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

Question

3.2 × 10^4 | 1.95 × 10^6 | 8.03 × 10^-8 | 2.56 × 10^-2 | 4.82 × 10^12
a. Choose two of the scientific numbers from the list and add them.
b. Choose two of the scientific numbers from the list and subtract them. Make it not too long and make it understandable none of that “ 3.2 \times 10^4 = 3.2 \times 10^4 \times \frac{100}{100} = 3.2 \times 10^4 \times 10^2 = 3.2 \times 10^6 \times 10^{-2}
\]
This can be rewritten as:” bs

Answer 1

Sure! Let's go through the operations clearly and concisely.

a. Addition

Let's add \(3.2 \times 10^4\) and \(2.56 \times 10^{-2}\).

1. Before adding, we need to express both numbers with the same exponent. We can convert \(3.2 \times 10^4\) into an equivalent form with \(10^{-2}\):

   • \(3.2 \times 10^4 = 3.2 \times 10^4 \times \frac{10^6}{10^6} = 3.2 \times 10^6 \times 10^{-2} = 3.2 \times 1000000 \times 0.01 = 3200000 \times 0.01 = 32000\)

2. Therefore, we can write:

   • \(3.2 \times 10^4 = 32000 \times 10^{-2}\)
   • \(2.56 \times 10^{-2}\) remains the same.

3. Now we can add: \[ 3.2 \times 10^4 + 2.56 \times 10^{-2} = 32000 \times 10^{-2} + 2.56 \times 10^{-2} = (32000 + 2.56) \times 10^{-2} = 32002.56 \times 10^{-2} \]

4. Converting back to scientific notation:

   • \(32.00256 \times 10^2\) (which is \(3.200256 \times 10^4\)).

Result of Addition:

\[ 3.200256 \times 10^4 \]

---

b. Subtraction

Let's subtract \(1.95 \times 10^6\) from \(4.82 \times 10^{12}\).

1. To perform the subtraction, it's best to express both numbers in the same base. We can convert \(4.82 \times 10^{12}\) into an equivalent form with \(10^{6}\):

   • \(4.82 \times 10^{12} = 4.82 \times 10^{12 - 6} \times 10^{6} = 4.82 \times 10^6 \times 10^6 = 4.82 \times 1000000\)

2. So, we can write:

   • \(4.82 \times 10^{12} = 4.82 \times 10^{6} \times 1000000 = 4820000000 \)

3. Now we can subtract: \[ 4.82 \times 10^{12} - 1.95 \times 10^{6} = 4820000000 - 1.95 \times 10^6 = 4820000000 - 1950000 = 4818050000 \]

4. Converting to scientific notation:

   • \(4.818050000 \times 10^9\) (which is \(4.81 \times 10^9\) rounded).

Result of Subtraction:

\[ 4.81 \times 10^9 \]

In summary:

• Addition Result: \(3.200256 \times 10^4\)
• Subtraction Result: \(4.81 \times 10^9\)

Answer 2

Make it shorter

Answer 3

Sure! Here’s a more concise version:

a. Addition

Add \(3.2 \times 10^4\) and \(2.56 \times 10^{-2}\).

1. Convert \(3.2 \times 10^4\) to match the exponent of \(10^{-2}\): \[ 3.2 \times 10^4 = 32000 \times 10^{-2} \]

2. Now add: \[ 32000 \times 10^{-2} + 2.56 \times 10^{-2} = (32000 + 2.56) \times 10^{-2} = 32002.56 \times 10^{-2} \]

3. Convert to scientific notation: \[ 3.200256 \times 10^4 \]

b. Subtraction

Subtract \(1.95 \times 10^6\) from \(4.82 \times 10^{12}\).

1. Convert \(4.82 \times 10^{12}\) to match \(10^6\): \[ 4.82 \times 10^{12} = 4.82 \times 10^6 \times 10^6 \] This is \(4820000000\).

2. Now subtract: \[ 4820000000 - 1950000 = 4818050000 \]

3. Convert to scientific notation: \[ 4.818050000 \times 10^9 \approx 4.81 \times 10^9 \]

Summary of Results:

• Addition: \(3.200256 \times 10^4\)
• Subtraction: \(4.81 \times 10^9\)