Chapter 8

EVALUATING EXPRESSIONS Evaluate the expression without using a calculator. Write the result in scientific notation and in decimal form. $$ \left(3 \times 10^{-2}\right)^{4} $$

Write your answer as a power or as a product of powers. $$ \left(6 a^{4}\right)^{2}\left(\frac{1}{4} a^{3}\right)^{2} $$

Simplify the expression. The simplified expression should have no negative exponents. $$ \frac{5 x^{-3} y^{2}}{x^{5} y^{-1}} \cdot \frac{\left(2 x y^{3}\right)^{-2}}{x y} $$

Solve the equation. Round the result to the nearest tenth if necessary. $$2.5(3.5 p+6.4)=18.2 p-6.5$$

Solve the equation. $$\frac{3}{2}(8 m-30)=-3 m$$

CALCULATOR Use a calculator to evaluate the expression. Write the result in scientific notation and in decimal form. $$ 2,000,000 \cdot 12,000 $$

Write your answer as a power or as a product of powers. $$ (2 t)^{3}\left(-t^{2}\right) $$

Simplify the expression. The simplified expression should have no negative exponents. $$ \left(\frac{2 x y^{-2} y^{4}}{3 x^{-1} y}\right)^{-2} \cdot\left(\frac{4 x y}{2 x^{-1} y^{-3}}\right)^{2} $$

A TV station's local news program has \(50,000\) viewers. The managers of the station plan to increase the number of viewers by \(2 \%\) per month. Write an exponential growth model to represent the number of viewers in \(t\) months.

Solve the equation. $$1.4(6.4 y-3.5)=-9.54 y+22.85$$

CALCULATOR Use a calculator to evaluate the expression. Write the result in scientific notation and in decimal form. $$ 6,000,000 \cdot 324,000 $$

Write your answer as a power or as a product of powers. $$ \left(-w^{3}\right)\left(3 w^{2}\right)^{2} $$

Use a calculator to evaluate the expression. Round your answer to the nearest hundred thousandth. $$2^{-5}$$

CALCULATOR Use a calculator to evaluate the expression. Write the result in scientific notation and in decimal form. $$ 0.000279 \cdot 3,940,000,000 $$

Write your answer as a power or as a product of powers. $$ (-y)^{3}(-y)^{4}(-y)^{5} $$

Use a calculator to evaluate the expression. Round your answer to the nearest hundred thousandth. $$(1.1)^{-2}$$

CALCULATOR Use a calculator to evaluate the expression. Write the result in scientific notation and in decimal form. $$ 654,000 \cdot 0.000042 $$

Write your answer as a power or as a product of powers. $$ (-x)^{4}(-x)^{3}(-x)^{2} $$

Use a calculator to evaluate the expression. Round your answer to the nearest hundred thousandth. $$55 \cdot 5^{-6}$$

The volume of a sphere is given by \(V=\frac{4}{3} \pi r^{3},\) where \(r\) is the radius of the sphere. Assuming that the radius of the Moon is \(\frac{1}{4}\) the radius of Earth, find the ratio of the volume of Earth to the volume of the Moon. Let \(r\) represent the radius of Earth. (image cannot copy)

CALCULATOR Use a calculator to evaluate the expression. Write the result in scientific notation and in decimal form. $$ \left(2.4 \times 10^{-4}\right)^{2} $$

Write your answer as a power or as a product of powers. $$ \left(a b c^{2}\right)^{3}\left(a^{2} b\right)^{2} $$

Use a calculator to evaluate the expression. Round your answer to the nearest hundred thousandth. $$3^{-5} \div 0.9$$

From 1994 to 1998 , the sales for a national clothing store increased by about the same percent each year. The sales \(S\) (in millions of dollars) for year \(t\) can be modeled by $$ S=3723\left(\frac{6}{5}\right)^{t} $$ where \(t=0\) corresponds to \(1994 .\) Find the ratio of 1998 sales to 1995 sales.

