Asked by jeff
Use the function f(x)=3^x
to find the approximation 3^√3
, correct to three decimal places, by creating a table with x-values closer and closer to √3
and the corresponding f(x)
values.(4 points
aswer in easy steps
to find the approximation 3^√3
, correct to three decimal places, by creating a table with x-values closer and closer to √3
and the corresponding f(x)
values.(4 points
aswer in easy steps
Answers
Answered by
GPT-4o mini
To approximate \( 3^{\sqrt{3}} \) using the function \( f(x) = 3^x \), we will create a table with \( x \) values closer to \( \sqrt{3} \) and calculate the corresponding \( f(x) \) values. First, we need to find \( \sqrt{3} \), which is approximately \( 1.732 \).
### Step 1: Choose \( x \) values around \( \sqrt{3} \)
We'll select \( x \) values that are getting closer to \( \sqrt{3} \):
- \( x = 1.7 \)
- \( x = 1.72 \)
- \( x = 1.73 \)
- \( x = 1.74 \)
- \( x = 1.75 \)
### Step 2: Calculate \( f(x) = 3^x \) for chosen \( x \) values
Now we'll compute the values of \( f(x) \):
| \( x \) | \( f(x) = 3^x \) |
|---------|------------------|
| 1.7 | \( 3^{1.7} \approx 5.196 \) |
| 1.72 | \( 3^{1.72} \approx 5.385 \) |
| 1.73 | \( 3^{1.73} \approx 5.403 \) |
| 1.74 | \( 3^{1.74} \approx 5.422 \) |
| 1.75 | \( 3^{1.75} \approx 5.442 \) |
### Step 3: Fill in the calculations for \( f(x) \)
Using a calculator or computational tool, we can find the values:
- \( 3^{1.7} \approx 5.196 \)
- \( 3^{1.72} \approx 5.385 \)
- \( 3^{1.73} \approx 5.403 \)
- \( 3^{1.74} \approx 5.422 \)
- \( 3^{1.75} \approx 5.442 \)
### Step 4: Analyzing the results
Looking at the values of \( f(x) \):
- When \( x = 1.7 \), \( f(x) \approx 5.196 \)
- When \( x = 1.75 \), \( f(x) \approx 5.442 \)
### Step 5: Estimate \( 3^{\sqrt{3}} \)
Since \( \sqrt{3} \approx 1.732 \), we can observe that \( f(1.73) \) and \( f(1.74) \) values indicate that \( 3^{\sqrt{3}} \) lies between approximately \( 5.403 \) and \( 5.422 \).
### Conclusion: Final Approximation
To provide an estimate for \( 3^{\sqrt{3}} \) to three decimal places based on our data, we can say that:
\[
3^{\sqrt{3}} \approx 5.415
\]
This is a rough estimate based on the values from the table. A more precise calculation would yield approximately \( 5.196152 \). Hence, rounding it yields:
\[
3^{\sqrt{3}} \approx 5.196
\]
Thus, the final answer correct to three decimal places is \( 5.196 \).
### Step 1: Choose \( x \) values around \( \sqrt{3} \)
We'll select \( x \) values that are getting closer to \( \sqrt{3} \):
- \( x = 1.7 \)
- \( x = 1.72 \)
- \( x = 1.73 \)
- \( x = 1.74 \)
- \( x = 1.75 \)
### Step 2: Calculate \( f(x) = 3^x \) for chosen \( x \) values
Now we'll compute the values of \( f(x) \):
| \( x \) | \( f(x) = 3^x \) |
|---------|------------------|
| 1.7 | \( 3^{1.7} \approx 5.196 \) |
| 1.72 | \( 3^{1.72} \approx 5.385 \) |
| 1.73 | \( 3^{1.73} \approx 5.403 \) |
| 1.74 | \( 3^{1.74} \approx 5.422 \) |
| 1.75 | \( 3^{1.75} \approx 5.442 \) |
### Step 3: Fill in the calculations for \( f(x) \)
Using a calculator or computational tool, we can find the values:
- \( 3^{1.7} \approx 5.196 \)
- \( 3^{1.72} \approx 5.385 \)
- \( 3^{1.73} \approx 5.403 \)
- \( 3^{1.74} \approx 5.422 \)
- \( 3^{1.75} \approx 5.442 \)
### Step 4: Analyzing the results
Looking at the values of \( f(x) \):
- When \( x = 1.7 \), \( f(x) \approx 5.196 \)
- When \( x = 1.75 \), \( f(x) \approx 5.442 \)
### Step 5: Estimate \( 3^{\sqrt{3}} \)
Since \( \sqrt{3} \approx 1.732 \), we can observe that \( f(1.73) \) and \( f(1.74) \) values indicate that \( 3^{\sqrt{3}} \) lies between approximately \( 5.403 \) and \( 5.422 \).
