Solve all values of x by factoring
X^-6x=-6x+4
To solve the equation by factoring, we need to rewrite it in the form of a quadratic equation.
First, let's add 6x to both sides of the equation to get rid of the term -6x on the right side:
x^(-6x) + 6x = 4
Next, since we have a negative exponent, we can rewrite the equation using the reciprocal:
1/(x^6 * x^x) + 6x = 4
Now, let's subtract 4 from both sides of the equation:
1/(x^6 * x^x) + 6x - 4 = 0
To factor this equation, we need to find the common denominator of the first two terms, which is x^6 * x^x:
(x * x^x + 6x * x^6 - 4 * x^6 * x^x) / (x^6 * x^x) = 0
We can simplify this equation further by multiplying out the terms in the numerator:
(x^(x+1) + 6x^(x+6) - 4x^(6+x)) / (x^6 * x^x) = 0
Now, we need to find the values of x that make the numerator equal to zero:
x^(x+1) + 6x^(x+6) - 4x^(6+x) = 0
Since the exponent is the same for all three terms, we can factor out x^(x+1):
x^(x+1) * (1 + 6x^(5) - 4) = 0
Simplifying the expression further:
x^(x+1) * (1 + 6x^5 - 4) = 0
x^(x+1) * (6x^5 - 3) = 0
Now, we have two factors multiplied together to equal zero. According to the zero product property, a product is equal to zero if and only if at least one of the factors is equal to zero.
Therefore, we have two possible equations to solve:
1) x^(x+1) = 0
2) 6x^5 - 3 = 0
For equation 1, x^(x+1) = 0, there is no real solution. This is because any positive number raised to any power will always be greater than zero.
Now, let's solve equation 2, 6x^5 - 3 = 0:
6x^5 = 3
x^5 = 1/2
Taking the fifth root of both sides:
x = (1/2)^(1/5)
So the values of x that satisfy the equation are:
x = (1/2)^(1/5)