7a) Find
the distance between the points 
and 
.
ANSWER
The distance between two points on the plane
and
is given by the formula
where
corresponds to and
corresponds to
so
You must always simplify radical expressions,
so
The final answer is
7b) Find
the distance between the points 
and 
ANSWER
corresponds to
and
corresponds to ,
so
To simplify the radical, you can factor
out the 
in 
so
The final answer is
Remember that
can not be further simplified because
That is,
Recall
the Rules for Working with Radicals!
8a) Write the
complex fraction 
in
the form 
,
where 
ANSWER
Multiply the numerator and the denominator
by the complex conjugate of the denominator.
Recall that 
,
so
But this is not yet in the form
, you must break up 
into two fractions
using the common denominator
.
The final answer is
where
and
8b)Write the complex
fraction 
in the form ,
where
ANSWER
Collect like terms in the numerator and
recall that
Writing this in in the standard form
,
that is, separating the real and the imaginary parts,
the final answer is
9) Solve
the equation
ANSWER
This is an example of an equation reducible
to a quadratic equation.
Using the Power to a Power Rule for Exponents,
the equation can re-written as
If you substitute 
, the equation becomes
In this form the equation is more obviously
a quadratic equation in the variable 
.
The left side of the equation can be factored
as
Now you use the Zero Product Principle that
if a product is equal to zero,
then some or all of the factors in the product
must be equal to zero.
In algebra notation,
implies that
or 
.
So
or
Solving each equation for
,
you get
or 
But
now you remember that you are solving the equation
for 
and that ,
so
is 
and
is
To get alone,
you take the 3rd power of both sides of the equations
and
Now you use the Power to a Power Rule for
Exponents to get alone
So the solutions of the equation are
or
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Review
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© edmond 2001