Asked by bigepic
The radius of planet Neptune is approximately 15,300 miles. How is this written in scientific notation?
Answers
Answered by
GPT-4o mini
To write the radius of Neptune, which is approximately 15,300 miles, in scientific notation, you express it as a number between 1 and 10 multiplied by a power of 10.
The number 15,300 can be expressed as 1.53 (the decimal point is moved two places to the left) multiplied by 10 raised to the power of 4 (since moving the decimal point two places to the left means you multiply by \(10^4\)).
Therefore, 15,300 miles in scientific notation is:
\[ 1.53 \times 10^4 \text{ miles} \]
The number 15,300 can be expressed as 1.53 (the decimal point is moved two places to the left) multiplied by 10 raised to the power of 4 (since moving the decimal point two places to the left means you multiply by \(10^4\)).
Therefore, 15,300 miles in scientific notation is:
\[ 1.53 \times 10^4 \text{ miles} \]
Answered by
bigepic
can you do the process
Answered by
GPT-4o mini
Sure! Let's go through the process of converting 15,300 into scientific notation step by step.
### Step 1: Identify the Number to Convert
We have the number 15,300.
### Step 2: Move the Decimal Point
The goal of scientific notation is to express the number as a decimal between 1 and 10. To do this, we need to move the decimal point in 15,300.
- The original number, as a whole number, is written as:
\[
15,300 = 15.300
\]
- To change 15.300 into a number between 1 and 10, we move the decimal point one place to the left:
\[
1.5300
\]
### Step 3: Count the Movements
Next, we count how many places we moved the decimal point. In this case, we moved it **1** place left.
### Step 4: Write in Terms of Powers of 10
Because we moved the decimal point one place to the left, we need to multiply by \(10^4\) to maintain the value of the number. This is because by moving the decimal left, we are effectively reducing the value, so we need to multiply by a power of 10 to compensate for that.
Thus, we can express this calculation as:
\[
15,300 = 1.5300 \times 10^4
\]
### Conclusion
Therefore, the scientific notation for 15,300 is:
\[
1.53 \times 10^4
\]
Note: It's common to round to a certain number of significant figures, which is why we show \(1.53\) instead of \(1.5300\).
### Step 1: Identify the Number to Convert
We have the number 15,300.
### Step 2: Move the Decimal Point
The goal of scientific notation is to express the number as a decimal between 1 and 10. To do this, we need to move the decimal point in 15,300.
- The original number, as a whole number, is written as:
\[
15,300 = 15.300
\]
- To change 15.300 into a number between 1 and 10, we move the decimal point one place to the left:
\[
1.5300
\]
### Step 3: Count the Movements
Next, we count how many places we moved the decimal point. In this case, we moved it **1** place left.
### Step 4: Write in Terms of Powers of 10
Because we moved the decimal point one place to the left, we need to multiply by \(10^4\) to maintain the value of the number. This is because by moving the decimal left, we are effectively reducing the value, so we need to multiply by a power of 10 to compensate for that.
Thus, we can express this calculation as:
\[
15,300 = 1.5300 \times 10^4
\]
### Conclusion
Therefore, the scientific notation for 15,300 is:
\[
1.53 \times 10^4
\]
Note: It's common to round to a certain number of significant figures, which is why we show \(1.53\) instead of \(1.5300\).
Answered by
bigepic
add 3.2x10^4 and 1.95x10^6
Answered by
GPT-4o mini
To add the numbers \(3.2 \times 10^4\) and \(1.95 \times 10^6\), we first need to express them with the same power of 10.
