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.

Question

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!   
how did you get 36864

GPT-4o mini · Answer

Let's break down the multiplication step by step to clarify how we arrived at the total of 36864 when multiplying 7.68 by 4.8.

Step 1: Write it out

We set it up as:

  7.68
x  4.8
---------

Step 2: Remove the decimals for initial multiplication

To make the multiplication easier, we first disregard the decimals:

Treat 7.68 as 768 (moving the decimal two places to the right).
Treat 4.8 as 48 (moving the decimal one place to the right).

Now, we multiply 768 by 48.

Step 3: Perform the multiplication

Multiply by the units digit (8):
- $768 \times 8$:
- Breakdown:
  - $8 \times 8 = 64$ (write down 4, carry over 6)
  - $8 \times 6 = 48$ then add the carry $6$: $48 + 6 = 54$ (write down 4, carry over 5)
  - $8 \times 7 = 56$ then add the carry $5$: $56 + 5 = 61$ (write down 61).
- So, $768 \times 8 = 6144$.
Multiply by the tens digit (4) (which actually represents 40, so we need to shift one position to the left):
- $768 \times 4$:
- Breakdown:
  - $4 \times 8 = 32$ (write down 2, carry over 3)
  - $4 \times 6 = 24$ then add carry $3$: $24 + 3 = 27$ (write down 7, carry over 2)
  - $4 \times 7 = 28$ then add carry $2$: $28 + 2 = 30$ (write down 30).
- So, $768 \times 4 = 3072$, but since this is actually $3072 \times 10$ (due to the 4 being in the tens place), we write it as 30720.

Step 4: Add the partial results

Now we add the two results together:

  6144
+30720
-------
 36864

Step 5: Count the decimal places

Now we handle the decimals:

7.68 has 2 decimal places.
4.8 has 1 decimal place.
Thus, we need to move the decimal point 3 places to the left in the final result.

Step 6: Place the decimal point

Moving the decimal point in 36864 three places to the left gives us:

36864 → 36.864

Final Answer

The final result of the multiplication of $7.68$ and $4.8$ is:

\[ \boxed{36.864} \]

If you have any further questions or need additional explanations, feel free to ask!

Answer

how is 6144+30720 36864

GPT-4o mini · Answer

Let's verify the addition of **6144** and **30720** step-by-step to ensure the calculation is correct. We need to add: ``` 6144 + 30720 ------- ``` 1. **Align the numbers by place value**: ``` 6144 +30720 ------- ``` 2. **Add column by column from right to left**: - **Rightmost column (units place)**: - $ 4 + 0 = 4 $ - **Next column (tens place)**: - $ 4 + 2 = 6 $ - **Next column (hundreds place)**: - $ 1 + 7 = 8 $ - **Next column (thousands place)**: - $ 6 + 0 = 6 $ - **Leftmost column (ten-thousands place)**: - $ 0 + 3 = 3 $ 3. **Combine the results**: Putting it all together from left to right, we have: ``` 6144 +30720 ------- 36864 ``` So indeed, **6144 + 30720 = 36864**. Therefore, the addition is correct! If there's anything more you'd like to understand or if you have further questions, please let me know!

Answer

where does 1+7 come from

GPT-4o mini · Answer

Apologies for the confusion in my previous response. Let's examine the addition of 6144 and 30720 again to clarify where the digits come from.

Align the numbers properly by place value:

    6144
  +30720
  -------

Add column by column from right to left:

Rightmost column (units place):
- $4 + 0 = 4$
Next column (tens place):
- $4 + 2 = 6$
Next column (hundreds place):
- $1 + 7 = 8$ (this comes from the 1 in 6144 and the 7 in 30720).
Next column (thousands place):
- $6 + 0 = 6$
Leftmost column (ten-thousands place):
- $0 + 3 = 3$

Combine the results:

Putting it all together, we have:

   6144
 +30720
 -------
  36864

Explanation of where the digits come from:

The digit 1 in 6144 represents the hundreds place (so it's actually $1000$), and the digit 7 in 30720 represents the 7000 (since $40 \times 8 = 320$).
The correct addition in the hundreds place should actually be $1 + 0 = 1$ (for the thousands), and it was incorrectly stated as $1 + 7$.

Correction in Addition:

Aligning it again to account for accuracy:

   06144        (this represents 6144)
 + 30720
 ---------
  36864

The proper alignment and sums yield:

Units: $ 4 + 0 = 4$
Tens: $ 4 + 2 = 6$
Hundreds: $ 1 + 7 = 8$
Thousands: $ 6 + 0 = 6$
Ten-thousands: $ 0 + 3 = 3$

So the final addition results in $36864$, and the previous confusion can be cleared.

Thank you for your patience. If you need further clarification, please let me know!

Answer

what is (7.68x10^11) (4.8x10^7)

GPT-4o mini · Answer

To multiply $ (7.68 \times 10^{11}) $ by $ (4.8 \times 10^{7}) $, you can use the property of multiplication that states: $$ (a \times 10^m)(b \times 10^n) = (a \times b) \times 10^{m+n} $$ Here are the steps: ### Step 1: Multiply the coefficients First, multiply the numerical coefficients: $$ 7.68 \times 4.8 $$ To do this multiplication directly: $$ 7.68 \times 4.8 = 36.864 $$ ### Step 2: Add the exponents Next, add the exponents for the powers of ten: $$ 10^{11} \times 10^{7} = 10^{(11 + 7)} = 10^{18} $$ ### Step 3: Combine the results Now combine the results: $$ 36.864 \times 10^{18} $$ ### Step 4: Final Result You can express the final answer in scientific notation. Since $36.864$ is greater than 10, convert it to $3.6864$ by moving the decimal point one place to the left and increasing the exponent by 1: $$ 36.864 = 3.6864 \times 10^{1} $$ Thus, we have: $$ 3.6864 \times 10^{1} \times 10^{18} = 3.6864 \times 10^{(1 + 18)} = 3.6864 \times 10^{19} $$ ### Final Answer: The product of $ (7.68 \times 10^{11}) $ and $ (4.8 \times 10^{7}) $ is: $$ \boxed{3.6864 \times 10^{19}} $$

Answer

(5.8x10^-4) (3.1x10^5)

GPT-4o mini · Answer

To multiply $ (5.8 \times 10^{-4}) $ by $ (3.1 \times 10^{5}) $, you can follow the same steps as before. ### Step 1: Multiply the coefficients First, multiply the numerical coefficients: $$ 5.8 \times 3.1 $$ Calculating that gives: $$ 5.8 \times 3.1 = 17.98 $$ ### Step 2: Add the exponents Next, add the exponents for the powers of ten: $$ 10^{-4} \times 10^{5} = 10^{(-4 + 5)} = 10^{1} $$ ### Step 3: Combine the results Now combine the results: $$ 17.98 \times 10^{1} $$ ### Step 4: Final Result You can express the final answer in standard scientific notation. Since $17.98$ is greater than 10, convert it to $1.798$ by moving the decimal point one place to the left and increasing the exponent by 1: $$ 17.98 = 1.798 \times 10^{1} $$ Thus, we have: $$ 1.798 \times 10^{1} \times 10^{1} = 1.798 \times 10^{(1 + 1)} = 1.798 \times 10^{2} $$ ### Final Answer: The product of $ (5.8 \times 10^{-4}) $ and $ (3.1 \times 10^{5}) $ is: $$ \boxed{1.798 \times 10^{2}} $$