13 answers
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:
\[ 1.53 \times 10^4 \text{ miles} \]
Sure! Let's go through the process of converting 15,300 into scientific notation step by step.
We have the number 15,300.
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: \[ 1.5300 \]
Next, we count how many places we moved the decimal point. In this case, we moved it 1 place left.
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: \[ 15,300 = 1.5300 \times 10^4 \]
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 \(1.5300\).
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.
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:
\[ 3.2 \times 10^4 = 0.032 \times 10^6 \]
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:
\[ 1.982 \times 10^6 \]
Thus, the result of adding \(3.2 \times 10^4\) and \(1.95 \times 10^6\) is:
\[ 1.982 \times 10^6 \]
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.
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:
\[ 2.56 \times 10^{-2} = 256000 \times 10^{-8} \]
Now we can rewrite both numbers using \(10^{-8}\):
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:
\[ 8.03 - 0.0000256 = 8.0299744 \]
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:
\[ \boxed{8.0299744 \times 10^{-8}} \]
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.
The two numbers are:
To make them have the same power of 10, we should rewrite \(2.56 \times 10^{-2}\) in terms of \(10^{-8}\).
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:
\[ 2.56 \times 10^{-2} = 256000 \times 10^{-8} \]
Now we have:
Now we subtract:
\[ 8.03 \times 10^{-8} - 256000 \times 10^{-8} = (8.03 - 256000) \times 10^{-8} \]
Calculating the subtraction:
\[ 8.03 - 256000 = -255991.97 \]
Now we write this in scientific notation:
\[ -255991.97 \times 10^{-8} = -2.5599197 \times 10^{-3} \text{ (since we move the decimal point 5 places left)} \]
So the result is approximately:
\[ \boxed{-2.56 \times 10^{-3}} \]
This presents a simplified view of the entire process.
Sure! Here’s a more concise way to subtract \(8.03 \times 10^{-8}\) and \(2.56 \times 10^{-2}\):
Convert \(2.56 \times 10^{-2}\) to \(10^{-8}\):
\[ 2.56 \times 10^{-2} = 2.56 \times 10^{-2} \times \frac{10^{6}}{10^{6}} = 2.56 \times 10^{4} \times 10^{-8} = 256000 \times 10^{-8} \]
Now subtract:
\[ 8.03 \times 10^{-8} - 256000 \times 10^{-8} = (8.03 - 256000) \times 10^{-8} \]
Calculating:
\[ 8.03 - 256000 = -255991.97 \]
Convert to scientific notation:
\[ -255991.97 \times 10^{-8} = -2.5599197 \times 10^{-3} \]
Thus, the result is:
\[ \boxed{-2.56 \times 10^{-3}} \]
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.
We need to convert \(2.56 \times 10^{-2}\) to the same power as \(8.03 \times 10^{-8}\).
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:
\[ 2.56 \times 10^{-2} = 25600 \times 10^{-8} \]
Now we can subtract:
\[ 8.03 \times 10^{-8} - 25600 \times 10^{-8} = (8.03 - 25600) \times 10^{-8} \]
Calculating:
\[ 8.03 - 25600 = -25591.97 \]
Now write the result in scientific notation:
\[ -25591.97 \times 10^{-8} = -2.559197 \times 10^{-4} \]
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!
