Question

let’s convert!
numbers in standard form and scientific notation
the objective of this notation is to express very large or very small numbers in a more concise and understandable form.
standard form ↔ scientific notation
3,700,000 = 3.7 × 10⁶
0.00003 = 3 × 10⁻⁵

➤ write each number in scientific notation.

1. 9,000,000 =
2. 5,100,014 =
3. 43,070,000 =
4. 709,090,000 =
5. 5,009,000,021 =
6. 0.00000000013 =
7. 80,000,000,000 =
8. 6,200,000,004 =
9. 0.0000000021 =
10. 5,000,300,000,000 =
11. 0.000000000062 =
12. 0.00000000000406 =

➤ write each number in standard form.

1. 2 × 10⁸ =
2. 3 × 10⁴ =
3. 4.22 × 10¹¹ =
4. 71 × 10¹⁰ =
5. 1.3 × 10¹⁴ =
6. 8 × 10⁵ =
7. 0.55 × 10⁻⁴ =
8. 7.43 × 10⁹ =
9. 6.04 × 10⁷ =
10. 3.08 × 10⁻¹⁰ =
11. 0.01 × 10¹⁷ =
12. 7.36 × 10⁻¹² =

Explanation:

Step1: Scientific notation rule: $a \times 10^n$, $1 \leq a < 10$

For numbers >1: $n$ = digits left of decimal after moving it to get $a$. For numbers <1: $n$ = negative of digits right of decimal before first non-zero digit.

Step2: Convert 1-12 to scientific notation

1. $9,000,000$: Move decimal 6 left: $9 \times 10^6$
2. $0.000814$: Move decimal 4 right: $8.14 \times 10^{-4}$
3. $43,090,000$: Move decimal 7 left: $4.309 \times 10^7$
4. $201,000,000$: Move decimal 8 left: $2.01 \times 10^8$
5. $0.000000021$: Move decimal 8 right: $2.1 \times 10^{-8}$
6. $0.00000000043$: Move decimal 10 right: $4.3 \times 10^{-10}$
7. $80,100,000,000$: Move decimal 10 left: $8.01 \times 10^{10}$
8. $1,001,000,000$: Move decimal 9 left: $1.001 \times 10^9$
9. $0.0000000021$: Move decimal 9 right: $2.1 \times 10^{-9}$
10. $5,000,000,000,000$: Move decimal 12 left: $5 \times 10^{12}$
11. $0.00000000062$: Move decimal 10 right: $6.2 \times 10^{-10}$
12. $0.0000000000405$: Move decimal 11 right: $4.05 \times 10^{-11}$

Step3: Convert 13-24 to standard form

For $a \times 10^n$: If $n>0$, move decimal $n$ right; if $n<0$, move decimal $|n|$ left.

1. $8 \times 10^{-6}$: Move decimal 6 left: $0.000008$
2. $9 \times 10^7$: Move decimal 7 right: $90,000,000$
3. $4.22 \times 10^5$: Move decimal 5 right: $422,000$
4. $7.1 \times 10^{-10}$: Move decimal 10 left: $0.00000000071$
5. $1.3 \times 10^{11}$: Move decimal 11 right: $130,000,000,000$
6. $8 \times 10^3$: Move decimal 3 right: $8,000$
7. $8.55 \times 10^{-4}$: Move decimal 4 left: $0.000855$
8. $7.43 \times 10^{10}$: Move decimal 10 right: $74,300,000,000$
9. $6.04 \times 10^7$: Move decimal 7 right: $60,400,000$
10. $3.08 \times 10^{-13}$: Move decimal 13 left: $0.000000000000308$
11. $9.8 \times 10^{12}$: Move decimal 12 right: $9,800,000,000,000$
12. $2.36 \times 10^{-9}$: Move decimal 9 left: $0.00000000236$

Answer:

1. $9 \times 10^6$
2. $8.14 \times 10^{-4}$
3. $4.309 \times 10^7$
4. $2.01 \times 10^8$
5. $2.1 \times 10^{-8}$
6. $4.3 \times 10^{-10}$
7. $8.01 \times 10^{10}$
8. $1.001 \times 10^9$
9. $2.1 \times 10^{-9}$
10. $5 \times 10^{12}$
11. $6.2 \times 10^{-10}$
12. $4.05 \times 10^{-11}$
13. $0.000008$
14. $90,000,000$
15. $422,000$
16. $0.00000000071$
17. $130,000,000,000$
18. $8,000$
19. $0.000855$
20. $74,300,000,000$
21. $60,400,000$
22. $0.000000000000308$
23. $9,800,000,000,000$
24. $0.00000000236$