# Scientific Notation

See how to write and solve a number

in scientific notation.

8 examples and their solutions.

## Scientific Notation

### Definition

a × 10

1 ≤ a < 10

n: integer

Scientific notation is a way to write a number^{n}1 ≤ a < 10

n: integer

that is too big or too small.

a shows how the number looks like.

n shows how big the number is.

1.23 = 1.23 × 10

^{0}

12.3 = 1.23 × 10

^{1}

123 = 1.23 × 10

^{2}

1230 = 1.23 × 10

^{3}

## Scientific Notation → Standard Notation

### Example

310200

Solution 310200● - [1] [2]

↓

3●10200◌ - [3]

5 digits

310200 = 3.102 × 10

↓

3●10200◌ - [3]

5 digits

310200 = 3.102 × 10

^{5}- [4][1]

First, find the a part:

numbers next to the consecutive 0s.

→ 3102

numbers next to the consecutive 0s.

→ 3102

[2]

Next, find the decimal point.

[3]

Move the decimal point

to make 3102

between 1 ≤ a < 10.

to make 3102

between 1 ≤ a < 10.

[4]

The decimal point moved 5 digits.

So the 10

310200 is greater than 1 (= 10

So the 10

So 310200 = 3.102 × 10

So the 10

^{n}part is either 10^{5}or 10^{-5}.310200 is greater than 1 (= 10

^{0}).So the 10

^{n}part is 10^{5}.So 310200 = 3.102 × 10

^{5}.Close

### Example

0.00509

Solution 0●00509 - [1]

↓

0◌005●09 - [2]

3 digits

0.00509 = 5.09 × 10

↓

0◌005●09 - [2]

3 digits

0.00509 = 5.09 × 10

^{-3}- [3][1]

First, find the a part:

numbers next to the consecutive 0s.

→ 509

numbers next to the consecutive 0s.

→ 509

[2]

Move the decimal point

to make 509

between 1 ≤ a < 10.

to make 509

between 1 ≤ a < 10.

[3]

The decimal point moved 3 digits.

So the 10

0.00509 is less than 1 (= 10

So the 10

So 0.00509 = 5.09 × 10

So the 10

^{n}part is either 10^{3}or 10^{-3}.0.00509 is less than 1 (= 10

^{0}).So the 10

^{n}part is 10^{-3}.So 0.00509 = 5.09 × 10

^{-3}.Close

## Standard Notation → Scientific Notation

### Example

7.95 × 10

Solution ^{6} 7●95 - [1]

↓

7◌950000● - [2]

7.95 × 10

↓

7◌950000● - [2]

7.95 × 10

^{6}= 7950000[1]

Write the a part.

[2]

7.95 × 10

So move the decimal point

6 digits →.

(to make the number greater than 1)

Fill the 0s, 0000, in the blank digits.

^{6}is greater than 1 (= 10^{0}).So move the decimal point

6 digits →.

(to make the number greater than 1)

Fill the 0s, 0000, in the blank digits.

Close

### Example

8.6 × 10

Solution ^{-4} 8●6 - [1]

↓

0●0008◌6 - [2]

8.6 × 10

↓

0●0008◌6 - [2]

8.6 × 10

^{-4}= 0.00086[1]

Write the a part.

[2]

8.6 × 10

So move the decimal point

4 digits →.

(to make the number less than 1)

Fill the 0s, 0.000, in the blank digits.

^{-4}is less than 1 (= 10^{0}).So move the decimal point

4 digits →.

(to make the number less than 1)

Fill the 0s, 0.000, in the blank digits.

Close

## Multiplying Scientific Notation

### Example

(2.15 × 10

Solution ^{3})⋅(1.98 × 10^{5}) (2.15 × 10

= (2.15⋅1.98)⋅(10

= 4.26 × 10

^{3})⋅(1.98 × 10^{5})= (2.15⋅1.98)⋅(10

^{3}⋅10^{5})= 4.26 × 10

^{8}- [1] [2][1]

2.15⋅1.98 = 4.2570

2.15 and 1.98 have 3 significant digits.

(215, 198)

So, to make the a part 3 significant digits,

round 4.2570 to the nearest hundredth.

4.2570 → 4.26

2.15 and 1.98 have 3 significant digits.

(215, 198)

So, to make the a part 3 significant digits,

round 4.2570 to the nearest hundredth.

4.2570 → 4.26

[2]

Close

### Example

(8.73 × 10

Solution ^{4})⋅(9.01 × 10^{2}) (8.73 × 10

= (8.73⋅9.01)⋅(10

= 78.7 × 10

= 7.87⋅10 × 10

= 7.87 × 10

^{4})⋅(9.01 × 10^{2})= (8.73⋅9.01)⋅(10

^{4}⋅10^{2})= 78.7 × 10

^{6}- [1]= 7.87⋅10 × 10

^{6}= 7.87 × 10

^{7}- [2][1]

8.73⋅9.01 = 78.6573

8.73 and 9.01 have 3 significant digits.

(873, 901)

So, to make the a part 3 significant digits,

round 78.6573 to the nearest hundredth.

78.6573 → 78.7

8.73 and 9.01 have 3 significant digits.

(873, 901)

So, to make the a part 3 significant digits,

round 78.6573 to the nearest hundredth.

78.6573 → 78.7

[2]

Make 78.7 between 1 ≤ a < 10.

78.7 → 7.87⋅10

78.7 → 7.87⋅10

Close

## Dividing Scientific Notation

### Example

(9.02 × 10

Solution ^{7}) ÷ (3.75 × 10^{4}) (9.02 × 10

= (9.02 ÷ 3.75)⋅(10

= 2.41 × 10

^{7}) ÷ (3.75 × 10^{4})= (9.02 ÷ 3.75)⋅(10

^{7}÷ 10^{4})= 2.41 × 10

^{3}- [1] [2][1]

9.02 ÷ 3.75 = 2.405...

9.02 and 3.75 have 3 significant digits.

(902, 375)

So, to make the a part 3 significant digits,

round 2.405... to the nearest hundredth.

2.405... → 2.41

9.02 and 3.75 have 3 significant digits.

(902, 375)

So, to make the a part 3 significant digits,

round 2.405... to the nearest hundredth.

2.405... → 2.41

[2]

Close

### Example

(1.57 × 10

Solution ^{8}) ÷ (8.06 × 10^{5}) (1.57 × 10

= (1.57 ÷ 8.06)⋅(10

= 0.174 × 10

= 1.74⋅10

= 1.74 × 10

^{8}) ÷ (8.06 × 10^{5})= (1.57 ÷ 8.06)⋅(10

^{8}÷ 10^{5})= 0.174 × 10

^{3}- [1]= 1.74⋅10

^{-1}× 10^{3}= 1.74 × 10

^{2}- [2][1]

1.57 ÷ 8.06 = 0.1742...

1.57 and 8.06 have 3 significant digits.

(157, 806)

So, to make the a part 3 significant digits,

round 0.1742... to the nearest hundredth.

0.1742... → 0.174

1.57 and 8.06 have 3 significant digits.

(157, 806)

So, to make the a part 3 significant digits,

round 0.1742... to the nearest hundredth.

0.1742... → 0.174

[2]

Make 0.174 between 1 ≤ a < 10.

0.174 → 1.74⋅10

0.174 → 1.74⋅10

^{-1}Close