$ (\sqrt{2 x+5}-1)^{2} $$

Find each power of \(i\). $$ i^{11} $$

Solve each equation. \((y+2)(5 y+4)=0\)

Assume that all variables represent positive real numbers. $$ \sqrt{\frac{4}{81}} $$

Use rational expressions to write as a single radical expression. $$ \frac{\sqrt[4]{a}}{\sqrt[5]{a}} $$

Use the quotient rule to divide. Then simplify if possible. Assume that all variables represent positive real numbers. \(\frac{\sqrt[5]{192 x^{6} y^{12}}}{\sqrt[5]{2 x^{-1} y^{-3}}}\)

Multiply. Then simplify if possible. Assume that all variables represent positive real numbers. $$ (\sqrt{x-6}-7)^{2} $$