Is the equation (4√8)^−24⋅(√8)^16=8^−48 true or false? Select the response that correctly answers the question and provides the appropriate justification.(1 point)
Responses
False, because the Negative Exponent Rule should be applied.
True, but the equation can be simplified further.
False, because the exponents should be added rather than multiplied.
True. When
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.