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Theory:

Important!
In the US, the scientific form is only known as the scientific notation. However, in England, the scientific form is known as both standard notation and scientific notation.

Despite us using British English, we consider the number form as the standard notation and the scientific form as the scientific notation, mainly to maintain TN book's consistency.
A number in standard notation is simply the number as we usually write it.
Example:
The number $$256$$ in standard form.

Scientific notation
A number is said to be in the scientific form when written as $$k \times 10^n$$, where $$1 \le k < 10$$ and $$n$$ is an integer. Scientific notation allows us to express a very small or very large number in a compact form.
Example:
The number $$256$$ in scientific form.

Procedure to write the number in scientific notation:
Step 1: A number to be indicated as the product of a number (integer or decimal) and a power of $$10$$.

Step 2: The number of steps moved is taken as the exponent of $$10$$.

Step 3: Move the decimal point after the highest place value of the given number.

Step 4: Now, add a power of $$10$$ that tells how many places you moved the decimal right or left.

Step 5: If you move the decimal place to the right, then the power will have a negative exponent. If you move the decimal places to the left, then the power will have a positive exponent.
Example:
1. Write the scientific form of the number $$5624.032$$.

Move the decimal place to after $$5$$ (left side).

Number of steps to be moved $$=$$ $$3$$

$$5624.032$$ $$=$$ $$5.624032 \times 10^3$$.

2. Write the scientific form of $$(3.7 \times 10^{-5}) (3 \times 10^{-3})$$.

$$(3.7 \times 10^{-5}) (3 \times 10^{-3})$$ $$=$$ $$(3.7 \times 3) \times (10^{-5-3})$$

$$=$$ $$11.1 \times (10^{-8})$$

$$=$$ $$1.11 \times (10^{-8 + 1})$$

Here, one decimal place moved to the left side, so $$1$$ is added in the exponent of $$10$$.

$$=$$ $$1.11 \times (10^{-7})$$

Therefore, $$(3.7 \times 10^{-5}) (3 \times 10^{-3})$$ $$=$$ $$1.11 \times 10^{-7}$$.

3. Write in scientific form $$(3.7 \times 10^{5}) (3 \times 10^{3})$$.

$$(3.7 \times 10^{5}) (3 \times 10^{3})$$ $$=$$ $$(3.7 \times 3) \times (10^{5 + 3})$$

$$=$$ $$11.1 \times (10^{8})$$

$$=$$ $$1.11 \times (10^{8+1})$$

Here, one decimal place moved to the left side, so $$1$$ is added in the exponent of $$10$$.

$$=$$ $$1.11 \times 10^{9}$$

Therefore, $$(3.7 \times 10^{5}) (3 \times 10^{3})$$ $$=$$ $$1.11 \times 10^{9}$$.
Important!
1. The positive exponents in scientific form indicates that it is a large number.

2. The negative exponents in scientific form indicates that it is a small number.