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Answer :
Sure! Let's break down the problem step-by-step to find the correct scientific notation for the number [tex]\(221,000,000,000,000,000,000\)[/tex].
### Understanding Scientific Notation
Scientific notation represents numbers as a product of a number between 1 and 10 and a power of 10. The general form is:
[tex]\[ a \times 10^n \][/tex]
where [tex]\(1 \leq a < 10\)[/tex] and [tex]\(n\)[/tex] is an integer.
### Converting [tex]\(221,000,000,000,000,000,000\)[/tex] to Scientific Notation
1. Identify the significant figures: The significant figures in [tex]\(221,000,000,000,000,000,000\)[/tex] are 221.
2. Determine the decimal point's new location: To get a number between 1 and 10, place the decimal after the first significant digit (2). That gives us: [tex]\(2.21\)[/tex].
3. Counting how many places the decimal point has moved: The original number is [tex]\(221,000,000,000,000,000,000\)[/tex] or [tex]\(2.21 \times 10^{20}\)[/tex], then we add one more zero.
Since the decimal point moves 21 places to the left to transform [tex]\(221,000,000,000,000,000,000\)[/tex] to [tex]\(2.21\)[/tex], the exponent will be 21.
Thus, [tex]\(221,000,000,000,000,000,000\)[/tex] in scientific notation is:
[tex]\[ 2.21 \times 10^{21} \][/tex].
### Identifying Correct Options
Now, we need to match this result with the provided options.
1. [tex]\(2.21 \times 10^{21}\)[/tex] — Correct.
2. [tex]\(2.21E20\)[/tex] — Incorrect, exponent is wrong.
3. [tex]\(2.21E21\)[/tex] — Correct (this format is another way to write [tex]\(2.21 \times 10^{21}\)[/tex]).
4. [tex]\(2.21E-20\)[/tex] — Incorrect, exponent is wrong.
5. [tex]\(221 \times 10^{-21}\)[/tex] — Incorrect, the format is not standard scientific notation and the exponent is wrong.
6. [tex]\(221 \times 10^{-20}\)[/tex] — Incorrect, both format and exponent are wrong.
7. [tex]\(221 \times 10^{20}\)[/tex] — Incorrect, the format is not proper scientific notation.
8. [tex]\(2.21E-21\)[/tex] — Incorrect, exponent is wrong.
### Conclusion
The correct scientific notations for the number [tex]\(221,000,000,000,000,000,000\)[/tex] are:
[tex]\[ 2.21 \times 10^{21} \][/tex]
and
[tex]\[ 2.21E21 \][/tex]
Thus, the correct answers are:
- [tex]\(2.21 \times 10^{21}\)[/tex]
- [tex]\(2.21E21\)[/tex]
### Understanding Scientific Notation
Scientific notation represents numbers as a product of a number between 1 and 10 and a power of 10. The general form is:
[tex]\[ a \times 10^n \][/tex]
where [tex]\(1 \leq a < 10\)[/tex] and [tex]\(n\)[/tex] is an integer.
### Converting [tex]\(221,000,000,000,000,000,000\)[/tex] to Scientific Notation
1. Identify the significant figures: The significant figures in [tex]\(221,000,000,000,000,000,000\)[/tex] are 221.
2. Determine the decimal point's new location: To get a number between 1 and 10, place the decimal after the first significant digit (2). That gives us: [tex]\(2.21\)[/tex].
3. Counting how many places the decimal point has moved: The original number is [tex]\(221,000,000,000,000,000,000\)[/tex] or [tex]\(2.21 \times 10^{20}\)[/tex], then we add one more zero.
Since the decimal point moves 21 places to the left to transform [tex]\(221,000,000,000,000,000,000\)[/tex] to [tex]\(2.21\)[/tex], the exponent will be 21.
Thus, [tex]\(221,000,000,000,000,000,000\)[/tex] in scientific notation is:
[tex]\[ 2.21 \times 10^{21} \][/tex].
### Identifying Correct Options
Now, we need to match this result with the provided options.
1. [tex]\(2.21 \times 10^{21}\)[/tex] — Correct.
2. [tex]\(2.21E20\)[/tex] — Incorrect, exponent is wrong.
3. [tex]\(2.21E21\)[/tex] — Correct (this format is another way to write [tex]\(2.21 \times 10^{21}\)[/tex]).
4. [tex]\(2.21E-20\)[/tex] — Incorrect, exponent is wrong.
5. [tex]\(221 \times 10^{-21}\)[/tex] — Incorrect, the format is not standard scientific notation and the exponent is wrong.
6. [tex]\(221 \times 10^{-20}\)[/tex] — Incorrect, both format and exponent are wrong.
7. [tex]\(221 \times 10^{20}\)[/tex] — Incorrect, the format is not proper scientific notation.
8. [tex]\(2.21E-21\)[/tex] — Incorrect, exponent is wrong.
### Conclusion
The correct scientific notations for the number [tex]\(221,000,000,000,000,000,000\)[/tex] are:
[tex]\[ 2.21 \times 10^{21} \][/tex]
and
[tex]\[ 2.21E21 \][/tex]
Thus, the correct answers are:
- [tex]\(2.21 \times 10^{21}\)[/tex]
- [tex]\(2.21E21\)[/tex]
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