Chapter 5

Change each radical to simplest radical form. \(\frac{\sqrt{11}}{\sqrt{24}}\)

Simplify each expression. Express final results without using zero or negative integers as exponents. \(\left(a^{3} b^{-3} c^{-2}\right)^{-5}\)

The Social Security program paid out approximately \(\$ 33,200,000,000\) in benefits in May 2000 . Express this number in scientific notation.

Write each of the following using positive rational exponents. For example, \(\sqrt{a b}=(a b)^{\frac{1}{2}}=a^{\frac{1}{2}} b^{\frac{1}{2}}\). \(\sqrt[6]{a b^{5}}\)

Solve each equation. Don't forget to check each of your potential solutions. \(\sqrt{2 x-1}-\sqrt{x+3}=1\)

Find the following products and express answers in simplest radical form. All variables represent nonnegative real numbers. \(3 \sqrt[3]{3}(4 \sqrt[3]{9}+5 \sqrt[3]{7})\)

Express each of the following in simplest radical form. All variables represent positive real numbers. \(\sqrt[3]{54 x^{3}}\)

Change each radical to simplest radical form. \(\frac{\sqrt{5}}{\sqrt{48}}\)

Simplify each expression. Express final results without using zero or negative integers as exponents. \(\left(a^{3} b^{-3} c^{-2}\right)^{-5}\)

Carlos's first computer had a processing speed of (1.6) \(\left(10^{6}\right)\) hertz. He recently purchased a laptop computer with a processing speed of \((1.33)\left(10^{9}\right)\) hertz. Approximately how many times faster is the processing speed of his laptop than that of his first computer? Express the result in decimal form.

Write each of the following using positive rational exponents. For example, \(\sqrt{a b}=(a b)^{\frac{1}{2}}=a^{\frac{1}{2}} b^{\frac{1}{2}}\). \(\sqrt[5]{(2 x-y)^{3}}\)

Solve each equation. Don't forget to check each of your potential solutions. \(\sqrt{n-4}+\sqrt{n+4}=2 \sqrt{n-1}\)

For Problems \(53-76\), rationalize the denominator and simplify. All variables represent positive real numbers. \(\frac{2}{\sqrt{7}+1}\)

Express each of the following in simplest radical form. All variables represent positive real numbers. \(\sqrt[3]{56 x^{6} y^{8}}\)

Change each radical to simplest radical form. \(\frac{\sqrt{18}}{\sqrt{27}}\)

Simplify each expression. Express final results without using zero or negative integers as exponents. \(\left(2 x^{3} y^{-4}\right)^{-3}\)

Alaska has an area of approximately \((6.15)\left(10^{5}\right)\) square miles. In 1999 the state had a population of approximately 619,000 people. Compute the population density to the nearest hundredth. Population density is the number of people per square mile. Express the result in decimal form rounded to the nearest hundredth.

Write each of the following using positive rational exponents. For example, \(\sqrt{a b}=(a b)^{\frac{1}{2}}=a^{\frac{1}{2}} b^{\frac{1}{2}}\). \(\sqrt[7]{(3 x-y)^{4}}\)

Solve each equation. Don't forget to check each of your potential solutions. \(\sqrt{n-3}+\sqrt{n+5}=2 \sqrt{n}\)

Rationalize the denominator and simplify. All variables represent positive real numbers. \(\frac{6}{\sqrt{5}+2}\)

Express each of the following in simplest radical form. All variables represent positive real numbers. \(\sqrt[3]{81 x^{5} y^{6}}\)

Change each radical to simplest radical form. \(\frac{\sqrt{10}}{\sqrt{20}}\)

Simplify each expression. Express final results without using zero or negative integers as exponents. \(\left(4 x^{5} y^{-2}\right)^{-2}\)

In the year 2000 the public debt of the United States was approximately \(\$ 5,700,000,000,000\). For July 2000 , the census reported that \(275,000,000\) people lived in the United States. Convert these figures to scientific notation, and compute the average debt per person. Express the result in scientific notation.

