SOLUTION: solve with rational exponents then check solution...???... (x^2-x-4)^3/4-2=6

Question 345185: solve with rational exponents then check solution...???...

(x^2-x-4)^3/4-2=6
Answer by jsmallt9(3758) (Show Source): You can put this solution on YOUR website!

First we'll isolate the exponentiated expression by adding 2 to each side:

Now we want to get rid of that exponent. Actually evyerthing has an exponent. But we don't see the exponent if it is a 1. So when we say "get rid of an exponent", we really mean "turn the exponent into a 1". So how can we turn the 3/4 into a one? Any number (except 0) can be turned into a 1 by multiplying it by its reciprocal. So if we can figure out an operation that will allow us to multiply the exponent, 3/4, by its reciprocal, 4/3 then the exponent will "disappear". Fortunately we have a rule that says if you raise a power to a power then multiply the exponents. So all we need to do is raise both sides of the equation to the 4/3 power!

On the left the exponent turns into a 1 and disappears. On the right we need to simplify . From what we should know about fractional exponents: . Since 8 is a perfect cube I will use the second expression. (Don't worry, the first expression still works. But it is harder to simplify.) So now our equation is:

At this point we have a quadratic equation to solve. So we want one side to be zero. Subtracting 16 from each side will work:

Now we factor (or use the Quadratic Formula). This factors fairly easily:

By the Zero Product property we know that this (or any) product can be zero only if one (or more) of the factors is zero. So:
or
Solving these we get:
or

Now we'll check our answers, using the original equation:

Checking x = -4:

Check!
Checking x = 5:

Check!

So there are two solutions to the original equation: x = -4 and x = 5.
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