Let's analyze the steps Jamie applied to evaluate the expression:
Step 1:
Expression: \(\frac{[(2^3)(2)]^4}{2^{10}}\)
- Transformation: Jamie rewrote it as \(\frac{(2^4)^4}{2^{10}}\).
Rule Applied: This step involves the Product of Powers Rule, which states that when you multiply two expressions with the same base, you add their exponents. In the original expression, \( (2^3)(2) \) simplifies to \( 2^{3+1} = 2^4 \).
Step 2:
Expression: \(\frac{(2^4)^4}{2^{10}}\)
- Transformation: Jamie simplified \((2^4)^4\) to \(2^{16}\).
Rule Applied: This step uses the Power of a Power Rule, which states that when you raise a power to another power, you multiply the exponents: \( (a^m)^n = a^{m \cdot n} \).
Step 3:
Expression: \(\frac{2^{16}}{2^{10}}\)
- Transformation: Jamie simplified the fraction to \(2^{16 - 10}\).
Rule Applied: This step applies the Quotient of Powers Rule, which states that when you divide two expressions with the same base, you subtract the exponents: \(\frac{a^m}{a^n} = a^{m-n}\).
Summary of Rules Applied:
- Step 1: Product of Powers Rule
- Step 2: Power of a Power Rule
- Step 3: Quotient of Powers Rule
Final Answer: Jamie made a computational error in Step 3. The correct computation would yield \(2^6\), which equals \(64\), confirming that Step 4 is correct.