Calculating the Value of n in the Given Equation

To find the value of n in the equation [√2(16^1/4 7776^1/5)^4]^n = 4√2, we will break down the equation step by step and simplify it. Let's proceed as follows:

Step 1: Simplify the expression within the brackets.
- Inside the brackets, we have two numbers, 16^1/4 and 7776^1/5.
- Simplifying each term separately, we get:
- 16^1/4 = √(16) = √(2^4) = 2
- 7776^1/5 = ∛(7776) = ∛(6^5) = 6
- Substituting these values back into the equation, we have:
[√2(2 6)^4]^n = 4√2

Step 2: Simplify the expression within the parentheses.
- Within the parentheses, we have (2 6)^4.
- This can be simplified as follows:
- (2 6)^4 = 12^4 = (2^2 3)^4 = 2^8 3^4 = 256 81
- Substituting this value back into the equation, we have:
[√2(256 81)]^n = 4√2

Step 3: Simplify the expression within the square brackets.
- Within the square brackets, we have √2(256 81).
- This can be simplified as follows:
- √2(256 81) = √2(256) √2(81) = √2(16^2) √2(9^2) = √2(4^2 2^2) √2(3^2) = 4√2 2√2 3
- Substituting this value back into the equation, we have:
(4√2 2√2 3)^n = 4√2

Step 4: Combine like terms.
- On the left side of the equation, we have (4√2 2√2 3)^n.
- Combining the terms inside the parentheses, we get:
- (4√2 2√2 3) = (4√2) + (2√2) + 3 = 6√2 + 2√2 + 3 = 8√2 + 3
- Substituting this value back into the equation, we have:
(8√2 + 3)^n = 4√2

Step 5: Set up an equation.
- Setting up the equation, we have:
(8√2 + 3)^n = 4√2

Step 6: Solve for n.
- To solve for n, we take the n-th root of both sides of the equation:
n-th root of [(8√2 + 3)^n] = n-th root of (4√2)
- Simplifying the right side