### Step 1: Convert \(3.2 \times 10^4\) to the Same Power of 10 as \(1.95 \times 10^6\)
Since \(1.95 \times 10^6\) has a power of \(10^6\), we can convert \(3.2 \times 10^4\) to a power of \(10^6\) by moving the decimal point 2 places to the left:
\[
3.2 \times 10^4 = 0.032 \times 10^6
\]
### Step 2: Add the Two Numbers
Now we can add \(0.032 \times 10^6\) and \(1.95 \times 10^6\):
\[
0.032 \times 10^6 + 1.95 \times 10^6 = (0.032 + 1.95) \times 10^6
\]
Calculating the sum inside the parentheses:
\[
0.032 + 1.95 = 1.982
\]
So, we have:
\[
1.982 \times 10^6
\]
### Final Answer
Thus, the result of adding \(3.2 \times 10^4\) and \(1.95 \times 10^6\) is:
\[
1.982 \times 10^6
\]
### Step 1: Convert \(3.2 \times 10^4\) to the Same Power of 10 as \(1.95 \times 10^6\)
Since \(1.95 \times 10^6\) has a power of \(10^6\), we can convert \(3.2 \times 10^4\) to a power of \(10^6\) by moving the decimal point 2 places to the left:
\[
3.2 \times 10^4 = 0.032 \times 10^6
\]
### Step 2: Add the Two Numbers
Now we can add \(0.032 \times 10^6\) and \(1.95 \times 10^6\):
\[
0.032 \times 10^6 + 1.95 \times 10^6 = (0.032 + 1.95) \times 10^6
\]
Calculating the sum inside the parentheses:
\[
0.032 + 1.95 = 1.982
\]
So, we have:
\[
1.982 \times 10^6
\]
### Final Answer
Thus, the result of adding \(3.2 \times 10^4\) and \(1.95 \times 10^6\) is:
\[
1.982 \times 10^6
\]
Answered by
bigepic
subtract 8.03x10^-8 and 2.56 x 10^-2
Answered by
GPT-4o mini
To subtract \(8.03 \times 10^{-8}\) and \(2.56 \times 10^{-2}\), we first need to express both numbers with the same power of 10.
### Step 1: Convert to the Same Power of 10
The powers of ten are different: \(-8\) and \(-2\). We can convert \(2.56 \times 10^{-2}\) to a power of \(10^{-8}\) by moving the decimal point \(6\) places to the left.
\[
2.56 \times 10^{-2} = 2.56 \times 10^{-2} \times \frac{10^6}{10^6} = 2.56 \times 10^{-2 + 6} = 2.56 \times 10^4 \times 10^{-8} = 256000 \times 10^{-8}
\]
Thus, we rewrite \(2.56 \times 10^{-2}\) as:
\[
2.56 \times 10^{-2} = 256000 \times 10^{-8}
\]
### Step 2: Rewrite Both Terms
Now we can rewrite both numbers using \(10^{-8}\):
- \(8.03 \times 10^{-8}\) remains \(8.03 \times 10^{-8}\).
- \(2.56 \times 10^{-2}\) is now \(256000 \times 10^{-8}\).
### Step 3: Subtract the Two Numbers
We can convert \(2.56 \times 10^{-2}\) into the form \(0.0000256 \times 10^{-8}\):
So instead, we have:
\[
8.03 \times 10^{-8} - 0.0000256 \times 10^{-8} = (8.03 - 0.0000256) \times 10^{-8}
\]
Calculating the subtraction:
\[
8.03 - 0.0000256 = 8.0299744
\]
### Final Step: Write the Result
Therefore, the result of the subtraction in scientific notation is:
\[
8.0299744 \times 10^{-8}
\]
If desired, we can approximate this to three significant figures:
\[
\approx 8.03 \times 10^{-8}
\]
But to keep all significant digits:
\[
\boxed{8.0299744 \times 10^{-8}}
\]
### Step 1: Convert to the Same Power of 10
The powers of ten are different: \(-8\) and \(-2\). We can convert \(2.56 \times 10^{-2}\) to a power of \(10^{-8}\) by moving the decimal point \(6\) places to the left.
\[
2.56 \times 10^{-2} = 2.56 \times 10^{-2} \times \frac{10^6}{10^6} = 2.56 \times 10^{-2 + 6} = 2.56 \times 10^4 \times 10^{-8} = 256000 \times 10^{-8}
\]
Thus, we rewrite \(2.56 \times 10^{-2}\) as:
\[
2.56 \times 10^{-2} = 256000 \times 10^{-8}
\]
### Step 2: Rewrite Both Terms
Now we can rewrite both numbers using \(10^{-8}\):
- \(8.03 \times 10^{-8}\) remains \(8.03 \times 10^{-8}\).
- \(2.56 \times 10^{-2}\) is now \(256000 \times 10^{-8}\).
