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Absolute value and scientific notation
Two short topics that show up often: absolute value, which is just a distance, and scientific notation, which is just a tidy way to write very big and very small numbers. Each has one rule and one trap.
Absolute value is a distance
The bars | | mean “how far from zero.” A number and its opposite are the same distance away: |−5| = |5| = 5. Absolute value just drops the sign.
Scientific notation: one digit, then a power of 10
A number like 4,500,000 is written 4.5 × 10⁶: a front number from 1 up to (not including) 10, times a power of 10 that says how far the decimal point moved.
How to solve them
-
1
|x| is the distance from zero
Positive numbers stay the same, negative numbers lose their minus sign, and zero stays zero.
Next step: Inside firstA negative|−8|
8
A positive|8|
8
A difference|3 − 10|
|−7| = 7
-
2
Work inside the bars first, then drop the sign
Absolute value bars act like parentheses: finish the calculation inside, then take the absolute value.
Next step: Two answers- |4 − 9| + |−2|
- |−5| + |−2|
- 5 + 2
- 7
-
3
|x| = 6 has two answers
Two numbers are 6 away from zero: 6 and −6. So an equation with an absolute value splits into two equations.
Next step: Scientific notationExampleSolve |x − 3| = 5
- Either x − 3 = 5 or x − 3 = −5.
- x = 8 or x = −2.
- Check: |8 − 3| = 5 and |−2 − 3| = |−5| = 5.
Answer x = 8 or x = −2
-
4
Writing a number in scientific notation
Move the decimal point until one nonzero digit is in front of it. Count the moves: that is the exponent. Big numbers get a positive exponent, small numbers (below 1) a negative one.
Next step: Multiply & divideA big number4,500,000
decimal moves 6 left → 4.5 × 10⁶
A small number0.00072
decimal moves 4 right → 7.2 × 10⁻⁴
Back to normal3.1 × 10³
move 3 right → 3,100
-
5
Multiply: multiply the fronts, add the exponents
Handle the front numbers and the powers of 10 separately. To divide, divide the fronts and subtract the exponents.
Next step: Fix the front- (3 × 10⁴)(2 × 10⁵)
- (3 × 2) × 10⁴⁺⁵
- 6 × 10⁹
Divide(8 × 10⁷) ÷ (2 × 10³)
(8 ÷ 2) × 10⁷⁻³ = 4 × 10⁴
-
6
Keep the front number between 1 and 10
If the front number ends up 10 or more, move its decimal one place left and add 1 to the exponent.
- (5 × 10³)(4 × 10²)
- 20 × 10⁵
- 2 × 10¹ × 10⁵
- 2 × 10⁶
4 traps in the answer choices
The test writes its wrong answers on purpose: each one is what you get if you make one of these mistakes.
Absolute value term by term
|3 − 8|:
Wrong|3| − |8| = −5
Right|−5| = 5
Only one answer to |x| = 6
Solve |x| = 6:
Wrongx = 6
Rightx = 6 or x = −6
Moving the decimal the wrong way
0.0045 in scientific notation:
Wrong4.5 × 10³
Right4.5 × 10⁻³
Multiplying the exponents
10³ × 10⁴:
Wrong10¹²
Right10⁷
5 questions at test level
Pick an answer, then read why. No timer, no account.
What is |−12| − |5|?
Show the answer
C. Take each absolute value first: |−12| = 12 and |5| = 5. Then 12 − 5 = 7. −17 subtracts before dropping the sign. The answer is 7.
Which number is a solution of |x + 2| = 9?
Show the answer
B. x + 2 = 9 gives x = 7, and x + 2 = −9 gives x = −11. Of the choices, only −11 works: |−11 + 2| = |−9| = 9. The answer is −11.
Write 0.00063 in scientific notation.
Show the answer
C. Move the decimal point 4 places right to get 6.3, and the number is small, so the exponent is negative: 6.3 × 10⁻⁴. 63 × 10⁻⁵ has the right value but a front number above 10. The answer is 6.3 × 10⁻⁴.
(4 × 10⁵) × (2 × 10³) = ?
Show the answer
B. Multiply the fronts: 4 × 2 = 8. Add the exponents: 5 + 3 = 8. 8 × 10¹⁵ multiplies the exponents, and 6 × 10⁸ adds the fronts. The answer is 8 × 10⁸.
(9 × 10⁸) ÷ (3 × 10²) = ?
Show the answer
D. Divide the fronts: 9 ÷ 3 = 3. Subtract the exponents: 8 − 2 = 6. 3 × 10⁴ divides the exponents, and 3 × 10¹⁰ adds them. The answer is 3 × 10⁶.
10 flashcards
Tap a card to flip it. Keep the ones you miss until they stick.
All 10 cards known.
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