18^(4-4y)=54(sqrt(2))^(3y-2)
step plz help me
18^(4-4y)=54(√2)^(3y-2)
take log of both sides
(4-4y)log18 = lof54 + (1/2)(3y-2)log2
4log18 - 4ylog18 = log54 + (3y/2)log2 - log2
4log18 - log54 + log2 = y( (3/2)log2 + 4log18)
3.58972... = (5.4726...) y
y = .655941
Sure, I'll try to help you step by step.
First, let's simplify the equation as much as possible.
We have: 18^(4-4y) = 54(sqrt(2))^(3y-2)
Now, we can simplify the right side of the equation.
54(sqrt(2))^(3y-2) can be rewritten as 54 * (2^(1/2))^(3y-2).
Using the property (a^b)^c = a^(b*c), we can simplify it further.
54 * (2^(1/2))^(3y-2) is the same as 54 * 2^((1/2)*(3y-2)).
Now, let's simplify the left side of the equation.
18^(4-4y) can be rewritten as (2*3^2)^(4-4y).
Using the property (a*b)^c = a^c * b^c, we can simplify it further.
(2*3^2)^(4-4y) is the same as 2^(4-4y) * (3^2)^(4-4y).
Now, the equation becomes: 2^(4-4y) * (3^2)^(4-4y) = 54 * 2^((1/2)*(3y-2)).
Since the bases on both sides are the same (2), we can set the exponents equal to each other:
4 - 4y = (1/2)*(3y-2).
Now, we have a simple equation to solve for the value of y.
Hope this helps! Let me know if you need further assistance or if you'd like a joke to lighten the mood!
Sure, I can help you. Let's break down the equation step-by-step.
Step 1: Simplify the right side of the equation.
The square root of 2 times the cube root of 2 is equal to the 6th root of 2 raised to the power of 6 divided by 2, which simplifies to 2^(6/2) = 2^3 = 8.
So the right side of the equation simplifies to 54 * 8^(3y-2).
Step 2: Rewrite the equation with the simplified right side.
The equation becomes: 18^(4-4y) = 54 * 8^(3y-2).
Step 3: Express both sides of the equation as powers of the base 2.
18 can be written as 2^3 and 54 can be written as 2 * 27.
So our equation becomes: (2^3)^(4-4y) = 2 * 27 * 8^(3y-2).
Step 4: Simplify the left side of the equation.
To simplify the left side, we need to apply the exponent rule for raising a power to a power. The rule states that (a^m)^n = a^(m * n).
Using this rule, we can simplify the left side of the equation to 2^(3 * (4-4y)) = 2^(12-12y).
Step 5: Simplify the right side of the equation.
2 * 27 * 8^(3y-2) can be further simplified as 2 * 27 * (2^3)^(3y-2).
Using the same exponent rule as before, (a^m)^n = a^(m * n), we can simplify the right side to 2 * 27 * 2^(3 * (3y-2)) = 54 * 2^(9y-6).
Step 6: Set the two sides of the equation equal to each other.
Now that both sides are simplified, we can set them equal to each other:
2^(12-12y) = 54 * 2^(9y-6).
This is the simplified form of the equation. If you have further steps or need assistance in solving for y, please let me know.
Sure! I can help you step by step.
The equation you provided is:
18^(4-4y) = 54(sqrt(2))^(3y-2)
To solve this equation, we can start by simplifying the expressions on both sides.
First, let's simplify the exponent on the left side of the equation. We can rewrite 18 as (3^2)^2, which gives us (3^2)^(2-4y). Using the exponent rule (a^m)^n = a^(m*n), we can simplify further to 3^(2*(2-4y)), which becomes 3^(4-8y).
Next, let's simplify the exponent on the right side of the equation. The square root of 2 can be rewritten as 2^(1/2), and we can rewrite (2^(1/2))^(3y-2) as 2^((1/2)*(3y-2)), which becomes 2^((3/2)y - 1).
Now the equation becomes:
3^(4-8y) = 54 * 2^((3/2)y - 1)
To simplify further, let's write 54 as 2 * 27 and rewrite 27 as 3^3:
3^(4-8y) = 2 * 3^3 * 2^((3/2)y - 1)
Next, we can combine the bases with the same exponent by using the rule a^m * a^n = a^(m + n):
3^(4-8y) = 2^(1 + (3/2)y - 1) * 3^3
Simplifying further:
3^(4-8y) = 2^(3/2)y * 3^3
Now we have exponential expressions with the same base. In order to solve for y, we can equate the exponents:
4-8y = 3/2y
Next, we can solve for y by isolating it on one side of the equation. Let's start by moving the y terms to one side and the constant terms to the other side:
4 = 3/2y + 8y
Combining like terms gives us:
4 = (3/2 + 8)y
Finally, divide both sides of the equation by the coefficient of y (3/2 + 8):
y = 4 / (3/2 + 8)
Now, you can simplify the final expression to find the solution for y.