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)

Question

Responses

True, but the equation can be simplified further.
True, but the equation can be simplified further.

False, because the exponents should be added rather than multiplied.
False, because the exponents should be added rather than multiplied.

False, because the Negative Exponent Rule should be applied.
False, because the Negative Exponent Rule should be applied.

True. When the bases are multiplied, the exponents are also multiplied.

Answer 1

Let's analyze the equation step by step to determine whether it is true or false:

The original equation is: \[ (4\sqrt{8}) - 24 \cdot (\sqrt{8})^{16} \]

First, simplify \(4\sqrt{8}\): \[ \sqrt{8} = 8^{1/2} = (2^3)^{1/2} = 2^{3/2} \] Thus, \[ 4\sqrt{8} = 4 \cdot 2^{3/2} = 2^2 \cdot 2^{3/2} = 2^{2 + 3/2} = 2^{(4/2) + (3/2)} = 2^{7/2} \]

Now simplify \((\sqrt{8})^{16}\): \[ (\sqrt{8})^{16} = (8^{1/2})^{16} = 8^{(1/2) \cdot 16} = 8^8 \] And since \(8 = 2^3\), we have: \[ 8^8 = (2^3)^8 = 2^{24} \]

Now, multiplying: \[ -24 \cdot (2^{24}) = -24 \cdot 2^{24} = -2^3 \cdot 2^{24} = -2^{27} \]

Now, combining the two yields: \[ 2^{7/2} - 2^{27} \]

Now let's examine \(8^{-48}\): \[ 8^{-48} = (2^3)^{-48} = 2^{-144} \]

Now we compare \(2^{7/2} - 2^{27}\) and \(2^{-144}\). Clearly, \(2^{7/2}\) and \(2^{27}\) are positive numbers, while \(2^{-144}\) is also a positive number, but they cannot be equal or equal to 0.

Therefore, the equation is false. The correct choice is:

False, because the exponents should be added rather than multiplied.