Question
what happens to the exponent when you are multiplying scientific notation
Answers
Answered by
GPT-4o mini
When you multiply numbers in scientific notation, you multiply their coefficients (the numbers in front) and add their exponents.
The general form for multiplication in scientific notation is:
\[
(a \times 10^m) \times (b \times 10^n) = (a \times b) \times 10^{(m+n)}
\]
For example, if you multiply \(3 \times 10^4\) by \(2 \times 10^3\):
1. Multiply the coefficients: \(3 \times 2 = 6\)
2. Add the exponents: \(4 + 3 = 7\)
So,
\[
(3 \times 10^4) \times (2 \times 10^3) = 6 \times 10^7
\]
Thus, when multiplying in scientific notation, you keep the base (which is 10) constant and simply add the exponents.
The general form for multiplication in scientific notation is:
\[
(a \times 10^m) \times (b \times 10^n) = (a \times b) \times 10^{(m+n)}
\]
For example, if you multiply \(3 \times 10^4\) by \(2 \times 10^3\):
1. Multiply the coefficients: \(3 \times 2 = 6\)
2. Add the exponents: \(4 + 3 = 7\)
So,
\[
(3 \times 10^4) \times (2 \times 10^3) = 6 \times 10^7
\]
Thus, when multiplying in scientific notation, you keep the base (which is 10) constant and simply add the exponents.
Answer
what if one is negative
Answered by
GPT-4o mini
If one of the numbers in scientific notation has a negative exponent, you still multiply the coefficients and add the exponents, just as you would with positive exponents. The key is to handle the exponent correctly based on its sign.
Here's how it works:
### Example 1: One Number with a Positive Exponent and One with a Negative Exponent
Suppose you want to multiply \(3 \times 10^4\) by \(2 \times 10^{-3}\):
1. **Multiply the coefficients**: \(3 \times 2 = 6\)
2. **Add the exponents**: \(4 + (-3) = 4 - 3 = 1\)
So the result is:
\[
(3 \times 10^4) \times (2 \times 10^{-3}) = 6 \times 10^{(4 + (-3))} = 6 \times 10^1
\]
This can also be written as:
\[
6 \times 10^1 = 60
\]
### Example 2: Both Numbers with Negative Exponents
Now consider multiplying \(5 \times 10^{-2}\) by \(4 \times 10^{-3}\):
1. **Multiply the coefficients**: \(5 \times 4 = 20\)
2. **Add the exponents**: \(-2 + (-3) = -2 - 3 = -5\)
So the result is:
\[
(5 \times 10^{-2}) \times (4 \times 10^{-3}) = 20 \times 10^{-5}
\]
This can also be simplified to:
\[
2.0 \times 10^{-4}
\]
### Summary
- When multiplying numbers in scientific notation, you multiply the coefficients and **add** the exponents, regardless of whether they are positive or negative.
- Be mindful of the signs when adding the exponents.
Here's how it works:
### Example 1: One Number with a Positive Exponent and One with a Negative Exponent
Suppose you want to multiply \(3 \times 10^4\) by \(2 \times 10^{-3}\):
1. **Multiply the coefficients**: \(3 \times 2 = 6\)
2. **Add the exponents**: \(4 + (-3) = 4 - 3 = 1\)
So the result is:
\[
(3 \times 10^4) \times (2 \times 10^{-3}) = 6 \times 10^{(4 + (-3))} = 6 \times 10^1
\]
This can also be written as:
\[
6 \times 10^1 = 60
\]
### Example 2: Both Numbers with Negative Exponents
Now consider multiplying \(5 \times 10^{-2}\) by \(4 \times 10^{-3}\):
1. **Multiply the coefficients**: \(5 \times 4 = 20\)
2. **Add the exponents**: \(-2 + (-3) = -2 - 3 = -5\)
So the result is:
\[
(5 \times 10^{-2}) \times (4 \times 10^{-3}) = 20 \times 10^{-5}
\]
This can also be simplified to:
\[
2.0 \times 10^{-4}
\]
### Summary
- When multiplying numbers in scientific notation, you multiply the coefficients and **add** the exponents, regardless of whether they are positive or negative.
- Be mindful of the signs when adding the exponents.
Answer
7.68
x 4.8
-------------
x 4.8
-------------
Answered by
GPT-4o mini
To multiply \(7.68\) by \(4.8\), you can follow these steps:
### Step 1: Perform the multiplication
You can do this by aligning the numbers and multiplying as you would with whole numbers, then considering the decimal places.
\[
\begin{array}{r}
7.68 \\
\times 4.8 \\
\hline
6144 \quad \text{(this is 7.68 * 8)} \\
+ 3840 \quad \text{(this is 7.68 * 4, shift one position to the left)} \\
\hline
36864 \\
\end{array}
\]
### Step 2: Adjust for the decimal places
Now, count the total number of decimal places in the original numbers:
- \(7.68\) has **2 decimal places**.
- \(4.8\) has **1 decimal place**.
So, the total number of decimal places is \(2 + 1 = 3\).
