Question

In this problem, you will practice adding, subtracting, multiplying, and dividing numbers written in scientific notation. For each part, use the following numbers:

3.2 x 10^4
1.95 x 10^6
8.03 x 10^-8
2.56 x 10^-2
4.82 x 10^12

a. choose two of the scientific numbers from the list and add them and show your work.

b. choose two of the scientific numbers from the list and subtract the. Show your work.

c. choose two of the scientific numbers from the list and multiply them. Show your work.

d. Choose two of the scientific numbers from the list and divide them. Show your work.

e. choose one of the scientific numbers from the list and add it to 32.8. Show your work

f. Choose one of the scientific numbers from the list and subtract it from 1,845.19 (or subtract 1,845.19 from the number you choose). Show your work.

g. choose one of the scientific numbers from the list and multiply it by 256.7. Show your work.

h. choose one of the scientific numbers from the list and divide it by 0.3 (or divide 0.3 by the number you choose). Show your work.

Answer 1

Sure! Let's go through each part step by step.

### a. Addition
**Numbers chosen:** \(3.2 \times 10^4\) and \(1.95 \times 10^6\)

**Step 1: Align the powers of 10.**
To add these numbers, we should express them with the same exponent. We will convert \(3.2 \times 10^4\) to have the same exponent as \(1.95 \times 10^6\):
\[
3.2 \times 10^4 = 0.032 \times 10^6
\]

**Step 2: Add the coefficients.**
Now we can add:
\[
0.032 \times 10^6 + 1.95 \times 10^6 = (0.032 + 1.95) \times 10^6 = 1.982 \times 10^6
\]

### b. Subtraction
**Numbers chosen:** \(1.95 \times 10^6\) and \(3.2 \times 10^4\)

**Step 1: Align the powers of 10.**
Convert \(3.2 \times 10^4\) to have the same exponent as \(1.95 \times 10^6\):
\[
3.2 \times 10^4 = 0.032 \times 10^6
\]

**Step 2: Subtract the coefficients.**
Now we can subtract:
\[
1.95 \times 10^6 - 0.032 \times 10^6 = (1.95 - 0.032) \times 10^6 = 1.918 \times 10^6
\]

### c. Multiplication
**Numbers chosen:** \(4.82 \times 10^{12}\) and \(8.03 \times 10^{-8}\)

**Step 1: Multiply the coefficients and add the exponents.**
\[
(4.82 \times 8.03) \times 10^{12 + (-8)} = 38.7456 \times 10^4
\]

**Step 2: Convert to proper scientific notation.**
\[
38.7456 \times 10^4 = 3.87456 \times 10^5
\]

### d. Division
**Numbers chosen:** \(1.95 \times 10^6\) and \(8.03 \times 10^{-8}\)

**Step 1: Divide the coefficients and subtract the exponents.**
\[
\frac{1.95}{8.03} \times 10^{6 - (-8)} = 0.2437 \times 10^{14}
\]

**Step 2: Convert to proper scientific notation.**
\[
0.2437 \times 10^{14} = 2.437 \times 10^{13}
\]

### e. Addition with a number
**Number chosen:** \(2.56 \times 10^{-2}\)

**Add to 32.8:**
We need to convert \(32.8\) to scientific notation:
\[
32.8 = 3.28 \times 10^1
\]

**Align the powers of 10.**
Convert \(2.56 \times 10^{-2}\) to match \(3.28 \times 10^1\):
\[
2.56 \times 10^{-2} = 0.000256 \times 10^1
\]

**Now add:**
\[
3.28 \times 10^1 + 0.000256 \times 10^1 = (3.28 + 0.000256) \times 10^1 = 3.280256 \times 10^1
\]

### f. Subtraction with a number
**Number chosen:** \(4.82 \times 10^{12}\)

**Subtract from \(1845.19\):**
Convert \(1845.19\) to scientific notation:
\[
1845.19 = 1.84519 \times 10^3
\]

**Step 1: Align the powers of 10.**
Convert \(4.82 \times 10^{12}\) to \(4.82 \times 10^{12} = 4820000000000 \times 10^{-3}\)
So,
\[
1.84519 \times 10^3 - 4.82 \times 10^{12}= 1.84519 \times 10^{3} - 4820000000000 \times 10^{-3}
\]

**This results in a large overwhelmingly negative value.**

### g. Multiply by a number
**Number chosen:** \(3.2 \times 10^4\)

**Multiply by 256.7:**
\[
(3.2 \times 256.7) \times 10^4 = (819.84) \times 10^4 = 8.1984 \times 10^6
\]

### h. Divide by a number
**Number chosen:** \(8.03 \times 10^{-8}\)

**Divide by \(0.3\):**
\[
\frac{8.03 \times 10^{-8}}{0.3} = \frac{8.03}{0.3} \times 10^{-8} = 26.76666667 \times 10^{-8}
\]

**Convert to proper scientific notation:**
\[
2.676666667 \times 10^{-7}
\]

### Summary
- a: \(1.982 \times 10^6\)
- b: \(1.918 \times 10^6\)
- c: \(3.87456 \times 10^5\)
- d: \(2.437 \times 10^{13}\)
- e: \(3.280256 \times 10^1\)
- f: \(1.84519 \times 10^3 - 4.82 \times 10^{12} \text{ results in a negative} \)
- g: \(8.1984 \times 10^6\)
- h: \(2.676666667 \times 10^{-7}\)