Standard Form (Scientific Notation)

What is Standard Form?

Standard form is a way of writing very large or very small numbers efficiently. It always looks like this:

$$ A \times 10^n $$

A must be between 1 and 10 (e.g., $1 \le A < 10$). It can be a decimal, but it cannot be 10 or larger, and it cannot be less than 1.
n must be an integer (whole number, positive or negative).

1. Converting Numbers TO Standard Form

Large Numbers (Positive Powers)

Move the decimal point to the left until you have a number between 1 and 10. The number of jumps is your power.

Convert 45,000 to Standard Form
1. Place decimal after the first digit: $4.5$
2. Count jumps from original decimal (at the end) to new position: 4 jumps left.
$4.5 \times 10^4$

Small Numbers (Negative Powers)

Move the decimal point to the right. The number of jumps is your power, but it will be negative.

Convert 0.0032 to Standard Form
1. Place decimal after the first non-zero digit: $3.2$
2. Count jumps from original decimal to new position: 3 jumps right.
$3.2 \times 10^{-3}$

2. Converting FROM Standard Form

Positive Power: $6.2 \times 10^5$
Move decimal 5 places right.
$6.2 \rightarrow 620,000$

Negative Power: $1.4 \times 10^{-4}$
Move decimal 4 places left.
$1.4 \rightarrow 0.00014$

3. Calculating with Standard Form

Multiplication

Multiply the numbers first, then add the powers using index laws. Check your answer is still in standard form!

$$(3 \times 10^4) \times (2 \times 10^3)$$ 1. Multiply numbers: $3 \times 2 = 6$
2. Add powers: $10^4 \times 10^3 = 10^{4+3} = 10^7$
Answer: $6 \times 10^7$

Division

Divide the numbers, then subtract the powers.

$$(8 \times 10^6) \div (2 \times 10^2)$$ 1. Divide numbers: $8 \div 2 = 4$
2. Subtract powers: $10^6 \div 10^2 = 10^{6-2} = 10^4$
Answer: $4 \times 10^4$

Practice Questions & Solutions

Try these yourself, then click to reveal the solution.

Question 1: Convert 0.0000705 into Standard Form

1. Identify the first non-zero digit (7). The decimal goes after it: $7.05$

2. Count how many places the decimal moved from the start to get between the 7 and 0.

It moved 5 places to the right.

3. Since the original number was small (less than 1), the power is negative.

Answer: $7.05 \times 10^{-5}$

Question 2: Calculate $(4 \times 10^5) \times (5 \times 10^3)$

Step 1: Multiply the lead numbers
$4 \times 5 = 20$

Step 2: Add the powers
$10^5 \times 10^3 = 10^8$

Step 3: Check Standard Form
We have $20 \times 10^8$.
This is NOT in standard form because 20 is not between 1 and 10.

Step 4: Adjustment
Convert 20 to $2 \times 10^1$.
So, $(2 \times 10^1) \times 10^8 = 2 \times 10^9$

Answer: $2 \times 10^9$

Question 3: Calculate $\frac{2.4 \times 10^8}{8 \times 10^2}$

Step 1: Divide the numbers
$2.4 \div 8 = 0.3$

Step 2: Subtract the powers
$10^8 \div 10^2 = 10^6$

Step 3: Adjust to Standard Form
We have $0.3 \times 10^6$. (0.3 is too small).
Move decimal right to make it 3. Subtract 1 from the power.

Answer: $3 \times 10^5$

Question 4: Convert $5.6 \times 10^{-3}$ to an ordinary number

The power is negative (-3), so the number is small. We move the decimal 3 places to the left.

Start: $5.6$

Jump 1: $0.56$

Jump 2: $0.056$

Jump 3: $0.0056$

Answer: 0.0056

Question 5: Addition Challenge - Calculate $(3 \times 10^4) + (2 \times 10^3)$

Warning: You cannot just add the powers for addition!

Method: Convert to ordinary numbers first

$3 \times 10^4 = 30,000$
$2 \times 10^3 = 2,000$

Add them: $30,000 + 2,000 = 32,000$

Convert back to Standard Form:

Answer: $3.2 \times 10^4$