To determine if the equation (4√8)^−24⋅(√8)^16=8^−48 is true or false, let's simplify it step by step:
Step 1: Simplify the expression on the left side of the equation.
(4√8)^−24⋅(√8)^16 = (4⋅√8)^−24⋅(√8)^16
Step 2: Simplify the radicals.
(4⋅√8)^−24⋅(√8)^16 = (4⋅2√2)^−24⋅2^16 = (8√2)^−24⋅2^16
Step 3: Apply the exponent to each factor inside the parentheses.
(8√2)^−24⋅2^16 = 8^−24(√2)^−24⋅2^16 = 8^−24⋅(√2)^−24⋅2^16
Step 4: Simplify the radical.
8^−24⋅(√2)^−24⋅2^16 = 8^−24⋅(2^−24)⋅2^16 = 8^−24⋅2^(−24+16)
Step 5: Simplify the exponent on 2.
8^−24⋅2^(−24+16) = 8^−24⋅2^−8
Step 6: Apply the exponent rule for multiplication.
8^−24⋅2^−8 = (8⋅2)^−24⋅2^−8 = 16^−24⋅2^−8
Step 7: Simplify the base of 16^−24.
16^−24⋅2^−8 = (2^4)^−24⋅2^−8 = 2^(4⋅−24)⋅2^−8 = 2^−96⋅2^−8
Step 8: Apply the exponent rule for multiplication.
2^−96⋅2^−8 = 2^(−96−8) = 2^−104
Now, let's compare this with the expression on the right side of the equation, which is 8^−48.
The equation (4√8)^−24⋅(√8)^16=8^−48 is False because 2^−104 is not equal to 8^−48.