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

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

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

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

Answer 1

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

To begin with, rewrite the terms in the left side of the equation using exponents:

1. \(^4√8\) can be expressed as \(8^{1/4}\).
2. \(\sqrt{8}\) can be expressed as \(8^{1/2}\).

Now, substituting these into the left side:

\[ (^4√8)^{-24} = (8^{1/4})^{-24} = 8^{-24 \cdot \frac{1}{4}} = 8^{-6} \]

\[ (\sqrt{8})^{16} = (8^{1/2})^{16} = 8^{16 \cdot \frac{1}{2}} = 8^{8} \]

Now combine these two results:

\[ 8^{-6} \cdot 8^{8} = 8^{-6 + 8} = 8^{2} \]

Now, examine the right side of the original equation:

The right side is:

\[ 8^{-48} \]

Thus we have:

\[ 8^{2} \neq 8^{-48} \]

Therefore the original equation is false.

The correct response is:

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