Standard Form
Table Of Contents
🎬 Math Angel Video: How to Write Numbers in Standard Form
What is Standard Form?
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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)
🛎️ How to Write a Number in Standard Form:
Example: Write 26000 in Standard Form
Move the decimal point so that you get a number between 1 and 10: 2.6
Write as standard form: Since you moved the decimal point 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, $26000$ in standard form is $2.6 \times 10^4$.
How to Write Very Small Number in Standard Form?
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🛎️ Example: Writing 0.007 in Standard Form
Move the decimal point so that you get a number between 1 and 10: 7
Write as standard form: Since you moved the decimal point 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?
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🛎️ Example: Writing 384000 in Standard Form
Move the decimal point so you get a number between 1 and 10:
3.84Write as standard form: Since you moved the decimal point 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, $384000$ in standard form is $3.84 \times 10^5$.
How to Multiply Numbers in Standard Form?
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🛎️ 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):
- 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 correct answer is $1.6 \times 10^{-4}$.
Practice: Multiplying Numbers in Standard Form
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🛎️ Example:
Multiply $3 \times 10^{-5}$ by $40,000,000$ and write your answer in standard form.
Step 1:
Convert $40,000,000$ into standard form:
- Write $40,000,000 = 4 \times 10^7$
Step 2:
Multiply the powers of 10 by adding their exponents:
$10^{-5} \times 10^7 = 10^{-5 + 7} = 10^2$
Step 3: (Don’t forget!)
Write your answer in standard form (make sure the number is between 1 and 10):
- Because $12$ is not between $1$ and $10$, we write it as $1.2 \times 10^1$.
Then we add the exponents: $10^1 \times 10^2 = 10^3$,
so the correct answer is $1.2 \times 10^3$.
🍪 Quiz: Test Your Skills with Standard Form
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