Question

scientific notation
scientists very often deal with very small and very large numbers, which can lead to a lot of confusion when counting zeros! we have learned to express these numbers as powers of 10.
scientific notation takes the form of m x 10^n where 1≤m < 10 and “n” represents the number of decimal places to be moved. positive n indicates the standard form is larger than zero whereas negative n would indicate a number smaller than zero.
example 1: convert 1,500,000 to scientific notation.
we move the decimal point so that there is only one digit to its left, a total of 6 places.
1,500,000 = 1.5 x 10^6
example 2: convert 0.000025 to scientific notation.
for this, we move the decimal point 5 places to the right.
0.000025 = 2.5 x 10^-5
(note that when a number starts out less than one, the exponent is always negative.)
convert the following to scientific notation.

1. 0.005 =
2. 5,050 =
3. 0.0008 =
4. 1,000 =
5. 1,000,000 =
6. 0.25 =
7. 0.025 =
8. 0.0025 =
9. 500 =
10. 5,000 =

convert the following to standard notation.

1. 1.5 x 10^3 =
2. 1.5 x 10^-3 =
3. 3.75 x 10^-2 =
4. 3.75 x 10^2 =
5. 2.2 x 10^5 =
6. 3.35 x 10^-1 =
7. 1.2 x 10^-4 =
8. 1 x 10^4 =
9. 1 x 10^-1 =
10. 4 x 10^0 =

Explanation:

Step1: Convert to scientific - notation rules

Move decimal to get $1\leq M<10$, exponent $n$ is number of places moved. Positive $n$ for large numbers, negative $n$ for small numbers.

Step2: Convert to standard - notation rules

Multiply $M$ by $10^n$. Move decimal $n$ places right for positive $n$ and left for negative $n$.

Convert to scientific notation:

1. For $0.005$, move decimal 3 places right: $5\times10^{- 3}$
2. For $5050$, move decimal 3 places left: $5.05\times10^{3}$
3. For $0.0008$, move decimal 4 places right: $8\times10^{-4}$
4. For $1000$, move decimal 3 places left: $1\times10^{3}$
5. For $1000000$, move decimal 6 places left: $1\times10^{6}$
6. For $0.25$, move decimal 1 place right: $2.5\times10^{-1}$
7. For $0.025$, move decimal 2 places right: $2.5\times10^{-2}$
8. For $0.0025$, move decimal 3 places right: $2.5\times10^{-3}$
9. For $500$, move decimal 2 places left: $5\times10^{2}$
10. For $5000$, move decimal 3 places left: $5\times10^{3}$

Convert to standard notation:

1. For $1.5\times10^{3}$, move decimal 3 places right: $1500$
2. For $1.5\times10^{-3}$, move decimal 3 places left: $0.0015$
3. For $3.75\times10^{-2}$, move decimal 2 places left: $0.0375$
4. For $3.75\times10^{2}$, move decimal 2 places right: $375$
5. For $2.2\times10^{5}$, move decimal 5 places right: $220000$
6. For $3.35\times10^{-1}$, move decimal 1 place left: $0.335$
7. For $1.2\times10^{-4}$, move decimal 4 places left: $0.00012$
8. For $1\times10^{4}$, move decimal 4 places right: $10000$
9. For $1\times10^{-1}$, move decimal 1 place left: $0.1$
10. For $4\times10^{0}$ ($10^{0}=1$), it is $4$

Answer:

Convert to scientific notation:

1. $5\times10^{-3}$
2. $5.05\times10^{3}$
3. $8\times10^{-4}$
4. $1\times10^{3}$
5. $1\times10^{6}$
6. $2.5\times10^{-1}$
7. $2.5\times10^{-2}$
8. $2.5\times10^{-3}$
9. $5\times10^{2}$
10. $5\times10^{3}$

Convert to standard notation:

1. $1500$
2. $0.0015$
3. $0.0375$
4. $375$
5. $220000$
6. $0.335$
7. $0.00012$
8. $10000$
9. $0.1$
10. $4$