Scientific notation is a method to express very large or very small numbers, is based on the powers of the base number 10. Scientific notation is just a short hand way of expressing gigantic numbers like 1,000,000 or incredibly small numbers like 0.000000001. When there are too large or too small numbers involved in calculations, then to ease the job scientific notations are used. Scientific notation converts a decimal number into a product of a number between 1 and 10, and a power of 10. In General Form, Scientific Notation of a Number is Written as:

a $\times$ $10^b$ , 1 $\leq$ a < 10

The "digits" part, a, is between 1 and 10 (it can be 1, but never 10)

and the "power" part, b, shows exactly how many places to move the decimal point.**Scientific Notation is a special way of writing small and big numbers.
to the base 10.**

To figure out the power of 10, let's see how many places do we move the decimal point.

and the power of 10 will be

Example

$10^3$ = 1000, So 5000 = 5 $\times$ $10^3$ is in scientific notation

Example

0.0005 = 5 $\times$ 0.0001 = 5 $\times$ $10^ {-4}$ in scientific notation.

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Converting a number from scientific notation to decimal notation, just remove the 10 and exponent, and move the decimal place to the right of the number of the times that was the value of the exponent.

**Steps For Converting Scientific Notation to Decimal:**

**Step 1: **Change the exponent of 10 to a multiple of 10.

**Step 2:** Multiply the number with the multiple of 10 obtained above.

### Solved Examples

**Question 1: **Convert 5.123 $\times$ $10^3$ to decimal form.

** Solution: **

Given 5.123 $\times$ $10^3$ in scientific notation

**Step 1:** Move the decimal point in 5.123 three places to the right because 10 is the raised to the power of 3.

5.123 $\times$ $10^3$ = 51.23 $\times$ $10^2$

= 512.3 $\times$ $10^1$

= 5123 $\times$ $10^0$

= 5123 $\times$ 1

= 5123

**Step 2:** Moving the decimal three places to the right gives us the original number of 5123.

**Question 2: **Convert 4.657 $\times$ $10^7$ to decimal form.

** Solution: **

Given 4.657 $\times$ $10^7$ in scientific form

**Step 1:** Move the decimal point in 7 places to the right because 10 is the raised to the power of 7.

4.657 $\times$ $10^7$ = 46.57 $\times$ $10^6$

= 465.7 $\times$ $10^5$

= 4657 $\times$ $10^4$

= 46570000

**Step 2:**

Moving the decimal 7 places to the right gives us the original number of

46570000 in decimal form.

A number in scientific notation is written as the product of a number and a power of 10. A decimal number smaller than 1 can be converted to scientific notation by decreasing the power of 10 by one for each place the decimal point is moved to the right. In a scientific notation, $a\times 10^b$ the term b indicates the number of digits the decimal was shifted.

**Steps For Converting decimal to Scientific Notation:**

Step 1:Move the decimal point to the right of the leftmost non zero digit.

If we have 42300, the decimal point will come to right of 3, that is, 42.300

**Step 2:** Have the count of number of digits the after the decimal.

In 42.300 there are three digits after decimal.

**Step 3:** Place a multiplication sign and the number 10 to the right of the integers and use exponent to indicate the number of places the decimal point has been moved. From the number remove all the zeroes at the end.

42.300 = $42.3 \times 10^3$

**Step 4:** Keep shifting decimal and adding number to the power of 10 till the given number has a value between 1 and 10.

$42.3\times10^3 = 4.23\times 10^4$.

But keep in mind these two points during**Step 3**,

(1) For numbers** greater than 10**, decimal moved to the left, use a **positive exponent.**

(2) For numbers** less than 10**, decimal moved to the right, use a** negative exponent.** ### Solved Examples

**Question 1: **Convert 0.00000056 in scientific notation.

** Solution: **

**Step 1: **Move the decimal point to the right of 0.00000056 up to 7 places.

**Step 2: **The decimal point was moved 7 places to the right to form the number 5.6

**Step 3: **Numbers** less than 10**, decimal moved to the right, use a** negative exponent.**

$\Rightarrow$ 0.00000056 = 5.6 $\times$ $10^{-7}$

**Question 2: **Convert 201000000 in scientific notation.

** Solution: **

Given 201000000 scientific notation.

**Step 1: **Move the decimal to the left 8 places so it is positioned to the right of the leftmost non zero digit 2.01000000

**Step 2: **The decimal point was moved 8 places to the left to form the number 2.01.

Drop all outside zeros and multiply the new number by 10.

