To solve this equation, we need to get rid of the exponents on both sides of the equation. We can start by simplifying the expressions inside the parentheses using the properties of exponents:
³√3^(x+1) = (⁴√9^x-2)^(x+1) Cube root of 3^(x+1) = Fourth root of 9^x-2^(x+1)
Now, let's simplify the exponents inside the parentheses:
3^(x+1) = (3^2)^x-2 (x+1) 3^(x+1) = 9^(x-2) (x+1)
Now, let's equate the expressions inside the parentheses:
3^(x+1) = 9^(x-2) * (x+1)
Since both sides are in terms of the same base, we can equate the exponents:
x+1 = (x-2)*(x+1)
Now let's solve for x:
x+1 = x^2 -2x + x -2 x+1 = x^2 -2 0 = x^2 - x - 3
Now, we have a quadratic equation that we can solve using the quadratic formula:
x = (-(-1) ± √((-1)^2 - 41(-3)))/(2*1) x = (1 ± √(1+12))/2 x = (1 ± √13)/2
Therefore, the solutions to the equation are: x = (1 + √13)/2 or x = (1 - √13)/2
To solve this equation, we need to get rid of the exponents on both sides of the equation. We can start by simplifying the expressions inside the parentheses using the properties of exponents:
³√3^(x+1) = (⁴√9^x-2)^(x+1)
Cube root of 3^(x+1) = Fourth root of 9^x-2^(x+1)
Now, let's simplify the exponents inside the parentheses:
3^(x+1) = (3^2)^x-2 (x+1)
3^(x+1) = 9^(x-2) (x+1)
Now, let's equate the expressions inside the parentheses:
3^(x+1) = 9^(x-2) * (x+1)
Since both sides are in terms of the same base, we can equate the exponents:
x+1 = (x-2)*(x+1)
Now let's solve for x:
x+1 = x^2 -2x + x -2
x+1 = x^2 -2
0 = x^2 - x - 3
Now, we have a quadratic equation that we can solve using the quadratic formula:
x = (-(-1) ± √((-1)^2 - 41(-3)))/(2*1)
x = (1 ± √(1+12))/2
x = (1 ± √13)/2
Therefore, the solutions to the equation are:
x = (1 + √13)/2
or
x = (1 - √13)/2