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What does a number in standard form look like?
a × 10ᵏ, with 1 ≤ a < 10 and k a whole number. Example: 4.53 × 10⁴.
In a × 10ᵏ, what is the allowed range for the coefficient a?
1 ≤ a < 10 (at least 1, less than 10). So 7 × 10³ ✓ but 12 × 10³ ✗.
Big number (10 or more): is the exponent positive or negative?
Positive. Example: 52 000 = 5.2 × 10⁴.
Small number (less than 1): is the exponent positive or negative?
Negative. Example: 0.0007 = 7 × 10⁻⁴.
How do you find the exponent k?
Count how many places the point moves to leave one non-zero digit in front. Left → positive, right → negative.
Write 73 000 in standard form.
7.3 × 10⁴.
Your GDC shows 6.1ᴇ-5. What is this in standard form?
6.1 × 10⁻⁵ — the ᴇ symbol means × 10.
Why is 45.3 × 10⁶ not standard form? Fix it.
The coefficient 45.3 is not between 1 and 10. Correct: 4.53 × 10⁷.
When you multiply powers of ten, what do you do to the exponents?
Add them. Example: 10³ × 10⁴ = 10⁷.
When you divide powers of ten, what do you do to the exponents?
Subtract them. Example: 10⁸ ÷ 10³ = 10⁵.
To raise a power of ten to a power, what do you do?
Multiply the exponents. Example: (10⁴)² = 10⁸.
How do you multiply two numbers in standard form?
Multiply the coefficients, add the powers of ten, then re-normalise. Example: (3×10⁴)(2×10³) = 6×10⁷.
Find (2 × 10³)² without a calculator.
4 × 10⁶ — square the coefficient (2² = 4) and double the exponent.
A cube has edge 3 × 10² cm. Find its volume in standard form.
(3×10²)³ = 27×10⁶ = 2.7 × 10⁷ cm³.
After multiplying you get 0.5 × 10⁻³. Fix it.
5 × 10⁻⁴ — 0.5 = 5 × 10⁻¹, so subtract 1 from the exponent.
After cubing you get 27 × 10⁶. Fix it.
2.7 × 10⁷ — 27 = 2.7 × 10¹, so add 1 to the exponent.
What makes a counting question an 'arrangement'?
Order matters — the position of each object counts, so ABC and CBA are different. Count by filling positions and multiplying.
Describe the box method for arrangements.
Draw one box per position, write how many choices go in each box, then multiply. Fill the most restricted box first.
How do you count r-digit numbers with no leading zero?
The first box has 9 choices (1–9, not 0); fill the rest from the remaining digits, then multiply.
How do you arrange ALL n distinct objects in a row?
n! = n × (n − 1) × … × 1. Example: 9 people in a line = 9! = 362880.
State the formula for ⁿPᵣ and what it counts.
ⁿPᵣ = n!/(n − r)! — the number of ways to arrange r objects out of n in a definite order.
How is ⁿPᵣ just the multiplication idea?
It multiplies r numbers counting down from n: n × (n − 1) × … (r factors). e.g. ⁸P₃ = 8 × 7 × 6 = 336.
Arrange ALL of them vs SOME of them — which formula?
All n → n!. Just r of them, in order → ⁿPᵣ = n!/(n − r)!.
Seats, finishing orders, codes — arrangement or not?
Arrangements — the position matters, so use n! (all) or ⁿPᵣ (some), filling positions and multiplying.
On Paper 2, where is nPr on the TI-84?
MATH, arrow right to PRB, then 2: nPr. Type n, choose nPr, type r, ENTER.
What is ⁿPₙ equal to?
n! — arranging all n in order (since n!/(n − n)! = n!/0! = n!).
First question to ask on any counting problem?
Does order matter? Arrange/rank/roles → yes (ⁿPᵣ); choose/team/committee → no (ⁿCᵣ). Then check for restrictions.
Trigger: 'together' / 'next to each other'?
Block method — glue them into one item, arrange, then multiply by the internal orders (k!).
Trigger: 'not together' / 'not adjacent'?
Total − together (for a pair), or the gap method (seat the others, drop the rest into separate gaps).
Trigger: 'must include' / 'always chosen'?
Fix that member in place, then choose the rest from those remaining.
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