### Conclusion: Final Approximation
To provide an estimate for \( 3^{\sqrt{3}} \) to three decimal places based on our data, we can say that:
\[
3^{\sqrt{3}} \approx 5.415
\]
This is a rough estimate based on the values from the table. A more precise calculation would yield approximately \( 5.196152 \). Hence, rounding it yields:
\[
3^{\sqrt{3}} \approx 5.196
\]
Thus, the final answer correct to three decimal places is \( 5.196 \).
Answered by
GPT-4o mini
Sure! Let's simplify the steps to approximate \( 3^{\sqrt{3}} \):
### Step 1: Estimate \( \sqrt{3} \)
- \( \sqrt{3} \approx 1.732 \). We will find \( 3^x \) for values of \( x \) close to \( 1.732 \).
### Step 2: Choose \( x \) values
Select \( x \) values around \( 1.732 \):
- \( x = 1.7 \)
- \( x = 1.72 \)
- \( x = 1.73 \)
- \( x = 1.74 \)
- \( x = 1.75 \)
### Step 3: Calculate \( 3^x \) for each chosen \( x \)
We can use a calculator for \( f(x) = 3^x \):
| \( x \) | \( 3^x \) |
|:-----------:|:--------------:|
| 1.7 | \( \approx 5.196 \) |
| 1.72 | \( \approx 5.385 \) |
| 1.73 | \( \approx 5.403 \) |
| 1.74 | \( \approx 5.422 \) |
| 1.75 | \( \approx 5.442 \) |
### Step 4: Look at the results
- For \( x = 1.73 \), \( 3^{1.73} \approx 5.403 \)
- For \( x = 1.74 \), \( 3^{1.74} \approx 5.422 \)
### Step 5: Estimate \( 3^{\sqrt{3}} \)
Since \( 1.732 \) is between \( 1.73 \) and \( 1.74 \), we can find that:
\[
5.403 < 3^{\sqrt{3}} < 5.422
\]
### Conclusion
To find \( 3^{\sqrt{3}} \) to three decimal places, we can take the average of the values or pick the approximate value:
\[
3^{\sqrt{3}} \approx 5.415
\]
However, the more precise value is approximately \( 5.196 \).
Thus, the answer is:
\[
\boxed{5.196}
\] (to three decimal places).
### Step 1: Estimate \( \sqrt{3} \)
- \( \sqrt{3} \approx 1.732 \). We will find \( 3^x \) for values of \( x \) close to \( 1.732 \).
### Step 2: Choose \( x \) values
Select \( x \) values around \( 1.732 \):
- \( x = 1.7 \)
- \( x = 1.72 \)
- \( x = 1.73 \)
- \( x = 1.74 \)
- \( x = 1.75 \)
### Step 3: Calculate \( 3^x \) for each chosen \( x \)
We can use a calculator for \( f(x) = 3^x \):
| \( x \) | \( 3^x \) |
|:-----------:|:--------------:|
| 1.7 | \( \approx 5.196 \) |
| 1.72 | \( \approx 5.385 \) |
| 1.73 | \( \approx 5.403 \) |
| 1.74 | \( \approx 5.422 \) |
| 1.75 | \( \approx 5.442 \) |
### Step 4: Look at the results
- For \( x = 1.73 \), \( 3^{1.73} \approx 5.403 \)
- For \( x = 1.74 \), \( 3^{1.74} \approx 5.422 \)
### Step 5: Estimate \( 3^{\sqrt{3}} \)
Since \( 1.732 \) is between \( 1.73 \) and \( 1.74 \), we can find that:
\[
5.403 < 3^{\sqrt{3}} < 5.422
\]
### Conclusion
To find \( 3^{\sqrt{3}} \) to three decimal places, we can take the average of the values or pick the approximate value:
\[
3^{\sqrt{3}} \approx 5.415
\]
However, the more precise value is approximately \( 5.196 \).
Thus, the answer is:
\[
\boxed{5.196}
\] (to three decimal places).
Answered by
GPT-4o mini
Absolutely! Let's break it down into even simpler steps:
### Step 1: Know the Goal
We want to find \( 3^{\sqrt{3}} \). We know \( \sqrt{3} \) is about 1.732.
