Standard Form

Standard form (also called scientific notation) is a way to write very large or very small numbers in a simple format:

$$ \large A \times 10^n$$

Importantly:

• $1 \leq A < 10$ (A is a number between 1 and 10)

• $n$ is an integer (positive or negative whole number)

🔮 Example of Writing a Number in Standard Form:

Example: Write 26,000 in Standard Form

1. Move the decimal point so you get a number between 1 and 10:
   2.6 (decimal moved 4 places to the left)

2. Write as standard form: Because you moved the decimal 4 places to the left, the number becomes $10^4$ times smaller. To get back to the original large number, you must multiply by $10^4$.

So, 26,000 in standard form is $2.6 \times 10^4$.

🔮 Example: Writing 0.007 in Standard Form

1. Move the decimal point so you get a number between 1 and 10:
   7 (decimal moved 3 places to the right)

2. Write as standard form: Because you moved the decimal 3 places to the right, the number becomes $10^3$ times bigger. To turn it back into the original small number, you must multiply by $10^{-3}$.

So, 0.007 in standard form is $7 \times 10^{-3}$.

Exam Tip: Moving the decimal to the right makes the number bigger, so you must use a negative power to shrink it back down.

🔮 Example: Writing 384,000 in Standard Form

1. Move the decimal point so you get a number between 1 and 10:
   3.84 (decimal moved 5 places to the left)

2. Write as standard form: Because you moved the decimal 5 places to the left, the number becomes $10^5$ times smaller. To get back to the original large number, you must multiply by $10^5$.

So, 384,000 in standard form is $3.84 \times 10^5$.

🔮 Example:

Multiply $2 \times 10^7$ by $8 \times 10^{-12}$ and write your answer in standard form.

Step 1:
Multiply the numbers and the powers of 10 separately:

• Multiply the numbers: $2 \times 8 = 16$
• Multiply the powers of 10 by adding their exponents:
  $10^7 \times 10^{-12} = 10^{7 + (-12)} = 10^{-5}$

Step 2 (Don’t forget!):
Write your answer in standard form (make sure the number is between 1 and 10):

• $16 \times 10^{-5} = 1.6 \times 10^{-4}$
• Because $16$ is not between $1$ and $10$, we write it as $1.6 \times 10^1$. Then we add the exponents: $10^1 \times 10^{-5} = 10^{-4}$, so the answer is $1.6 \times 10^{-4}$.