HOW TO WRITE NUMBERS IN SCIENTIFIC NOTATION

Scientific notation helps us to represent the numbers which are very huge or very tiny in a form of multiplication of single-digit numbers and 10 raised to the power of the respective exponent.

• The exponent is positive if the number is very large.
• It is negative if the number is very small.

a × 10^b ; 1 ≤ a < 10

Convert the following into scientific notation :

Problem 1 :

92100

Solution :

In the given number, we could not see the decimal point. So, we have to consider the decimal point is at last.

92100.

Now, we have to shift this decimal point to the left side, four digits. So, we have to take positive power.

9.2100 × 10⁴

= 9.21 × 10⁴

Problem 2 :

0.000084

Solution :

Now we have to shift this decimal point to the right side, five digits. So, we have to use negative power.

8.4 × 10^-5

Problem 3 :

700000

Solution :

In the given number, we could not see the decimal point. So, we have to consider the decimal point is at last.

700000.

Now, we have to shift this decimal point to the left side, five digits. So, we have to use positive power.

7.00000 × 10⁵

= 7 × 10⁵

Problem 4 :

9000

Solution :

In the given number, we could not see the decimal point. So, we have to consider the decimal point is at last.

9000.

Now, we have to shift this decimal point to the left side, three digits. So, we have to use positive power.

9.000 × 10³

= 9 × 10³

Problem 5 :

9.03

Solution :

9.03 × 10⁰

Problem 6 :

20

Solution :

In the given number, we could not see the decimal point. So, we have to consider the decimal point is at last.

20.

Now, we have to shift this decimal point to the left side, one digit. So, we have to take positive power.

2.0 × 10¹

= 2 × 10¹

Problem 7 :

0.000056

Solution :

Now we have to shift this decimal point to the right side, five digits. So, we have to use negative power.

5.6 × 10^-5

Problem 8 :

0.0002

Solution :

Now we have to shift this decimal point to the right side, four digits. So, we have to use negative power.

2 × 10^-4

Problem 9 :

4.12

Solution :

4.12 × 10⁰

Problem 10 :

160

Solution :

In the given number, we could not see the decimal point. So, we have to consider the decimal point is at last.

160.

Now, we have to shift this decimal point to the left side, one digit. So, we have to use positive power.

1.60 × 10²

= 1.6 × 10²

Problem 11 :

0.66

Solution :

Now we have to shift this decimal point to the right side, one digits. So, we have to use negative power.

6.6 × 10^-1

Problem 12 :

Mercury’s distance from the Sun is approximately 5.79 × 10⁷ kilometers. What is this distance in standard form?

a) 5,790,000,000 km       b) 579,000,000 km

c)  57,900,000 km       d) 5,790,000 km

Solution :

Distance of Mercury from the Sun = 5.79 × 10⁷ kilometers

Since we have power 7, we have to move the decimal 7 digits to the right. We have only 2 digits after the decimal, then we have to add 5 zeros.

= 57900000

Problem 13 :

What is the diameter of a human hair written in scientific notation?

Solution :

Diameter = 0.000099 meter

Moving the decimal 5 digits to the right,

= 9.9 x 10⁵

Problem 14 :

What is the circumference of Earth written in scientifi c notation?

Solution :

Circumference at the equator = 40100000 meter

= 4.01 x 10⁷ meter

Problem 15 :

The total power of a space shuttle during launch is the sum of the power from its solid rocket boosters and the power from its main engines. The power from the solid rocket boosters is 9,750,000,000 watts. What is the power from the main engines?

Solution :

The power from the solid rocket boosters = 9,750,000,000 watts.

= 9.75 x 10⁹

Total power = 1.174 x 10¹⁰ watts

Total power = power from its solid rocket boosters + the power from its main engines

1.174 x 10¹⁰ = 9.75 x 10⁹ + the power from its main engines

The power from its main engines = 1.174 x 10¹⁰ - 9.75 x 10⁹

= 1.174 x 10 x 10⁹ - 9.75 x 10⁹

= 11.74 x 10⁹ - 9.75 x 10⁹

Factoring 10⁹,

= (11.74 - 9.75) x 10⁹

= 1.99 x 10⁹

Since we have two digits after the decimal, we have to add seven more zeros.

= 1990000000

Problem 16 :

Write the number 0.853 in scientific notation.

Solution :

Given number, 0.853

Moving the decimal one digit to the right.

= 8.53 x 10¹

Problem 17 :

0.36² in standard form is

(a) 0.01296 ×10¹      (b) 1.296 × 10^-1       (c) 0.1296 ×10⁰

(d) 12.96 ×10^-2

Solution :

= 0.36²

= 0.36 x 0.36

= 0.1296

Since we have four digits after the decimal, we have to move the decimal one digit to the right.

= 1.296 x 10¹

So, option a is correct.