Reading and Writing Scientific Notation

Scientific notation is a mathematical shorthand used to write very large or very small numbers. Instead of writing out long strings of zeros, we express the number as a product of a number and a power of $10$.

The standard format is: $a \times 10^n$ Where:

• $a$ is a number greater than or equal to $1$ but less than $10$ ($1 \leq a < 10$).
• $n$ is an integer (positive or negative).

Writing Numbers in Scientific Notation

To convert a standard number into scientific notation, follow these steps:

1. Find $a$: Move the decimal point to the left or right until you have a number between $1$ and $10$.
2. Find $n$: Count how many places you moved the decimal point. If you moved it left (for a large number), $n$ is positive. If you moved it right (for a small number), $n$ is negative.

Example 1: Large Numbers Write $45{,}000{,}000$ in scientific notation.

• Move the decimal point $7$ places to the left to get $4.5$.
• Since we moved left, the exponent is positive $7$.
• Answer: $4.5 \times 10^7$

Example 2: Small Numbers Write $0.00081$ in scientific notation.

• Move the decimal point $4$ places to the right to get $8.1$.
• Since we moved right, the exponent is negative $4$.
• Answer: $8.1 \times 10^{-4}$

Converting to Standard Form

To change scientific notation back to standard form, simply move the decimal point based on the exponent $n$:

• If $n$ is positive, move the decimal point $n$ places to the right.
• If $n$ is negative, move the decimal point $n$ places to the left.

Example 3: Convert $3.2 \times 10^{-4}$ to standard form.

• The exponent is $-4$, so move the decimal $4$ places to the left. Add zeros as placeholders.
• Answer: $0.00032$

Multiplying Numbers in Scientific Notation

You can easily multiply numbers in scientific notation by grouping the numbers and the powers of $10$ separately. Multiply the $a$ values together, and add the exponents for the powers of $10$.

Example 4: Multiply $(2 \times 10^3)(4 \times 10^5)$.

• Multiply the front numbers: $2 \times 4 = 8$
• Add the exponents: $10^3 \times 10^5 = 10^{3+5} = 10^8$
• Answer: $8 \times 10^8$