Question
Jamie evaluated this expression.
StartFraction left-bracket (2 cubed) (2) right-bracket Superscript 4 Baseline Over 2 Superscript 10 Baseline Endfraction
step 1: StartFraction left-bracket (2 Superscript 4 Baseline) Superscript 4 Baseline Over 2 Superscript 10 EndFraction
step 2: StartFraction 2 Superscript 16 Over 2 Superscript 10 EndFraction
step 3: 26
step 4: 64
Analyze the steps Jamie applied to evaluate the expression.
Which rule of exponents was applied in each step?
Step 1:
Step 2:
Step 3:
StartFraction left-bracket (2 cubed) (2) right-bracket Superscript 4 Baseline Over 2 Superscript 10 Baseline Endfraction
step 1: StartFraction left-bracket (2 Superscript 4 Baseline) Superscript 4 Baseline Over 2 Superscript 10 EndFraction
step 2: StartFraction 2 Superscript 16 Over 2 Superscript 10 EndFraction
step 3: 26
step 4: 64
Analyze the steps Jamie applied to evaluate the expression.
Which rule of exponents was applied in each step?
Step 1:
Step 2:
Step 3:
Answers
GPT-4o mini
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.
### 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.