Chapter 5

The mass of an electron is (9.11) \(\left(10^{-31}\right)\) kilogram, and the mass of a proton is \((1.67)\left(10^{-27}\right)\) kilogram. Approximately how many times more is the weight of a proton than the weight of an electron? Express the result in decimal form.

For Problems 59-80, simplify each of the following. Express final results using positive exponents only. For example,\(\left(2 x^{\frac{1}{2}}\right)\left(3 x^{\frac{1}{3}}\right)=6 x^{\frac{5}{6}}\). \(\left(2 x^{\frac{2}{5}}\right)\left(6 x^{\frac{1}{4}}\right)\)

Rationalize the denominator and simplify. All variables represent positive real numbers. \(\frac{\sqrt{2}}{\sqrt{10}-\sqrt{3}}\)

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. \(-\frac{4 \sqrt{12}}{\sqrt{5}}\)

Simplify each expression. Express final results without using zero or negative integers as exponents. \(\frac{x^{-6}}{x^{-4}}\)

A square pixel on a computer screen has a side of length (1.17) \(\left(10^{-2}\right)\) inches. Find the approximate area of the pixel in inches. Express the result in decimal form.

Simplify each of the following. Express final results using positive exponents only. For example,\(\left(2 x^{\frac{1}{2}}\right)\left(3 x^{\frac{1}{3}}\right)=6 x^{\frac{5}{6}}\). \(\left(3 x^{\frac{1}{4}}\right)\left(5 x^{\frac{1}{3}}\right)\)

Explain the concept of extraneous solutions.

Rationalize the denominator and simplify. All variables represent positive real numbers. \(\frac{\sqrt{3}}{\sqrt{7}-\sqrt{2}}\)

Express each of the following in simplest radical form. All variables represent positive real numbers. \(\frac{5}{\sqrt[3]{9 x y^{2}}}\)

Change each radical to simplest radical form. \(\frac{-6 \sqrt{5}}{\sqrt{18}}\)

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

Explain the importance of scientific notation.

Explain why possible solutions for radical equations must be checked.

Rationalize the denominator and simplify. All variables represent positive real numbers. \(\frac{\sqrt{3}}{2 \sqrt{5}+4}\)

Express each of the following in simplest radical form. All variables represent positive real numbers. \(\sqrt{8 x+12 y} \quad[\) Hint: \(\sqrt{8 x+12 y}=\sqrt{4(2 x+3 y)}]\)

Change each radical to simplest radical form. \(\frac{3 \sqrt{2}}{4 \sqrt{3}}\)

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

Why do we need scientific notation even when using calculators and computers?

Simplify each of the following. Express final results using positive exponents only. For example,\(\left(2 x^{\frac{1}{2}}\right)\left(3 x^{\frac{1}{3}}\right)=6 x^{\frac{5}{6}}\). \(\left(y^{\frac{3}{4}}\right)\left(y^{-\frac{1}{2}}\right)\)

Your friend makes an effort to solve the equation \(3+2 \sqrt{x}=x\) as follows: $$ \begin{array}{r} (3+2 \sqrt{x})^{2}=x^{2} \\ 9+12 \sqrt{x}+4 x=x^{2} \end{array} $$ At this step he stops and doesn't know how to proceed. What help would you give him?

Rationalize the denominator and simplify. All variables represent positive real numbers. \(\frac{\sqrt{7}}{3 \sqrt{2}-5}\)

Express each of the following in simplest radical form. All variables represent positive real numbers. \(\sqrt{4 x+4 y}\)

Change each radical to simplest radical form. \(\frac{6 \sqrt{5}}{5 \sqrt{12}}\)

Simplify each expression. Express final results without using zero or negative integers as exponents. \(\frac{x^{-3} y^{-4}}{x^{2} y^{-1}}\)

Sometimes it is more convenient to express a number as a product of a power of 10 and a number that is not between 1 and 10 . For example, suppose that we want to calculate \(\sqrt{640,000}\). We can proceed as follows: $$ \begin{aligned} \sqrt{640,000} &=\sqrt{(64)(10)^{4}} \\ &=\left((64)(10)^{4}\right)^{\frac{1}{2}} \\ &=(64)^{\frac{1}{2}}\left(10^{4}\right)^{\frac{1}{2}} \\ &=(8)(10)^{2} \\ &=8(100)=800 \end{aligned} $$ Compute each of the following without a calculator, and then use a calculator to check your answers. (a) \(\sqrt{49,000,000}\) (b) \(\sqrt{0.0025}\) (c) \(\sqrt{14,400}\) (d) \(\sqrt{0.000121}\) (e) \(\sqrt[3]{27,000}\) (f) \(\sqrt[3]{0.000064}\)

