SOLUTION: I have no done math in years. I am seeking out help for a friend. What is the rule to an equation where the x has an exponent? For example, x^3/4=6 with 3/4 being the exponent?

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Question 1145610: I have no done math in years. I am seeking out help for a friend. What is the rule to an equation where the x has an exponent? For example, x^3/4=6 with 3/4 being the exponent?
Found 3 solutions by Theo, Edwin McCravy, AnlytcPhil:
Answer by Theo(13342)   (Show Source): You can put this solution on YOUR website!
x^(3/4) = 6 can be solved as follows:

raise both sides of the equation to the power of (4/3) to get:
x^(3/4)^(4/3) = 6^(4/3)
this is equivalent to:
x^(3/4 * 4/3) = 6^(4/3)
simplify to get x= 6^(4/3) = 10.90272356
confirm by replacing x with that to get:
10.90272356^(3/4) = 6
result is 6 = 6, confirming the solution is correct.

x^(3/4) is the same as the fourth foot of x^3 and is also the same as the fourth root of x raised to the third power.

this looks like (x^3)^(1/4) or (x^(1/4))^3

the basic rule for exponent arithmatic is:

(x^a)^b = x^(a*b)
if a is 3 and b is (1/4), you get (x^3)^(1/4) = x^(3*1/4) = x^(3/4)
if a is 1/4 and b is 3, you get (x^(1/4)^3 = x^((1/4)*3) = x^(3/4)



Answer by Edwin McCravy(20060)   (Show Source): You can put this solution on YOUR website!

Answer by AnlytcPhil(1807)   (Show Source): You can put this solution on YOUR website!
CORRECTED VERSION OF MY OTHER SOLUTION:

The first solution above is a decimal approximation using a calculator, 
and I'm sure your teacher would count that wrong.  Here is what your
teacher wants, which does not use a calculator:



Raise both sides to the 4 power to cause the exponent
to be a whole number after we multiply:



Multiply the exponents on the left to remove the parentheses
and make the exponent a whole number:



Take cube roots of both sides:

 

On the left side, taking the cube root of a cube takes
away both the cube and the cube root!

On the right side, we break the 64 up as 6361
so we can take the cube root of part of the right side:



Since the cube root of a product is the product of the cube roots,
we take cube roots of both factors:



On the right side, we use the fact again that taking the cube root of 
a cube takes away both the cube and the cube root!:



Edwin
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