Question

standards: 8.eea.3, 8.eea.4
directions: for numbers 1-20, convert each number to its oppos

1. 82,900,000

8.29×10⁵

1. 0.0000068

6.8×10⁻⁶

1. 12,850.

1.285×10⁴

1. 0.00085
2. 10,500,000,000

1.05×10¹⁰

1. 0.0000005
2. 1,200,000

1.2×10⁶

1. 0.0033
2. 5,280
3. 0.4

Answer:

Let's solve each problem step by step (assuming "opposite" here means converting between standard and scientific notation, or vice - versa. Let's take each number:

Problem 1: 82,900,000 (convert to scientific notation)

Step 1: Move the decimal point

We need to move the decimal point to get a number between 1 and 10. For 82900000, moving the decimal 7 places to the left gives 8.29.

Step 2: Determine the exponent

Since we moved the decimal 7 places to the left, the exponent of 10 is 7. So in scientific notation, it is $8.29\times10^{7}$ (the original handwritten answer of $8.29\times 10^{5}$ is incorrect).

Problem 3: 0.0000068 (convert to scientific notation)

Step 1: Move the decimal point

Move the decimal point 6 places to the right to get 6.8.

Step 2: Determine the exponent

Since we moved the decimal 6 places to the right, the exponent of 10 is - 6. So it is $6.8\times10^{-6}$ (the handwritten answer is correct).

Problem 5: 12,850 (convert to scientific notation)

Step 1: Move the decimal point

Move the decimal point 4 places to the left to get 1.285.

Step 2: Determine the exponent

Since we moved the decimal 4 places to the left, the exponent of 10 is 4. So it is $1.285\times10^{4}$ (the handwritten answer is correct).

Problem 7: 0.00085 (convert to scientific notation)

Step 1: Move the decimal point

Move the decimal point 4 places to the right to get 8.5.

Step 2: Determine the exponent

Since we moved the decimal 4 places to the right, the exponent of 10 is - 4. So it is $8.5\times10^{-4}$.

Problem 9: 10,500,000,000 (convert to scientific notation)

Step 1: Move the decimal point

Move the decimal point 10 places to the left to get 1.05.

Step 2: Determine the exponent

Since we moved the decimal 10 places to the left, the exponent of 10 is 10. So it is $1.05\times10^{10}$ (the handwritten answer has a typo, the correct exponent is 10, not 10 with a decimal error. The number 10,500,000,000 has 11 digits, so when we write it as $a\times10^{n}$, $a = 1.05$ and $n=10$).

Problem 11: 0.0000005 (convert to scientific notation)

Step 1: Move the decimal point

Move the decimal point 7 places to the right to get 5.

Step 2: Determine the exponent

Since we moved the decimal 7 places to the right, the exponent of 10 is - 7. So it is $5\times10^{-7}$.

Problem 13: 1,200,000 (convert to scientific notation)

Step 1: Move the decimal point

Move the decimal point 6 places to the left to get 1.2.

Step 2: Determine the exponent

Since we moved the decimal 6 places to the left, the exponent of 10 is 6. So it is $1.2\times10^{6}$ (the handwritten answer is correct).

Problem 15: 0.0033 (convert to scientific notation)

Step 1: Move the decimal point

Move the decimal point 3 places to the right to get 3.3.

Step 2: Determine the exponent

Since we moved the decimal 3 places to the right, the exponent of 10 is - 3. So it is $3.3\times10^{-3}$.

Problem 17: 5,280 (convert to scientific notation)

Step 1: Move the decimal point

Move the decimal point 3 places to the left to get 5.28.

Step 2: Determine the exponent

Since we moved the decimal 3 places to the left, the exponent of 10 is 3. So it is $5.28\times10^{3}$.

Problem 19: 0.4 (convert to scientific notation)

Step 1: Move the decimal point

Move the decimal point 1 place to the right to get 4.

Step 2: Determine the exponent

Since we moved the decimal 1 place to the right, the exponent of 10 is - 1. So it is $4\times10^{-1}$.

If we assume the "opposite" is converting scientific notation to standard notation (for the hand - written scientific notations):

For $8.29\times10^{5}$ (from problem 1 hand - written)

Step 1: Multiply by 10^5

$10^{5}=100000$, so $8.29\times100000 = 829000$ (but the original number was 82,900,000, so the hand - written scientific notation was wrong).

For $6.8\times10^{-6}$ (from problem 3 hand - written)

Step 1: Multiply by 10^-6

$10^{-6}=\frac{1}{1000000}$, so $6.8\times\frac{1}{1000000}=0.0000068$ (correct, matches the original number).

For $1.285\times10^{4}$ (from problem 5 hand - written)

Step 1: Multiply by 10^4

$10^{4} = 10000$, so $1.285\times10000=12850$ (correct, matches the original number).

For $1.05\times10^{10}$ (from problem 9 hand - written, assuming the decimal was a typo)

Step 1: Multiply by 10^10

$10^{10}=10000000000$, so $1.05\times10000000000 = 10500000000$ (correct, matches the original number).

For $1.2\times10^{6}$ (from problem 13 hand - written)

Step 1: Multiply by 10^6

$10^{6}=1000000$, so $1.2\times1000000 = 1200000$ (correct, matches the original number).

If you want the answers for converting to scientific notation (correcting errors):

1. $8.29\times10^{7}$
2. $6.8\times10^{-6}$ (correct as is)
3. $1.285\times10^{4}$ (correct as is)
4. $8.5\times10^{-4}$
5. $1.05\times10^{10}$
6. $5\times10^{-7}$
7. $1.2\times10^{6}$ (correct as is)
8. $3.3\times10^{-3}$
9. $5.28\times10^{3}$
10. $4\times10^{-1}$

If you want the answers for converting the hand - written scientific notation to standard notation:

1. $8.29\times10^{5}=829000$ (but original number was 82,900,000, so the scientific notation was wrong)
2. $6.8\times10^{-6}=0.0000068$ (correct)
3. $1.285\times10^{4}=12850$ (correct)
4. $1.05\times10^{10}=10500000000$ (correct)
5. $1.2\times10^{6}=1200000$ (correct)