Is the equation (\root(4)(8))^(-24)* (\sqrt(8))^(16) =8^(-48) true or false? select the response that correctly answers the question and provides the appropriate justification

Question

false, bc the exponents should be added rather than multiplied
True, when the base are multiplied, the exponents are also multiplied
false, bc the negative exponent rule should be applied
true but the equation can be simplified further

Answer 1

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

Let's break down each term:

1. Convert the expressions to powers of 8:

   • The fourth root can be expressed as an exponent: \(\sqrt[4]{8} = 8^{1/4}\). Thus, \((\sqrt[4]{8})^{-24} = (8^{1/4})^{-24} = 8^{-24/4} = 8^{-6}\).
   • The square root can also be expressed similarly: \(\sqrt{8} = 8^{1/2}\). Therefore, \((\sqrt{8})^{16} = (8^{1/2})^{16} = 8^{16/2} = 8^8\).

2. Combine both terms:

   • Now we have: \[ 8^{-6} \cdot 8^{8} = 8^{(-6 + 8)} = 8^2 \]

3. Compare with the right side:

   • The right side of the equation is \(8^{-48}\).

Now we can see: \[ 8^2 \neq 8^{-48} \]

Thus, the equation is false.

Correct Response:

false, bc the negative exponent rule should be applied.

This justification is based on the simplification process and comparison of the resulting exponents, demonstrating that both sides of the equation are not equal.