Is the equation (4√8)^−24⋅(√8)^16=8^−48

Question

true or false? Select the response that correctly answers the question and provides the appropriate justification.(1 point)
Responses

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

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, but the equation can be simplified further.

Answer 1

To determine if the equation \((4\sqrt{8})^{-24} \cdot (\sqrt{8})^{16} = 8^{-48}\) is true or false, let's simplify both sides.

Firstly, we can rewrite the terms using exponent rules.

1. Rewriting the expressions:

   • Note that \(\sqrt{8} = 8^{1/2}\).
   • Therefore, \((\sqrt{8})^{16} = (8^{1/2})^{16} = 8^{16 \cdot 1/2} = 8^8\).

2. Calculating \((4\sqrt{8})^{-24}\):

   • We can write \(4\) as \(8^{2/3}\) (since \(4 = 2^2\) and \(2 = 8^{1/3}\)).
   • Thus, \(4\sqrt{8} = 8^{2/3} \cdot 8^{1/2} = 8^{2/3 + 1/2} = 8^{4/6 + 3/6} = 8^{7/6}\).
   • Therefore, \((4\sqrt{8})^{-24} = (8^{7/6})^{-24} = 8^{-7 \cdot 24/6} = 8^{-28}\).

3. Combining the two terms:

   • Now we have: \[ (4\sqrt{8})^{-24} \cdot (\sqrt{8})^{16} = 8^{-28} \cdot 8^{8} = 8^{-28 + 8} = 8^{-20}. \]

4. Comparing to the right side:

   • The right side is \(8^{-48}\).

Thus, we find that the left-hand side simplifies to \(8^{-20}\) and the right-hand side is \(8^{-48}\).

Since \(-20 \neq -48\), the equation is false.

Therefore, the correct response is: False, because the exponents should be added rather than multiplied.