Question
Tasha used the pattern in the table to find the value of 4 Superscript negative 4.
Powers of 4
Value
4 squared
16
4 Superscript 1
4
4 Superscript 0
1
4 Superscript negative 1
One-fourth
4 Superscript negative 2
StartFraction 1 Over 16 EndFraction
She used these steps.
Step 1 Find a pattern in the table.
The pattern is to divide the previous value by 4 when the exponent decreases by 1.
Step 2 Find the value of 4 Superscript negative 3.
4 Superscript negative 3 = StartFraction 1 Over 16 EndFraction divided by 4 = StartFraction 1 Over 16 EndFraction times one-fourth = StartFraction 1 Over 64 EndFraction
Step 3 Find the value of 4 Superscript negative 4.
4 Superscript negative 4 = StartFraction 1 Over 64 EndFraction divided by 4 = StartFraction 1 Over 64 EndFraction times one-fourth = StartFraction 1 Over 256 EndFraction
Step 4 Rewrite the value for 4 Superscript negative 4.
StartFraction 1 Over 256 EndFraction = negative StartFraction 1 Over 4 Superscript negative 4 EndFraction
In which step did Tasha make the first error?
Step 1
Step 2
Step 3
Step 4
Powers of 4
Value
4 squared
16
4 Superscript 1
4
4 Superscript 0
1
4 Superscript negative 1
One-fourth
4 Superscript negative 2
StartFraction 1 Over 16 EndFraction
She used these steps.
Step 1 Find a pattern in the table.
The pattern is to divide the previous value by 4 when the exponent decreases by 1.
Step 2 Find the value of 4 Superscript negative 3.
4 Superscript negative 3 = StartFraction 1 Over 16 EndFraction divided by 4 = StartFraction 1 Over 16 EndFraction times one-fourth = StartFraction 1 Over 64 EndFraction
Step 3 Find the value of 4 Superscript negative 4.
4 Superscript negative 4 = StartFraction 1 Over 64 EndFraction divided by 4 = StartFraction 1 Over 64 EndFraction times one-fourth = StartFraction 1 Over 256 EndFraction
Step 4 Rewrite the value for 4 Superscript negative 4.
StartFraction 1 Over 256 EndFraction = negative StartFraction 1 Over 4 Superscript negative 4 EndFraction
In which step did Tasha make the first error?
Step 1
Step 2
Step 3
Step 4
Answers
GPT-4o mini
Let's analyze the steps provided to identify where Tasha made the first error.
1. **Step 1:** Tasha correctly identifies the pattern, which is to divide the previous value by 4 as the exponent decreases by 1.
2. **Step 2:** Tasha finds \(4^{-3}\) by dividing \(4^{-2} = \frac{1}{16}\) by 4. The calculation is:
\[
4^{-3} = \frac{1}{16} \div 4 = \frac{1}{16} \times \frac{1}{4} = \frac{1}{64}
\]
This step is correct.
3. **Step 3:** Tasha finds \(4^{-4}\) by dividing \(4^{-3} = \frac{1}{64}\) by 4:
\[
4^{-4} = \frac{1}{64} \div 4 = \frac{1}{64} \times \frac{1}{4} = \frac{1}{256}
\]
This step is also correct.
4. **Step 4:** Tasha rewrites \(4^{-4}\) as:
\[
\frac{1}{256} = -\frac{1}{4^{-4}}
\]
This is where the error occurs, because:
\[
-\frac{1}{4^{-4}} \neq \frac{1}{256}
\]
In fact, \(-\frac{1}{4^{-4}} = -\frac{1}{\frac{1}{256}} = -256\), which is incorrect.
Thus, the first error made by Tasha is in **Step 4**.
1. **Step 1:** Tasha correctly identifies the pattern, which is to divide the previous value by 4 as the exponent decreases by 1.
2. **Step 2:** Tasha finds \(4^{-3}\) by dividing \(4^{-2} = \frac{1}{16}\) by 4. The calculation is:
\[
4^{-3} = \frac{1}{16} \div 4 = \frac{1}{16} \times \frac{1}{4} = \frac{1}{64}
\]
This step is correct.
3. **Step 3:** Tasha finds \(4^{-4}\) by dividing \(4^{-3} = \frac{1}{64}\) by 4:
\[
4^{-4} = \frac{1}{64} \div 4 = \frac{1}{64} \times \frac{1}{4} = \frac{1}{256}
\]
This step is also correct.
4. **Step 4:** Tasha rewrites \(4^{-4}\) as:
\[
\frac{1}{256} = -\frac{1}{4^{-4}}
\]
This is where the error occurs, because:
\[
-\frac{1}{4^{-4}} \neq \frac{1}{256}
\]
In fact, \(-\frac{1}{4^{-4}} = -\frac{1}{\frac{1}{256}} = -256\), which is incorrect.
Thus, the first error made by Tasha is in **Step 4**.