SOLUTION: Hi, would you help me solve this problem, please? Directions: Solve the equation by making the appropriate substitution. (meaning "u" for "x"). {{{ x^(7/2)-6x^(7/4)+9=0 }}}

Algebra ->  Polynomials-and-rational-expressions -> SOLUTION: Hi, would you help me solve this problem, please? Directions: Solve the equation by making the appropriate substitution. (meaning "u" for "x"). {{{ x^(7/2)-6x^(7/4)+9=0 }}}       Log On


   

Question 906528: Hi, would you help me solve this problem, please?
Directions: Solve the equation by making the appropriate substitution. (meaning "u" for "x").
+x%5E%287%2F2%29-6x%5E%287%2F4%29%2B9=0+
I let +u=x+ and +u%5E2=x%5E%287%2F2%29+, but am unsure of how to take it from there.
**All of the numerators of the exponents are 7, just in case that got cut off in the formula plotting system.


Answer by Edwin McCravy(20056) About Me  (Show Source):
You can put this solution on YOUR website!
There is a way to make the plotting system handle fraction exponents,
Place the expression in a 2x1 matrix with the blank character "" as the
upper element and the fraction exponent as the lower element.

Don't let u=x. Let matrix%282%2C1%2C%22%22%2Cu=x%5E%287%2F4%29%29 and matrix%282%2C1%2C%22%22%2Cu%5E2=x%5E%287%2F2%29%29+. You had that right. 

+matrix%282%2C1%2C%22%22%2Cx%5E%287%2F2%29-6x%5E%287%2F4%29%2B9=0%29+

becomes

u%5E2-6u%2B9=0

Then factor:

%28u-3%29%28u-3%29=0

And there's just one solution, u=3

Then matrix%282%2C1%2C%22%22%2Cu=x%5E%287%2F4%29%29 becomes

matrix%282%2C1%2C%22%22%2C3=x%5E%287%2F4%29%29

Then we raise both sides to the 'reciprocal'th power, 
the 4%2F7ths power


matrix%282%2C1%2C%22%22%2C3%5E%284%2F7%29=%28x%5E%287%2F4%29%29%5E%284%2F7%29%29

When you multiply those exponents on the right you get


matrix%282%2C1%2C%22%22%2C3%5E%284%2F7%29=x%5E1%29  

matrix%282%2C1%2C%22%22%2C3%5E%284%2F7%29=x%29

The answer is 'the seventh root of three to the fourth power'.
 
root%287%2C3%5E4%29%22%22=%22%22root%287%2C81%29or approximately 1.873444005

Edwin