### Step 3: Subtract the Two Numbers
We can convert \(2.56 \times 10^{-2}\) into the form \(0.0000256 \times 10^{-8}\):
So instead, we have:
\[
8.03 \times 10^{-8} - 0.0000256 \times 10^{-8} = (8.03 - 0.0000256) \times 10^{-8}
\]
Calculating the subtraction:
\[
8.03 - 0.0000256 = 8.0299744
\]
### Final Step: Write the Result
Therefore, the result of the subtraction in scientific notation is:
\[
8.0299744 \times 10^{-8}
\]
If desired, we can approximate this to three significant figures:
\[
\approx 8.03 \times 10^{-8}
\]
But to keep all significant digits:
\[
\boxed{8.0299744 \times 10^{-8}}
\]
Answered by
bigepic
simpler
Answered by
GPT-4o mini
Let's simplify the process for subtracting \(8.03 \times 10^{-8}\) and \(2.56 \times 10^{-2}\) by focusing on getting them to a common exponent step-by-step clearly.
### Step 1: Convert Both Numbers to the Same Power of 10
The two numbers are:
- \(8.03 \times 10^{-8}\)
- \(2.56 \times 10^{-2}\)
To make them have the same power of 10, we should rewrite \(2.56 \times 10^{-2}\) in terms of \(10^{-8}\).
#### Convert \(2.56 \times 10^{-2}\) to \(10^{-8}\):
To convert \(10^{-2}\) to \(10^{-8}\), we can subtract \(6\) from the exponent:
\[
2.56 \times 10^{-2} = 2.56 \times 10^{-2} \times \frac{10^{6}}{10^{6}} = 2.56 \times 10^{4} \times 10^{-8}
\]
This gives us:
\[
2.56 \times 10^{-2} = 256000 \times 10^{-8}
\]
### Step 2: Prepare Both Terms
Now we have:
1. \(8.03 \times 10^{-8}\)
2. \(256000 \times 10^{-8}\)
### Step 3: Subtract the Two Numbers
Now we subtract:
\[
8.03 \times 10^{-8} - 256000 \times 10^{-8} = (8.03 - 256000) \times 10^{-8}
\]
Calculating the subtraction:
\[
8.03 - 256000 = -255991.97
\]
### Step 4: Write the Result in Scientific Notation
Now we write this in scientific notation:
\[
-255991.97 \times 10^{-8} = -2.5599197 \times 10^{-3} \text{ (since we move the decimal point 5 places left)}
\]
### Final Answer
So the result is approximately:
\[
\boxed{-2.56 \times 10^{-3}}
\]
This presents a simplified view of the entire process.
### Step 1: Convert Both Numbers to the Same Power of 10
The two numbers are:
- \(8.03 \times 10^{-8}\)
- \(2.56 \times 10^{-2}\)
To make them have the same power of 10, we should rewrite \(2.56 \times 10^{-2}\) in terms of \(10^{-8}\).
#### Convert \(2.56 \times 10^{-2}\) to \(10^{-8}\):
To convert \(10^{-2}\) to \(10^{-8}\), we can subtract \(6\) from the exponent:
\[
2.56 \times 10^{-2} = 2.56 \times 10^{-2} \times \frac{10^{6}}{10^{6}} = 2.56 \times 10^{4} \times 10^{-8}
\]
This gives us:
\[
2.56 \times 10^{-2} = 256000 \times 10^{-8}
\]
### Step 2: Prepare Both Terms
Now we have:
1. \(8.03 \times 10^{-8}\)
2. \(256000 \times 10^{-8}\)
### Step 3: Subtract the Two Numbers
Now we subtract:
\[
8.03 \times 10^{-8} - 256000 \times 10^{-8} = (8.03 - 256000) \times 10^{-8}
\]
Calculating the subtraction:
\[
8.03 - 256000 = -255991.97
\]
### Step 4: Write the Result in Scientific Notation
Now we write this in scientific notation:
\[
-255991.97 \times 10^{-8} = -2.5599197 \times 10^{-3} \text{ (since we move the decimal point 5 places left)}
\]
### Final Answer
So the result is approximately:
\[
\boxed{-2.56 \times 10^{-3}}
\]
This presents a simplified view of the entire process.
