scientific notation
8TH GRADE MATH: SCIENTIFIC NOTATION
WHAT IS SCIENTIFIC NOTATION?
Scientific notation is a way to write VERY LARGE or VERY SMALL numbers more easily.
The basic form is:
a × 10ⁿ
The number a must be:
1 ≤ a < 10
Examples:
3 × 10⁵
7.2 × 10⁻⁴
9.81 × 10⁶
WHY USE SCIENTIFIC NOTATION?
Instead of writing:
450,000,000
we can write:
4.5 × 10⁸
Instead of:
0.0000032
we can write:
3.2 × 10⁻⁶
THE EXPONENT TELLS YOU HOW FAR TO MOVE THE DECIMAL
POSITIVE EXPONENT → MOVE DECIMAL TO THE RIGHT
NEGATIVE EXPONENT → MOVE DECIMAL TO THE LEFT
Example:
4.2 × 10⁵
Move the decimal 5 places RIGHT:
4.2 → 42 → 420 → 4,200 → 42,000 → 420,000
Answer:
420,000
Example:
4.2 × 10⁻⁵
Move the decimal 5 places LEFT:
4.2 → 0.42 → 0.042 → 0.0042 → 0.00042 → 0.000042
Answer:
0.000042
MEMORY TRICK:
POSITIVE = RIGHT
NEGATIVE = LEFT
STANDARD FORM → SCIENTIFIC NOTATION
To convert a regular number into scientific notation:
STEP 1:
Move the decimal until you have ONE nonzero digit on the left.
STEP 2:
Count how many places you moved the decimal.
STEP 3:
Large number → POSITIVE exponent.
Small number → NEGATIVE exponent.
Example:
45,000
Move the decimal 4 places LEFT:
45,000 → 4.5
Therefore:
45,000 = 4.5 × 10⁴
Example:
0.00072
Move the decimal 4 places RIGHT:
0.00072 → 7.2
Therefore:
0.00072 = 7.2 × 10⁻⁴
SCIENTIFIC NOTATION → STANDARD FORM
Example:
6.3 × 10⁴
Positive exponent means move RIGHT 4 places:
6.3 → 63 → 630 → 6,300 → 63,000
Answer:
63,000
Example:
6.3 × 10⁻⁴
Negative exponent means move LEFT 4 places:
6.3 → 0.63 → 0.063 → 0.0063 → 0.00063
Answer:
0.00063
MULTIPLYING SCIENTIFIC NOTATION
When multiplying:
STEP 1:
Multiply the numbers in front.
STEP 2:
ADD the exponents.
Example:
(3 × 10⁴)(2 × 10³)
Multiply the coefficients:
3 × 2 = 6
Add the exponents:
4 + 3 = 7
Answer:
6 × 10⁷
Another example:
(4 × 10⁵)(3 × 10²)
4 × 3 = 12
5 + 2 = 7
12 × 10⁷
BUT scientific notation requires the first number to be between 1 and 10.
So:
12 × 10⁷
= 1.2 × 10⁸
DIVIDING SCIENTIFIC NOTATION
When dividing:
STEP 1:
Divide the numbers in front.
STEP 2:
SUBTRACT the exponents.
Example:
(8 × 10⁶) ÷ (2 × 10²)
8 ÷ 2 = 4
6 - 2 = 4
Answer:
4 × 10⁴
Another example:
(6 × 10⁵) ÷ (3 × 10²)
6 ÷ 3 = 2
5 - 2 = 3
Answer:
2 × 10³
ADDING SCIENTIFIC NOTATION
You CANNOT simply add the exponents.
First, make the powers of 10 the SAME.
Example:
3 × 10⁴ + 2 × 10⁴
The exponents are already the same.
Add the coefficients:
3 + 2 = 5
Answer:
5 × 10⁴
Example:
4 × 10⁵ + 3 × 10⁴
The exponents are different.
Rewrite 3 × 10⁴:
3 × 10⁴ = 0.3 × 10⁵
Now:
4 × 10⁵ + 0.3 × 10⁵
= 4.3 × 10⁵
SUBTRACTING SCIENTIFIC NOTATION
Just like addition, the exponents must be the SAME.
Example:
7 × 10⁶ - 2 × 10⁶
Subtract the coefficients:
7 - 2 = 5
Answer:
5 × 10⁶
Example:
5 × 10⁵ - 2 × 10⁴
Rewrite:
2 × 10⁴ = 0.2 × 10⁵
Then:
5 × 10⁵ - 0.2 × 10⁵
= 4.8 × 10⁵
COMPARING SCIENTIFIC NOTATION
When the exponents are the SAME:
Compare the coefficients.
Example:
3.5 × 10⁶ vs. 7.2 × 10⁶
7.2 is larger than 3.5.