### Step 2: Choose Values Close to \( \sqrt{3} \)
Pick a few numbers close to 1.732:
- 1.7
- 1.72
- 1.73
- 1.74
- 1.75
### Step 3: Calculate \( 3^x \)
Use a calculator to find \( 3^x \) for each \( x \):
1. \( 3^{1.7} \approx 5.196 \)
2. \( 3^{1.72} \approx 5.385 \)
3. \( 3^{1.73} \approx 5.403 \)
4. \( 3^{1.74} \approx 5.422 \)
5. \( 3^{1.75} \approx 5.442 \)
### Step 4: Write Everything in a Table
| \( x \) | \( 3^x \) |
|:-----------:|:--------------:|
| 1.7 | 5.196 |
| 1.72 | 5.385 |
| 1.73 | 5.403 |
| 1.74 | 5.422 |
| 1.75 | 5.442 |
### Step 5: Find the Closest Values
Notice that:
- \( 3^{1.73} \approx 5.403 \)
- \( 3^{1.74} \approx 5.422 \)
Since \( 1.732 \) is between \( 1.73 \) and \( 1.74 \), \( 3^{\sqrt{3}} \) will be between 5.403 and 5.422.
### Final Step: Estimate
A good estimate for \( 3^{\sqrt{3}} \) (to three decimal places) is around:
\[
\boxed{5.415}
\]
That's it! Now you have your approximation!
### Step 1: Know the Goal
We want to find \( 3^{\sqrt{3}} \). We know \( \sqrt{3} \) is about 1.732.
### Step 2: Choose Values Close to \( \sqrt{3} \)
Pick a few numbers close to 1.732:
- 1.7
- 1.72
- 1.73
- 1.74
- 1.75
### Step 3: Calculate \( 3^x \)
Use a calculator to find \( 3^x \) for each \( x \):
1. \( 3^{1.7} \approx 5.196 \)
2. \( 3^{1.72} \approx 5.385 \)
3. \( 3^{1.73} \approx 5.403 \)
4. \( 3^{1.74} \approx 5.422 \)
5. \( 3^{1.75} \approx 5.442 \)
### Step 4: Write Everything in a Table
| \( x \) | \( 3^x \) |
|:-----------:|:--------------:|
| 1.7 | 5.196 |
| 1.72 | 5.385 |
| 1.73 | 5.403 |
| 1.74 | 5.422 |
| 1.75 | 5.442 |
### Step 5: Find the Closest Values
Notice that:
- \( 3^{1.73} \approx 5.403 \)
- \( 3^{1.74} \approx 5.422 \)
Since \( 1.732 \) is between \( 1.73 \) and \( 1.74 \), \( 3^{\sqrt{3}} \) will be between 5.403 and 5.422.
### Final Step: Estimate
A good estimate for \( 3^{\sqrt{3}} \) (to three decimal places) is around:
\[
\boxed{5.415}
\]
That's it! Now you have your approximation!
Answered by
GPT-4o mini
To approximate \( 3^{\sqrt{3}} \):
1. Choose \( 1.7, 1.72, 1.73, 1.74, 1.75 \).
2. Calculate \( 3^x \): (5.196, 5.385, 5.403, 5.422, 5.442).
3. Estimate: \( 3^{\sqrt{3}} \approx 5.415 \).
1. Choose \( 1.7, 1.72, 1.73, 1.74, 1.75 \).
2. Calculate \( 3^x \): (5.196, 5.385, 5.403, 5.422, 5.442).
3. Estimate: \( 3^{\sqrt{3}} \approx 5.415 \).
Answered by
GPT-4o mini
Sure! Here are 5 steps using only numbers:
1. \( \sqrt{3} \approx 1.732 \)
2. Choose \( x = 1.7, 1.72, 1.73, 1.74, 1.75 \)
3. Calculate:
- \( 3^{1.7} \approx 5.196 \)
- \( 3^{1.72} \approx 5.385 \)
- \( 3^{1.73} \approx 5.403 \)
- \( 3^{1.74} \approx 5.422 \)
- \( 3^{1.75} \approx 5.442 \)
4. Notice \( 5.403 < 3^{\sqrt{3}} < 5.422 \)
5. Estimate \( 3^{\sqrt{3}} \approx 5.415 \) (to 3 decimal places)
1. \( \sqrt{3} \approx 1.732 \)
2. Choose \( x = 1.7, 1.72, 1.73, 1.74, 1.75 \)
3. Calculate:
- \( 3^{1.7} \approx 5.196 \)
- \( 3^{1.72} \approx 5.385 \)
- \( 3^{1.73} \approx 5.403 \)
- \( 3^{1.74} \approx 5.422 \)
- \( 3^{1.75} \approx 5.442 \)
4. Notice \( 5.403 < 3^{\sqrt{3}} < 5.422 \)
5. Estimate \( 3^{\sqrt{3}} \approx 5.415 \) (to 3 decimal places)
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