Simplify each of the following. Express final results using positive exponents only. For example,\(\left(2 x^{\frac{1}{2}}\right)\left(3 x^{\frac{1}{3}}\right)=6 x^{\frac{5}{6}}\). \(\left(x^{\frac{2}{5}}\right)\left(4 x^{-\frac{1}{2}}\right)\)

Rationalize the denominator and simplify. All variables represent positive real numbers. \(\frac{6}{3 \sqrt{7}-2 \sqrt{6}}\)

Express each of the following in simplest radical form. All variables represent positive real numbers. \(\sqrt{16 x+48 y}\)

Change each radical to simplest radical form. \(\frac{-8 \sqrt{18}}{10 \sqrt{50}}\)

For Problems \(63-74\), find the indicated products and quotients. Express final results using positive integral exponents only. \(\left(2 x y^{-1}\right)\left(3 x^{-2} y^{4}\right)\)

Use your calculator to evaluate each of the following. Express final answers in ordinary notation. (a) \((27,000)^{2}\) (b) \((450,000)^{2}\) (c) \((14,800)^{2}\) (d) \((1700)^{3}\)(e) \((900)^{4}\) (f) \((60)^{5}\) (g) \((0.0213)^{2}\) (h) \((0.000213)^{2}\) (i) \((0.000198)^{2}\) (j) \((0.000009)^{3}\)

Simplify each of the following. Express final results using positive exponents only. For example,\(\left(2 x^{\frac{1}{2}}\right)\left(3 x^{\frac{1}{3}}\right)=6 x^{\frac{5}{6}}\). \(\left(2 x^{\frac{1}{3}}\right)\left(x^{-\frac{1}{2}}\right)\)

Rationalize the denominator and simplify. All variables represent positive real numbers. \(\frac{5}{2 \sqrt{5}+3 \sqrt{7}}\)

Change each radical to simplest radical form. \(\frac{4 \sqrt{45}}{-6 \sqrt{20}}\)

Find the indicated products and quotients. Express final results using positive integral exponents only. \(\left(-4 x^{-1} y^{2}\right)\left(6 x^{3} y^{-4}\right)\)

Use your calculator to estimate each of the following. Express final answers in scientific notation with the number between 1 and 10 rounded to the nearest onethousandth. (a) \((4576)^{4}\) (b) \((719)^{10}\) (c) \((28)^{12}\) (d) \((8619)^{6}\) (e) \((314)^{5}\) (f) \((145,723)^{2}\)

Simplify each of the following. Express final results using positive exponents only. For example,\(\left(2 x^{\frac{1}{2}}\right)\left(3 x^{\frac{1}{3}}\right)=6 x^{\frac{5}{6}}\). \(\left(4 x^{\frac{1}{2}} y\right)^{2}\)

Rationalize the denominator and simplify. All variables represent positive real numbers. \(\frac{\sqrt{6}}{3 \sqrt{2}+2 \sqrt{3}}\)

For Problems \(65-74\), use the distributive property to help simplify each of the following. All variables represent positive real numbers. \(-3 \sqrt{4 x}+5 \sqrt{9 x}+6 \sqrt{16 x}\)

Change each radical to simplest radical form. \(\sqrt[3]{16}\)

Find the indicated products and quotients. Express final results using positive integral exponents only. \(\left(-7 a^{2} b^{-5}\right)\left(-a^{-2} b^{7}\right)\)

Use your calculator to estimate each of the following. Express final answers in ordinary notation rounded to the nearest one-thousandth. (a) \((1.09)^{5}\) (b) \((1.08)^{10}\) (c) \((1.14)^{7}\) (d) \((1.12)^{20}\) (e) \((0.785)^{4}\) (f) \((0.492)^{5}\)

Simplify each of the following. Express final results using positive exponents only. For example,\(\left(2 x^{\frac{1}{2}}\right)\left(3 x^{\frac{1}{3}}\right)=6 x^{\frac{5}{6}}\). \(\left(3 x^{\frac{1}{4}} y^{\frac{1}{5}}\right)^{3}\)

Rationalize the denominator and simplify. All variables represent positive real numbers. \(\frac{3 \sqrt{6}}{5 \sqrt{3}-4 \sqrt{2}}\)

Use the distributive property to help simplify each of the following. All variables represent positive real numbers. \(-2 \sqrt{25 x}-4 \sqrt{36 x}+7 \sqrt{64 x}\)

Change each radical to simplest radical form. \(\sqrt[3]{40}\)

Find the indicated products and quotients. Express final results using positive integral exponents only. \(\left(-9 a^{-3} b^{-6}\right)\left(-12 a^{-1} b^{4}\right)\)

Simplify each of the following. Express final results using positive exponents only. For example,\(\left(2 x^{\frac{1}{2}}\right)\left(3 x^{\frac{1}{3}}\right)=6 x^{\frac{5}{6}}\). \(\left(8 x^{6} y^{3}\right)^{\frac{1}{3}}\)