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

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

False, because the Negative Exponent Rule should be applied.

True, but the equation can be simplified further.

Answer 1

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

1. Simplifying \( 4\sqrt{8} \): \[ \sqrt{8} = \sqrt{4 \times 2} = 2\sqrt{2} \] Therefore: \[ 4\sqrt{8} = 4 \cdot 2\sqrt{2} = 8\sqrt{2} \] Thus: \[ (4\sqrt{8})^{-24} = (8\sqrt{2})^{-24} = 8^{-24} \cdot (\sqrt{2})^{-24} \]

2. Expressing \( \sqrt{2} \) in terms of base 2: \[ \sqrt{2} = 2^{1/2} \implies (\sqrt{2})^{-24} = (2^{1/2})^{-24} = 2^{-12} \]

3. Combining the terms: Therefore: \[ (4\sqrt{8})^{-24} = 8^{-24} \cdot 2^{-12} \]

Expressing \( 8 \) as \( 2^3 \): \[ 8^{-24} = (2^3)^{-24} = 2^{-72} \] Thus: \[ (4\sqrt{8})^{-24} = 2^{-72} \cdot 2^{-12} = 2^{-84} \]

4. Now simplifying \( (\sqrt{8})^{16} \): \[ \sqrt{8} = 2^{3/2} \implies (\sqrt{8})^{16} = (2^{3/2})^{16} = 2^{3 \cdot 8} = 2^{24} \]

5. Combining both sides of the original equation: Now we have: \[ (4\sqrt{8})^{-24} \cdot (\sqrt{8})^{16} = 2^{-84} \cdot 2^{24} = 2^{-60} \]

6. Simplifying the right-hand side: The right side \( 8^{-48} \): \[ 8^{-48} = (2^3)^{-48} = 2^{-144} \]

Now comparing both sides: Left side: \( 2^{-60} \) Right side: \( 2^{-144} \)

Since \( -60 \neq -144 \), the equation is:

False.

The correct justification is that the bases with exponents did not equate after simplification.

Answer 2

pick one of the answers

Answer 3

The answer is:

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