Chapter 7: Power, Roots, and Radicals
Chapter 7.1: Nth Roots and Rational Exponents Evaluating nth Roots:
Relating Indices and Powers √ √
Real nth Roots: Let be an integer greater than 1 and let be a real number. If is odd, then has one real nth root: If is even and , then has two real nth roots: If is even and , then has one nth root: If is even and , then has no real nth roots.
Example: Find the indicated nth root(s) of . n = 5, a = -32 n = 4, a = 625
n = 3, a = 64 n = 3, a = -27
Rational Exponents: Let ⁄ be an nth root of a, and let m be a positive integer.
⁄ ⁄ ( ) ( √ ) ⁄ ⁄ ( ⁄ ) ( √ )
Example: Evaluating Expressions with Rational Exponents
⁄ ⁄
⁄ ⁄
Example Use a calculator to approximate:
√ √
Solving Equations:
Example: Solve each equation.
( ) ( )
Example: The rate at which an initial deposit will grow to a balance in years with interest
⁄ compounded n times a year is given by the formula [( ) ]. Find if ,
, years, and .
Example: A basketball has a volume of about 455.6 cubic inches. The formula for the volume of a basketball is . Find the radius of the basketball.
Chapter 7.2: Properties of Rational Exponents Properties of Rational Exponents and Radicals
Example: Simplify.
⁄ ⁄ ( ) ⁄
⁄ ⁄ ( ) ⁄
Example: Simplify. ⁄ ⁄ ⁄ ( ) ⁄ ( ) ⁄
( ⁄ ⁄ )
⁄
Example: Simplify.
√ √ √
√
√ √ √
√
Simplest Form:
Checklist for writing radicals in simplest form:
Example: Write in simplest form.
√ √
Combining Like Radicals:
Example: Perform the indicated operation.
( ⁄ ) ( ⁄ ) √ √
( ⁄ ) ( ⁄ ) √ √
Properties with Variables:
Example: Simplify the expression. Assume all variables are positive.
√ √
( ) ⁄
⁄
⁄
Example: Write the expression in simplest form. Assume all variables are positive.
√ √
Example: Simplify the expression. Assume all variables are positive.
√ √
( ) ⁄ ⁄
Example: Perform the indicated operations. Assume all variables are positive. ⁄ ⁄ √ √ √ √
Example: Simplify the expression. Assume all variables are positive. ⁄ ⁄ √ √ √ √ √
Example: The weight W in tons of a whale as a function of length L in feet can be approximated by the model ⁄ . Approximate the weight of a humpback whale with length 49.17 feet.
Example: A pilot whale is about ⁄ the length of a blue whale. Is its weight also ⁄ the weight of the blue whale?
Chapter 7.3: Power Functions and Function Operations Concept Summary: Four basic operations with polynomial functions:
Power Functions:
Example: Let ( ) ⁄ and ( ) ⁄ . Find the following: The sum
The difference
The domains
Example: Let ( ) ⁄ and ( ) ⁄ . Find the following: The product
The quotient
The domains Composition of Two Functions:
Example: Let ( ) and ( ) . Find the following and their domains: ( ( ))
( ( ))
( ( ))
Example: Let ( ) and ( ) . Find the following and their domains: ( ( ))
( ( ))
( ( ))
Example: You do an experiment on bacteria and find that the growth rate G of the bacteria can be modeled by ( ) , and that the death rate D is ( ) , where t is the time in hours. Find an expression for the number N of bacteria living at time t.
Example: A computer catalog offers computers at a savings of 15% off the retail price. At the end of the month, it offers an additional 10% off its own price. Use composition of function to find the total percent discount.
What would be the sale price of a $899 computer?
Chapter 7.4: Inverse Functions Finding Inverses of Linear Functions:
Example: Find the inverse of .
Inverse Functions:
Example: Verify that ( ) and ( ) are inverses.
Example: Find the inverse of . Verify that this function and your answer are inverses.
Example: A model for a salary is , where S is the total salary (in dollars) for one week and h is the number of hours worked. Find the inverse function for the model.
If a person’s salary is $533, how many hours does the person work?
Example: A model for a telephone bill is , where T is the total bill, and m is the number of minutes used. Find the inverse model.
If the total bill is $54.15, how many minutes were used?
Example: Find the inverse of the function ( )
Example: Find the inverse of the function ( ) ,
Horizontal Line Test:
Example: Consider the function ( ) . Determine whether the inverse of f is a function and then find the inverse.
Example: The volume of a sphere is given by , where V is the volume and r is the radius.
Write the inverse function that gives the radius as a function of the volume. Then determine the radius of a volleyball given that its volume is about 293 cubic inches.
Example: The volume of a cylinder with height 10 feet is given by , where V is the volume, and r is the radius. Write the inverse model that gives the radius as a function of the volume. Then determine the radius of a cylinder given that its volume is 1050 cubic feet.
Chapter 7.5: Graphing Square root and Cube Root Functions Graphing Radical Functions:
Example: Describe how to obtain the graph of √ from the graph of √ .
Example: Graph √
Example: Describe how to obtain the graph of √ from the graph of √ . Then graph.
Example: Graph √ .
Example: Graph √ .
Example: State the domain and range of the function in all of the examples.
Example: A model for the period of a simple pendulum as measured in time units is given by
√ , where is the time in seconds, is the length of the pendulum in feet, and is 32 ft/sec2.
Use a graphing calculator to graph the model. Then use the graph to estimate the period of a pendulum that is 3 feet long.
Example: The length of a whale can be modeled by √ , where is the length in feet, and is the weight in tons. Graph the model, then use the graph to find the weight of a what that is 60 ft long.
Chapter 7.6: Solving Radical Equations Solving a Radical Equation:
Example: Solve √ √
√ √ √
⁄ √ √
⁄
Extraneous Solutions:
Example: Solve √ √
Example: The strings of guitars and pianos are under tension. The speed v of a wave on the string depends on the force (tension) F on the string and the mass M per unit length L according to the formula √ . A wave travels through a string with a mass of 0.2 kilograms at a speed of 9 meters per second. It is stretched by a force of 19.6 Newtons. Find the length of the string.
Example: Solve √ for x if R = 19.8.
Chapter 7.7: Statistics and Statistical Graphs Measures of Central Tendency:
Example: The number of games won by teams in the Eastern Conference for the 1997-1998 regular season of the NHL is shown on the chart. Find the mean, median, and mode for the data set. Eastern Conference
36, 39, 40, 34, 48, 33, 25, 30, 37, 17, 42, 40, 24
Measures of Dispersion:
Example: Find the range and standard deviation for the number of wins in the NHL Eastern Conference.
Box-and-Whisker Plots:
Example: Draw a box-and-whisker plot for the NHL Easter Conference. Eastern Conference
36, 39, 40, 34, 48, 33, 25, 30, 37, 17, 42, 40, 24
Histogram:
Example: Draw a histogram for the NHL Eastern Conference.