Write each of the following using positive rational exponents. For example, \(\sqrt{a b}=(a b)^{\frac{1}{2}}=a^{\frac{1}{2}} b^{\frac{1}{2}}\). \(5 x \sqrt{y}\)

Solve each equation. Don't forget to check each of your potential solutions. \(\sqrt{t+3}-\sqrt{t-2}=\sqrt{7-t}\)

Rationalize the denominator and simplify. All variables represent positive real numbers. \(\frac{3}{\sqrt{2}-5}\)

Express each of the following in simplest radical form. All variables represent positive real numbers. \(\sqrt[3]{\frac{7}{9 x^{2}}}\)

Change each radical to simplest radical form. \(\frac{\sqrt{35}}{\sqrt{7}}\)

Simplify each expression. Express final results without using zero or negative integers as exponents. \(\left(\frac{x^{-1}}{y^{-4}}\right)^{-3}\)

The space shuttle can travel at approximately 410,000 miles per day. If the shuttle could travel to Mars, and Mars was \(140,000,000\) miles away, how many days would it take the shuttle to travel to Mars? Express the result in decimal form.

Write each of the following using positive rational exponents. For example, \(\sqrt{a b}=(a b)^{\frac{1}{2}}=a^{\frac{1}{2}} b^{\frac{1}{2}}\). \(4 y \sqrt[3]{x}\)

Solve each equation. Don't forget to check each of your potential solutions. \(\sqrt{t+7}-2 \sqrt{t-8}=\sqrt{t-5}\)

Rationalize the denominator and simplify. All variables represent positive real numbers. \(\frac{-4}{\sqrt{6}-3}\)

Express each of the following in simplest radical form. All variables represent positive real numbers. \(\sqrt[3]{\frac{5}{2 x}}\)

Change each radical to simplest radical form. \(\frac{\sqrt{42}}{\sqrt{6}}\)

Simplify each expression. Express final results without using zero or negative integers as exponents. \(\left(\frac{y^{3}}{x^{-4}}\right)^{-2}\)

Atomic masses are measured in atomic mass units (amu). The amu, \((1.66)\left(10^{-27}\right)\) kilograms, is defined as \(\frac{1}{12}\) the mass of a common carbon atom. Find the mass of a carbon atom in kilograms. Express the result in scientific notation.

Write each of the following using positive rational exponents. For example, \(\sqrt{a b}=(a b)^{\frac{1}{2}}=a^{\frac{1}{2}} b^{\frac{1}{2}}\). \(-\sqrt[3]{x+y}\)

Rationalize the denominator and simplify. All variables represent positive real numbers. \(\frac{1}{\sqrt{2}+\sqrt{7}}\)

Express each of the following in simplest radical form. All variables represent positive real numbers. \(\frac{\sqrt[3]{3 y}}{\sqrt[3]{16 x^{4}}}\)

Change each radical to simplest radical form. \(\frac{2 \sqrt{3}}{\sqrt{7}}\)

Simplify each expression. Express final results without using zero or negative integers as exponents. \(\left(\frac{3 a^{-2}}{2 b^{-1}}\right)^{-2}\)

The field of view of a microscope is \((4)\left(10^{-4}\right)\) meters. If a single cell organism occupies \(\frac{1}{5}\) of the field of view, find the length of the organism in meters. Express the result in scientific notation.

Write each of the following using positive rational exponents. For example, \(\sqrt{a b}=(a b)^{\frac{1}{2}}=a^{\frac{1}{2}} b^{\frac{1}{2}}\). \(-\sqrt[5]{(x-y)^{2}}\)

Solve the formula \(T=2 \pi \sqrt{\frac{L}{32}}\) for \(L\). (Remember that in this formula, which was used in Section \(5.2, T\) represents the period of a pendulum expressed in seconds, and \(L\) represents the length of the pendulum in feet.)

Rationalize the denominator and simplify. All variables represent positive real numbers. \(\frac{3}{\sqrt{3}+\sqrt{10}}\)

Express each of the following in simplest radical form. All variables represent positive real numbers. \(\frac{\sqrt[3]{12 x y}}{\sqrt[3]{3 x^{2} y^{5}}}\)

Change each radical to simplest radical form. \(\frac{3 \sqrt{2}}{\sqrt{6}}\)

Simplify each expression. Express final results without using zero or negative integers as exponents. \(\left(\frac{2 x y^{2}}{5 a^{-1} b^{-2}}\right)^{-1}\)