### Step 3: Place the decimal point in the result
Now, place the decimal point in \(36864\) such that there are 3 decimal places:
\[
36864 \to 36.864
\]
### Final Answer
Thus, \(7.68 \times 4.8 = 36.864\).
### Step 1: Perform the multiplication
You can do this by aligning the numbers and multiplying as you would with whole numbers, then considering the decimal places.
\[
\begin{array}{r}
7.68 \\
\times 4.8 \\
\hline
6144 \quad \text{(this is 7.68 * 8)} \\
+ 3840 \quad \text{(this is 7.68 * 4, shift one position to the left)} \\
\hline
36864 \\
\end{array}
\]
### Step 2: Adjust for the decimal places
Now, count the total number of decimal places in the original numbers:
- \(7.68\) has **2 decimal places**.
- \(4.8\) has **1 decimal place**.
So, the total number of decimal places is \(2 + 1 = 3\).
### Step 3: Place the decimal point in the result
Now, place the decimal point in \(36864\) such that there are 3 decimal places:
\[
36864 \to 36.864
\]
### Final Answer
Thus, \(7.68 \times 4.8 = 36.864\).
Answer
do long multiplication with decimals
Answered by
GPT-4o mini
Sure! Let's go through the long multiplication of \(7.68\) and \(4.8\) step by step while clearly handling the decimals.
### Step 1: Set up the numbers without the decimals
Since we are multiplying, we'll temporarily ignore the decimals and treat them as whole numbers:
- \(7.68\) can be treated as \(768\) (moving the decimal point two places to the right).
- \(4.8\) can be treated as \(48\) (moving the decimal point one place to the right).
### Step 2: Multiply as whole numbers
Now, we multiply \(768\) by \(48\) using long multiplication.
```
768
x 48
---------
6144 (this is 768 * 8)
+ 3072 (this is 768 * 4, shifted one position to the left)
-----------
36864
```
### Step 3: Count the decimal places
Now, we determine the total number of decimal places in the original numbers:
- \(7.68\) has **2 decimal places**.
- \(4.8\) has **1 decimal place**.
Adding these together gives:
\[
2 + 1 = 3 \text{ decimal places}
\]
### Step 4: Place the decimal in the result
Next, we will place the decimal point in \(36864\) to account for the 3 decimal places:
\[
36864 \to 36.864
\]
### Final Answer
Thus, the result of \(7.68 \times 4.8\) is:
\[
\boxed{36.864}
\]
### Step 1: Set up the numbers without the decimals
Since we are multiplying, we'll temporarily ignore the decimals and treat them as whole numbers:
- \(7.68\) can be treated as \(768\) (moving the decimal point two places to the right).
- \(4.8\) can be treated as \(48\) (moving the decimal point one place to the right).
### Step 2: Multiply as whole numbers
Now, we multiply \(768\) by \(48\) using long multiplication.
```
768
x 48
---------
6144 (this is 768 * 8)
+ 3072 (this is 768 * 4, shifted one position to the left)
-----------
36864
```
### Step 3: Count the decimal places
Now, we determine the total number of decimal places in the original numbers:
- \(7.68\) has **2 decimal places**.
- \(4.8\) has **1 decimal place**.
Adding these together gives:
\[
2 + 1 = 3 \text{ decimal places}
\]
### Step 4: Place the decimal in the result
Next, we will place the decimal point in \(36864\) to account for the 3 decimal places:
\[
36864 \to 36.864
\]
### Final Answer
Thus, the result of \(7.68 \times 4.8\) is:
\[
\boxed{36.864}
\]
Answer
DECIMALS
Answered by
GPT-4o mini
Sure! Let's go through **long multiplication with decimals** clearly, including every step. We'll multiply **7.68** by **4.8** while keeping track of the decimal placement.
### Step 1: Write it out
Align the numbers as follows, disregarding the decimals for the moment:
```
7.68
x 4.8
---------
```
### Step 2: Multiply as if there are no decimals
Starting from the right (the units place of the bottom number), we will multiply \(7.68\) by \(8\) and then by \(4\).
#### Multiply by 8:
Starting with \(8\) (from \(4.8\)):
```
7.68
x 8
---------
6144 (This is 768 * 8)
```
#### Multiply by 4:
Now, we multiply by \(4\). Since \(4.8\) is in the tenths place, we need to add one zero placeholder to the result:
```
7.68
x 4
---------
3072 (This is 768 * 4, but shifted one position to the left because it represents 40)
```
### Step 3: Add the partial results
Now we add the two results together:
```
6144
+ 30720 (Shifted to the left)
---------
36864
```
### Step 4: Count the decimal places
Now, let's determine the placement of the decimal in the final answer:
- **7.68** has **2 decimal places**.
- **4.8** has **1 decimal place**.
- Therefore, you need to place the decimal point 2 + 1 = **3 places** from the right in the final product.
### Step 5: Place the decimal point
The result before placing the decimal point is **36864**.
Placing the decimal point:
- Move the decimal point 3 places to the left:
\[
36864 \rightarrow 36.864
\]
### Final Answer
The result of multiplying \(7.68\) by \(4.8\) is:
\[
\boxed{36.864}
\]
This method thoroughly demonstrates how to perform long multiplication with decimals and how to ensure that the decimal point is correctly placed in the final answer. If you have any additional questions or another example you want to discuss, feel free to ask!
