Question

pablo simplified the expression \\((2^{-4})^{-2}\\) as shown.

\\(2^{-4})^{-2} = 2^{-4 \cdot -2} = 2^{-8}\\

which statement explains pablos error?

• he should have added the exponents instead of multiplying them.
• he should have divided the exponents instead of multiplying them.
• he should have found 8 as the exponent, not -8.
• he should have written \\(2^{-8}\\) as \\(\frac{1}{2^8}\\).

Explanation:

Identify the given expression and Pablo's work

$$ (2^{-4})^{-2} = 2^{-4 \cdot -2} = 2^{-8} $$

Apply the power of a power rule

$$ (a^m)^n = a^{m \cdot n} $$

$$ (2^{-4})^{-2} = 2^{(-4) \cdot (-2)} $$

Calculate the product of the exponents

$$ (-4) \cdot (-2) = 8 $$

$$ (2^{-4})^{-2} = 2^8 $$

Pablo wrote the exponent as \(-8\) instead of \(8\).

Answer:

• He should have added the exponents instead of multiplying them.
• He should have divided the exponents instead of multiplying them.
• He should have found 8 as the exponent, not -8. (Correct answer)
• He should have written \(2^{-8}\) as \(\frac{1}{2^8}\).