What is Standard Form (Scientific Notation)? [pptime time="0:01"](0:01) [/pptime]

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

How to Write Very Small Number in Standard Form? [pptime time="0:50"](0:50) [/pptime]

🔮 Example: Writing 0.007 in Standard Form Move the decimal point so you get a number between 1 and 10: 7 (decimal moved 3 places to the right) 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.

How to Write Very Large Number in Standard Form? [pptime time="1:17"](1:17) [/pptime]

🔮 Example: Writing 384,000 in Standard Form Move the decimal point so you get a number between 1 and 10: 3.84 (decimal moved 5 places to the left) 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 $3.84 \times 10^5$.

How to Multiply Numbers in Standard Form? [pptime time="1:40"](1:40) [/pptime]

🔮 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 $1.6 \times 10^{-4}$.

Standard Form

🎬 Video: How to Write Numbers in Standard Form

What is Standard Form (Scientific Notation)? (0:01)

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

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

How to Write Very Small Number in Standard Form? (0:50)

🔮 Example: Writing 0.007 in Standard Form

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

How to Write Very Large Number in Standard Form? (1:17)

🔮 Example: Writing 384,000 in Standard Form

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

How to Multiply Numbers in Standard Form? (1:40)

🔮 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 $1.6 \times 10^{-4}$.