very challenging exponentical equations!! ... note these are not to be done using logarithms?

Question

1. 27^x / 9^(2x-1) = 3^(x+4) answer= -1

2. 8^1/4 * (1/4)^x/2 = 16^3/4 answer is-9/4

i tried both of these and well I am unsure where to begin as i tried getting the bases the same and reducing as well following all the exponent rules but can't geet these answers Please Help me!

Answer 1

(1)
27^x = (3^3)^x = 3^3x
9^(2x-1) = (3^2)^(2x-1) = 3^(4x-2)

3^(3x) / 3^(4x-2) = 3^(3x - 4x + 2) = 3^(-x + 2)
3^(-x + 2) = 3^(x+4)
-x + 2 = x + 4
2x = -2
x = -1

(2)
8^1/4 = (2^3)^1/4 = 2^(3/4)
1/4^(x/2) = (2^-2)^(x/2) = 2^-x
16^3/4 = (2^4)^(3/4) = 2^3

2^(3/4) * 2^-x = 2^3
3/4 - x = 3
x = 3/4 - 3 = 3/4 - 12/4 = -9/4

Hope this helps! :)

Answer 2

Since you've attempted these before, you probably know the laws of indices that I will be using. No need to repeat them.

1. 27^x / 9^(2x-1) = 3^(x+4)

The first thing to do is reduce them all to the same base, which is 3
27^x = (3^3)^x = 3^3x
9^(2x-1) = (3^2)^(2x-1) = 3^(4x-2)

Put these back into the equation
3^3x / 3^(4x-2) = 3^(x+4)

Now you can do two things: you can either subtract the exponents on the left side, or cross multiply the denominator on the left and then add the exponents on the right side. Let's subtract the exponents on the left side. We wind up with
3^(3x-(4x-2)) = 3^(x+4)
3^(3x-4x+2) = 3^(x+4)
Now the bases are the same, ignore them.
3x-4x+2 = x+4
-x+2 = x+4
Collect like terms
-x-x = 4-2
-2x = 2
x = -1

2. 8^1/4 * (1/4)^x/2 = 16^3/4

Again, reduce to the same base, which is 2.
8^1/4 = (2^3)^1/4 = 2^(3/4)
(1/4)^x/2 = (2^-2)^x/2 = 2^(-x)
16^3/4 = (2^4)^3/4 = 2^3

Put all this back into the original equation to get
2^(3/4) * 2^(-x) = 2^3

Add exponents on the left side
2^(3/4 + (-x)) = 2^3
2^(3/4 - x) = 2^3
Ignore the bases now that they are the same
3/4 - x = 3
3/4 - 3 = x
3/4 - 12/4 = x
x = -9/4

Answer 3

You're right, the first step is to write everything using a common base.

1. 27^x/9^(2x-1) = 3^(x+4)
3^(3x)/3^(2*(2x-1)) = 3^(x+4)
3^(3x-4x+2) = 3^(x+4)
=>
-x+2 = x+4
x=-1

2. 8^(1/4) * (1/4)^(x/2) = 16^(3/4)
2^(3/4) * 2^(-2*x/2) = 2^(4*3/4)
2^(3/4-x) = 2^3
x = 3/4-3
x = -9/4

Answer 4

Question 1
3^(3x) = 3^(x + 4) 3^(4x - 2)
3^(3x) = 3^(5x + 2)
3x = 5x + 2
2x = - 2
x = - 1

Question 2
8^(1/4) (1/4)^(x/2) = 16^(3/4)
8^(1/4) (1/4)^(x/2) = 8
(1/4)^(x/2) = 8^(3/4)
(1 / 2²)^(x/2) = (2 ³)^(3/4)
(2^(-2))^(x/2) = 2 ^ (9/4)
2^(- x) = 2^(9/4)
x = - 9/4