Solve:
3^x^2 : 3^x = 9 : 1
3^x^2 : 3^x = 9 : 1
3^(x^2) / 3^x = 9
3^x = 3^2
x = 2
check
3^4 : 9
= 81 : 9
= 9:1
To solve the equation 3^(x^2) / 3^x = 9/1, we can use the rules of exponents.
First, let's simplify the left side of the equation using the quotient rule of exponents:
3^(x^2 - x) = 9/1
Next, let's simplify the right side of the equation:
9/1 = 9
So now we have:
3^(x^2 - x) = 9
To solve for x, we need to apply the logarithm function to both sides of the equation.
Taking the logarithm (base 3) of both sides:
log3 (3^(x^2 - x)) = log3(9)
Using the power rule of logarithms, we can bring the exponent down as a coefficient:
(x^2 - x) log3(3) = log3(9)
Since log3(3) is equal to 1, the equation becomes:
x^2 - x = log3(9)
Now, we can simplify the right side by evaluating log3(9):
x^2 - x = log3(3^2)
Using the power rule of logarithms, we have:
x^2 - x = 2 log3(3)
Since log3(3) is equal to 1, the equation becomes:
x^2 - x = 2
Rearranging the equation into a quadratic form:
x^2 - x - 2 = 0
Now, we can solve this quadratic equation by factoring, completing the square, or using the quadratic formula. Let's use factoring:
(x - 2)(x + 1) = 0
Setting each factor equal to zero:
x - 2 = 0 or x + 1 = 0
Solving for x:
x = 2 or x = -1
So, the solutions to the equation are x = 2 or x = -1.