**Step 3: **Numbers** greater than 10**, decimal moved to the left, use a** positive exponent.**

$\Rightarrow$ 201000000 = 2.01 $\times$ $10^8$

Scientific notation is just a different way of expressing the standard notation of the number. Standard notation is the normal way of writing numbers. we need to move the decimal point a number of spaces and in the same direction as the sign of the exponent on the 10.

**Convert Scientific Notation to Stranded Notation:**

Move the decimal point a number of spaces and in the same direction as the sign of the exponent.

- If the exponent is negative, the decimal moves to the left.

- If the exponent is positive, it moves to the right.### Solved Examples

**Question 1: **Convert 1.86 $\times$ $10^7$ from scientific notation to standard notation

** Solution: **

**Step 1:** Given 1.86 $\times$ $10^7$ in scientific notation

Exponent = 7 (Positive)

Step 2:Move the decimal place 7 places to the right because exponent is positive

1.86 $\times$ $10^7$ = 1.86 $\times$ 10,000,000 = 18,600,000.

**Question 2: **Convert 2.234 $\times$ $10^{-8}$ from scientific notation to standard notation

** Solution: **

**Step 1: **Given 2.234 $\times$ $10^{-8}$ in scientific notation

Exponent = - 8 (negative)

**Step 2: **Move the decimal place 8 places to the left

2.234 $\times$ $10^{-8}$ = 2.234 $\times$ .00000001

= .000000000002234

**Question 3: **Convert 5.432 $\times$ $10^{-7}$ to standard notation

** Solution: **

**Step 1: **Given 5.432 $\times$ $10^{-7}$ in scientific notation

Exponent = - 7 (negative)

**Step 2: **Move the decimal place 7 places to the left

5.432 $\times$ $10^{-7}$ = 5.432 $\times$ .0000001

= .0000005432

**Question 4: **Convert 6.782 $\times$ $10^5$ from scientific notation to standard notation

** Solution: **

**Step 1:** Given 6.782 $\times$ $10^5$^{ }in scientific notation

Exponent = 5 (Positive)

Step 2:Move the decimal place 5 places to the right because exponent is positive

6.782 $\times$ $10^5$^{ }= 6.782 $\times$ 100000

= 678200.

**Problem 1:** Convert the given into scientific notations:

1. 298000000

2. 8756600000

3. 0.000088766

**Problem 2: **Write the given numbers in decimal notations:

1. $2.5\times 10^8$

2. $1.8\times 10^-4$

3. $8.6\times 10^7$

Given 5.123 $\times$ $10^3$ in scientific notation

5.123 $\times$ $10^3$ = 51.23 $\times$ $10^2$

= 512.3 $\times$ $10^1$

= 5123 $\times$ $10^0$

= 5123 $\times$ 1

= 5123

Given 4.657 $\times$ $10^7$ in scientific form

4.657 $\times$ $10^7$ = 46.57 $\times$ $10^6$

= 465.7 $\times$ $10^5$

= 4657 $\times$ $10^4$

= 46570000

Moving the decimal 7 places to the right gives us the original number of

46570000 in decimal form.

Step 1:

If we have 42300, the decimal point will come to right of 3, that is, 42.300

In 42.300 there are three digits after decimal.

42.300 = $42.3 \times 10^3$

$42.3\times10^3 = 4.23\times 10^4$.

But keep in mind these two points during

(1) For numbers

(2) For numbers

$\Rightarrow$ 0.00000056 = 5.6 $\times$ $10^{-7}$

Given 201000000 scientific notation.

Drop all outside zeros and multiply the new number by 10.

$\Rightarrow$ 201000000 = 2.01 $\times$ $10^8$

Scientific notation is just a different way of expressing the standard notation of the number. Standard notation is the normal way of writing numbers. we need to move the decimal point a number of spaces and in the same direction as the sign of the exponent on the 10.

Move the decimal point a number of spaces and in the same direction as the sign of the exponent.

- If the exponent is negative, the decimal moves to the left.

- If the exponent is positive, it moves to the right.

Exponent = 7 (Positive)

Step 2:

1.86 $\times$ $10^7$ = 1.86 $\times$ 10,000,000 = 18,600,000.

Exponent = - 8 (negative)

2.234 $\times$ $10^{-8}$ = 2.234 $\times$ .00000001

= .000000000002234

Exponent = - 7 (negative)

5.432 $\times$ $10^{-7}$ = 5.432 $\times$ .0000001

= .0000005432

Exponent = 5 (Positive)

Step 2:

6.782 $\times$ $10^5$

= 678200.

1. 298000000

2. 8756600000

3. 0.000088766

1. $2.5\times 10^8$

2. $1.8\times 10^-4$

3. $8.6\times 10^7$

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