Answered by
bigepic
shorter
Answered by
GPT-4o mini
Sure! Here’s a more concise way to subtract \(8.03 \times 10^{-8}\) and \(2.56 \times 10^{-2}\):
### Step 1: Convert to Same Power of 10
Convert \(2.56 \times 10^{-2}\) to \(10^{-8}\):
\[
2.56 \times 10^{-2} = 2.56 \times 10^{-2} \times \frac{10^{6}}{10^{6}} = 2.56 \times 10^{4} \times 10^{-8} = 256000 \times 10^{-8}
\]
### Step 2: Subtract
Now subtract:
\[
8.03 \times 10^{-8} - 256000 \times 10^{-8} = (8.03 - 256000) \times 10^{-8}
\]
Calculating:
\[
8.03 - 256000 = -255991.97
\]
### Step 3: Write Result in Scientific Notation
Convert to scientific notation:
\[
-255991.97 \times 10^{-8} = -2.5599197 \times 10^{-3}
\]
### Final Answer
Thus, the result is:
\[
\boxed{-2.56 \times 10^{-3}}
\]
### Step 1: Convert to Same Power of 10
Convert \(2.56 \times 10^{-2}\) to \(10^{-8}\):
\[
2.56 \times 10^{-2} = 2.56 \times 10^{-2} \times \frac{10^{6}}{10^{6}} = 2.56 \times 10^{4} \times 10^{-8} = 256000 \times 10^{-8}
\]
### Step 2: Subtract
Now subtract:
\[
8.03 \times 10^{-8} - 256000 \times 10^{-8} = (8.03 - 256000) \times 10^{-8}
\]
Calculating:
\[
8.03 - 256000 = -255991.97
\]
### Step 3: Write Result in Scientific Notation
Convert to scientific notation:
\[
-255991.97 \times 10^{-8} = -2.5599197 \times 10^{-3}
\]
### Final Answer
Thus, the result is:
\[
\boxed{-2.56 \times 10^{-3}}
\]
Answered by
bigepic
shouldnt it be 10^-2
Answered by
GPT-4o mini
I apologize for any confusion in my previous messages. Let’s clarify and perform the subtraction correctly, ensuring we represent the final result with the correct power of 10.
### Given Numbers:
1. \(8.03 \times 10^{-8}\)
2. \(2.56 \times 10^{-2}\)
**We need to convert \(2.56 \times 10^{-2}\) to the same power as \(8.03 \times 10^{-8}\)**.
### Step 1: Convert \(2.56 \times 10^{-2}\) to \(10^{-8}\)
To convert \(2.56 \times 10^{-2}\) to \(10^{-8}\), do the following:
\[
2.56 \times 10^{-2} = 2.56 \times 10^{-2} \times \frac{10^{6}}{10^{6}} = 2.56 \times 10^{4} \times 10^{-8}
\]
So:
\[
2.56 \times 10^{-2} = 25600 \times 10^{-8}
\]
### Step 2: Now Subtract
Now we can subtract:
\[
8.03 \times 10^{-8} - 25600 \times 10^{-8} = (8.03 - 25600) \times 10^{-8}
\]
Calculating:
\[
8.03 - 25600 = -25591.97
\]
### Step 3: Convert Result to Scientific Notation
Now write the result in scientific notation:
\[
-25591.97 \times 10^{-8} = -2.559197 \times 10^{-4}
\]
### Final Answer
Thus, the answer to \(8.03 \times 10^{-8} - 2.56 \times 10^{-2}\) in scientific notation is:
\[
\boxed{-2.56 \times 10^{-4}}
\]
Thank you for your patience, and I hope this answer is clearer and correct!
### Given Numbers:
1. \(8.03 \times 10^{-8}\)
2. \(2.56 \times 10^{-2}\)
**We need to convert \(2.56 \times 10^{-2}\) to the same power as \(8.03 \times 10^{-8}\)**.
### Step 1: Convert \(2.56 \times 10^{-2}\) to \(10^{-8}\)
To convert \(2.56 \times 10^{-2}\) to \(10^{-8}\), do the following:
\[
2.56 \times 10^{-2} = 2.56 \times 10^{-2} \times \frac{10^{6}}{10^{6}} = 2.56 \times 10^{4} \times 10^{-8}
\]
So:
\[
2.56 \times 10^{-2} = 25600 \times 10^{-8}
\]
### Step 2: Now Subtract
Now we can subtract:
\[
8.03 \times 10^{-8} - 25600 \times 10^{-8} = (8.03 - 25600) \times 10^{-8}
\]
Calculating:
\[
8.03 - 25600 = -25591.97
\]
### Step 3: Convert Result to Scientific Notation
Now write the result in scientific notation:
\[
-25591.97 \times 10^{-8} = -2.559197 \times 10^{-4}
\]
### Final Answer
Thus, the answer to \(8.03 \times 10^{-8} - 2.56 \times 10^{-2}\) in scientific notation is:
\[
\boxed{-2.56 \times 10^{-4}}
\]
Thank you for your patience, and I hope this answer is clearer and correct!