Therefore:
7.2 × 10⁶ > 3.5 × 10⁶
If the exponents are DIFFERENT:
The number with the larger positive exponent is generally larger.
Example:
4 × 10⁷ vs. 4 × 10⁵
10⁷ is larger than 10⁵.
Therefore:
4 × 10⁷ > 4 × 10⁵
NEGATIVE EXPONENTS
Negative exponents represent SMALL numbers.
Examples:
10⁻¹ = 0.1
10⁻² = 0.01
10⁻³ = 0.001
10⁻⁴ = 0.0001
10⁻⁵ = 0.00001
MEMORY TRICK:
10ⁿ → BIG number when n is positive
10⁻ⁿ → SMALL number when n is negative
SCIENTIFIC NOTATION EQUATIONS
Sometimes you will have to solve an equation involving scientific notation.
Example:
x × 10⁴ = 5 × 10⁴
Since the powers of 10 are the same:
x = 5
Example:
x × 10³ = 7 × 10⁵
Divide both sides by 10³:
x = 7 × 10²
x = 700
Example:
x × 10⁻² = 4 × 10³
Divide by 10⁻²:
x = 4 × 10⁵
x = 400,000
SCIENTIFIC NOTATION AND EXPONENT RULES
Remember your exponent rules:
MULTIPLY → ADD EXPONENTS
(2 × 10³)(4 × 10⁵)
2 × 4 = 8
3 + 5 = 8
Answer:
8 × 10⁸
DIVIDE → SUBTRACT EXPONENTS
(8 × 10⁷) ÷ (2 × 10³)
8 ÷ 2 = 4
7 - 3 = 4
Answer:
4 × 10⁴
VERY IMPORTANT: SCIENTIFIC NOTATION MUST HAVE ONE DIGIT BEFORE THE DECIMAL
Correct:
4.5 × 10⁶
Correct:
7.2 × 10⁻³
Incorrect:
45 × 10⁵
Incorrect:
0.72 × 10⁶
The number in front must be:
1 ≤ a < 10
QUICK REFERENCE
CONVERT LARGE NUMBER:
Move decimal LEFT
→ POSITIVE exponent
CONVERT SMALL NUMBER:
Move decimal RIGHT
→ NEGATIVE exponent
MULTIPLY:
Multiply coefficients
+
ADD exponents
DIVIDE:
Divide coefficients
SUBTRACT exponents
ADD:
Make exponents the SAME
→ Add coefficients
SUBTRACT:
Make exponents the SAME
→ Subtract coefficients
COMPARE:
Check the exponent first
→ Then compare coefficients if needed
PRACTICE: CONVERT TO SCIENTIFIC NOTATION
45,000
7,200,000
0.00056
0.0000092
380,000,000
0.00000041
PRACTICE: CONVERT TO STANDARD FORM
3.5 × 10⁴
8.2 × 10⁶
4.7 × 10⁻³
9.1 × 10⁻⁶
PRACTICE: MULTIPLY
(2 × 10³)(4 × 10⁵)
(3 × 10⁴)(5 × 10²)
(6 × 10⁻³)(2 × 10⁴)
PRACTICE: DIVIDE
(8 × 10⁶) ÷ (2 × 10²)
(9 × 10⁷) ÷ (3 × 10³)
(6 × 10⁻²) ÷ (2 × 10³)
PRACTICE: ADD AND SUBTRACT
3 × 10⁵ + 2 × 10⁵
7 × 10⁶ - 3 × 10⁶
4 × 10⁵ + 2 × 10⁴
8 × 10⁶ - 3 × 10⁵
ANSWERS
4.5 × 10⁴
7.2 × 10⁶
5.6 × 10⁻⁴
9.2 × 10⁻⁶
3.8 × 10⁸
4.1 × 10⁻⁷
35,000
8,200,000
0.0047
0.0000091
8 × 10⁸
1.5 × 10⁷
1.2 × 10²
4 × 10⁴
3 × 10⁴
3 × 10⁻⁵
5 × 10⁵
4 × 10⁶
4.2 × 10⁵
7.7 × 10⁶
THE MOST IMPORTANT THINGS TO MEMORIZE
SCIENTIFIC NOTATION:
a × 10ⁿ
The first number must be between 1 and 10.
POSITIVE EXPONENT:
Move decimal RIGHT.
NEGATIVE EXPONENT:
Move decimal LEFT.
MULTIPLY:
Multiply coefficients.
ADD exponents.
DIVIDE:
Divide coefficients.
SUBTRACT exponents.
ADD/SUBTRACT:
Make the exponents the SAME first.
MEMORY TRICK:
BIG → POSITIVE
SMALL → NEGATIVE
MULTIPLY → ADD
DIVIDE → SUBTRACT
ADD/SUBTRACT → SAME EXPONENTS FIRST
ONE DIGIT BEFORE DECIMAL