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Elementary Algebra Review
Exercises
Reduce each rational expression to lowest terms
Perform the indicated operations
Solve each of the following equations.
11. The speed of the current in a river is 5mph. If a boat
can travel 198 miles with the current in the same
time it could travel 138 miles against the current, what is the speed of the
boat in still water?
12. An inlet pipe can fill a tank in 10 minutes. A drain can empty the tank in
12 minutes. If the tank is
empty and both the pipe and the drain are open, how long will it take before the
tank overflows?
13. Connie can type 600 words in 5 minutes less than it takes Katie to type 600
words. If Connie can
type at a rate of 20 words per minute faster than Katie types, find Connie’s
typing rate.
14. The ratio of the squares to two consecutive even integers is 9/16. Find
these integers.
ROOTS AND RADICALS
Simplifying Radical Expressions
Radical expressions are algebraic expressions which contain a radical. A radical
expression is simplified
when the radicand (expression under the radical) does not contain any powers
greater than or equal to the
index, there are no radicals in the denominator and all fractions are in the
lowest terms.
Example:  
Write factors as perfect squares if possible  
Remove the squares 
Adding and Subtracting Radical Expressions
Adding and Subtracting can be done on terms with like radical portions. When the
terms have like racial
parts, add or subtract in the same manner as when adding or subtracting like
terms.
Example:  
Write as product of squares  
Remove squares  
Combine like terms 
Multiplying and Dividing Radical Expressions
Radical expressions are multiplied in the same way as other expressions.
Coefficients are multiplied
together and stay coefficients; radicands are multiplied together and remain
radicands .The result is then
simplified.
Example:  
Multiply  
Simplify  
Example:  
FOIL  
Simplify  
Combine like terms  
To divide one radical expression by another, write the division as a fraction and simplify.
Example:  
Write each division separately  
Write as one radical  
Simplify  
To simplify a fraction that has a binomial radical
expression in the denominator, multiply the numerator
and denominator by the conjugate form of the denominator and simplify.
Example:  
Multiply by the conjugate  
Solving Equations with Radical Expressions
Equations that contain radicals can be solved by using the equality property of
powers , which states that if
two real numbers are equal, then any natural power of these numbers is also
equal. To use this property,
first isolate the radical on one side of the equation. Next raise both sides of
the equation to the same
power and solve as usual. Always check all answers to eliminate “imposter roots”
that may be introduced
by the squaring process.
Example:  
Isolate the radical  
Square both sides  
FOIL  
Subtract X^{2} from both sides  
Combine like terms  
Divide by 7  
Example:  
The root is true 
Exercises
Solve each of the following:
9. An object is dropped from an airplane. Find the
distance the object has fallen when the speed reaches
400 feet per second. Use the equation ,
where V is the speed of the object and d is the distance.
10. How fare would a submarine periscope have to be above water to locate a ship
4.2 miles away? The
equation of the distance in miles that the lookout can see is
, where the height in feet above the
surface of the water and distance.
11. Find the width of a rectangle that has a diagonal of 10 feet if the length
is 2 feet longer than the width.
12. When trying to put out a fire, fire fighters are interested in the flow rate
of the water available at the
site of the fire. The flow rate is determined by the equation
where Q is flow rate in
gallons per minute, d is diameter of the hose nozzle in inches and P is static
pressure in pounds per inch.
What is the minimum static pressure to get at least 594 gal /min from a 2in fire
hose?
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