### Step 1: Write it out
Align the numbers as follows, disregarding the decimals for the moment:
```
7.68
x 4.8
---------
```
### Step 2: Multiply as if there are no decimals
Starting from the right (the units place of the bottom number), we will multiply \(7.68\) by \(8\) and then by \(4\).
#### Multiply by 8:
Starting with \(8\) (from \(4.8\)):
```
7.68
x 8
---------
6144 (This is 768 * 8)
```
#### Multiply by 4:
Now, we multiply by \(4\). Since \(4.8\) is in the tenths place, we need to add one zero placeholder to the result:
```
7.68
x 4
---------
3072 (This is 768 * 4, but shifted one position to the left because it represents 40)
```
### Step 3: Add the partial results
Now we add the two results together:
```
6144
+ 30720 (Shifted to the left)
---------
36864
```
### Step 4: Count the decimal places
Now, let's determine the placement of the decimal in the final answer:
- **7.68** has **2 decimal places**.
- **4.8** has **1 decimal place**.
- Therefore, you need to place the decimal point 2 + 1 = **3 places** from the right in the final product.
### Step 5: Place the decimal point
The result before placing the decimal point is **36864**.
Placing the decimal point:
- Move the decimal point 3 places to the left:
\[
36864 \rightarrow 36.864
\]
### Final Answer
The result of multiplying \(7.68\) by \(4.8\) is:
\[
\boxed{36.864}
\]
This method thoroughly demonstrates how to perform long multiplication with decimals and how to ensure that the decimal point is correctly placed in the final answer. If you have any additional questions or another example you want to discuss, feel free to ask!
Answer
where did the 0 at the end of 30720 come from
Answered by
GPT-4o mini
Great question! The zero at the end of \(30720\) comes from the fact that we are multiplying by \(4\) in the **tens** place when we multiply by \(4.8\). Let's clarify that step.
### Breakdown of the Multiplication Steps
When multiplying \(7.68\) by \(4.8\):
1. We first multiply \(7.68\) by the **8** in the **tenths** place of \(4.8\):
- This gives us:
\[
7.68 \times 8 = 61.44
\]
(Or when treated as whole numbers: \(768 \times 8 = 6144\))
2. Next, we multiply \(7.68\) by the **4** in the **whole number** place of \(4.8\):
- Since this \(4\) actually represents **40** (not just 4), we must account for that when we multiply.
- So, we calculate:
\[
7.68 \times 4 = 30.72
\]
When treated as whole numbers:
\[
768 \times 4 = 3072
\]
And because this is the tens place, we add a zero placeholder to shift the result one place to the left, effectively counting it as \(30.72\) or \(3072\) with a placeholder **0** at the end.
Hence, when we write \(3072\) in our multiplication array, it must have the zero added to emphasize it is from the tens place:
\[
3072 \rightarrow 30720 \text{ (shifting one place left)}
\]
### Final Addition of Results
So in summary, the zero at the end of \(30720\) comes from the fact that we are multiplying by the **4** from the \(4.8\), which represents \(40\), and we must shift the result \(3072\) one place to the left.
When you add the two partial products together:
\[
\begin{array}{r}
6144 \\
+ 30720 \\
\hline
36864 \\
\end{array}
\]
### Making Sense of the Decimals
Finally, after adding \(6144\) and \(30720\), we determine where to place the decimal point based on the original numbers, resulting in \(36.864\).
If you have more questions or need further clarification, feel free to ask!
### Breakdown of the Multiplication Steps
When multiplying \(7.68\) by \(4.8\):
1. We first multiply \(7.68\) by the **8** in the **tenths** place of \(4.8\):
- This gives us:
\[
7.68 \times 8 = 61.44
\]
(Or when treated as whole numbers: \(768 \times 8 = 6144\))
2. Next, we multiply \(7.68\) by the **4** in the **whole number** place of \(4.8\):
- Since this \(4\) actually represents **40** (not just 4), we must account for that when we multiply.
- So, we calculate:
\[
7.68 \times 4 = 30.72
\]
When treated as whole numbers:
\[
768 \times 4 = 3072
\]
And because this is the tens place, we add a zero placeholder to shift the result one place to the left, effectively counting it as \(30.72\) or \(3072\) with a placeholder **0** at the end.
Hence, when we write \(3072\) in our multiplication array, it must have the zero added to emphasize it is from the tens place:
\[
3072 \rightarrow 30720 \text{ (shifting one place left)}
\]
### Final Addition of Results
So in summary, the zero at the end of \(30720\) comes from the fact that we are multiplying by the **4** from the \(4.8\), which represents \(40\), and we must shift the result \(3072\) one place to the left.
When you add the two partial products together:
\[
\begin{array}{r}
6144 \\
+ 30720 \\
\hline
36864 \\
\end{array}
\]
### Making Sense of the Decimals
Finally, after adding \(6144\) and \(30720\), we determine where to place the decimal point based on the original numbers, resulting in \(36.864\).
If you have more questions or need further clarification, feel free to ask!