CALCULATOR Use a calculator to evaluate the expression. Write the result in scientific notation and in decimal form. $$ 0.000094^{3} $$

Write your answer as a power or as a product of powers. $$ -\left(r^{2} s t^{3}\right)^{2}\left(s^{4} t\right)^{3} $$

Does the graph of the function contain the point \((0,1) ?\) $$y=-3^{x}$$

The average salary for a professional baseball player in the United States can be approximated by \(y=283(1.2)^{t},\) where \(t=0\) represents the year \(1984 .\) Using this approximation, find the ratio of an average salary in 1988 to an average salary in 1994

SCIENTIFIC NOTATION IN REAL LIFE In Exercises \(53-57\), write the number in scientific notation. LIGHTNING The speed of a lightning bolt is \(120,000,000\) feet per second.

Write your answer as a power or as a product of powers. $$ \left(-3 x y^{2}\right)^{3}\left(-2 x^{2} y\right)^{2} $$

Does the graph of the function contain the point \((0,1) ?\) $$y=4^{x}$$

Use the following information. The average weight \(w\) (in pounds) of an Atlantic cod \(t\) years old can be modeled by the equation \(w=1.16(1.44)^{t} .\) (image cannot copy) Find the ratio of the weight of a 5 -year-old cod to the weight of a 2 -year- old cod. Express this ratio as a power of 1.44

SCIENTIFIC NOTATION IN REAL LIFE In Exercises \(53-57\), write the number in scientific notation. WORLD POPULATION In \(1997,\) the population of the world was estimated at $$ 5,852,000,000 $$

Does the graph of the function contain the point \((0,1) ?\) $$y=4 \cdot 1^{x}$$

Use the following information. The average weight \(w\) (in pounds) of an Atlantic cod \(t\) years old can be modeled by the equation \(w=1.16(1.44)^{t} .\) (image cannot copy) A 5 -year-old cod weighs how many times as much as a 2 -year-old cod?

Does the graph of the function contain the point \((0,1) ?\) $$y=60^{x}$$

Use the following information. The average weight \(w\) (in pounds) of an Atlantic cod \(t\) years old can be modeled by the equation \(w=1.16(1.44)^{t} .\) (image cannot copy) According to the model, what is the average weight of an Atlantic cod when it is hatched? How did you get your answer?

CHEMISTRY The mass of a carbon atom is 0.00000000000000000000002 gram.

Then evaluate the expression when \(a=1\) and \(b=2\). $$ \left(a^{3}\right)^{2} $$

Graph the exponential function. $$y=\left(\frac{1}{3}\right)^{x}$$

From 1995 through 1999 , the sales for a national furniture store increased by about the same percent each year. The sales \(s\) (in millions of dollars) for year \(t\) can be modeled by \(s=476(1.13)^{t},\) where \(t=0\) represents \(1995 .\) Find the ratio of 1997 sales to 1999 sales.

SIZE OF JUPITER Jupiter, the largest planet in our solar system, has a radius of about \(4.4 \times 10^{4}\) miles. Use the equation \(V=\frac{4}{3} \pi r^{3}\) to find Jupiter's volume.

Graph the exponential function. $$y=\left(\frac{1}{5}\right)^{x}$$

In Exercises \(58-60\), use the following information. In \(1803,\) the Louisiana Purchase added \(8.28 \times 10^{5}\) square miles to the United States. The cost of this land was \(\$ 15\) million. In \(1853,\) the Gadsden Purchase added \(2.94 \times 10^{4}\) square miles, and the cost was \(\$ 10\) million. Find the average cost of a square mile for each of the purchases.

Then evaluate the expression when \(a=1\) and \(b=2\). $$ \left(a \cdot b^{2}\right)^{2} $$

Graph the exponential function. $$y=4^{-x}$$

Then evaluate the expression when \(a=1\) and \(b=2\). $$ \left(a^{2} b\right)^{4} $$

Graph the exponential function. $$y=5^{x}$$

You roll a die eight times. What is the probability that you will roll eight sixes in a row?

In Exercises \(58-60\), use the following information. In \(1803,\) the Louisiana Purchase added \(8.28 \times 10^{5}\) square miles to the United States. The cost of this land was \(\$ 15\) million. In \(1853,\) the Gadsden Purchase added \(2.94 \times 10^{4}\) square miles, and the cost was \(\$ 10\) million. Writing Describe a factor that you think might explain the difference in the price per acre for these two purchases.