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Math 30 min

Number & Quantity

Exponents, roots, ratios, units, complex numbers, vectors and matrices.

Board Buddy
Number & Quantity is the “rules of numbers” skill: exponents, roots, units, and a few oddballs like complex numbers and matrices. Learn the rules once and these become free points.

What it tests

This skill checks how well you handle numbers beyond basic arithmetic: exponent and root rules, scientific notation, factors and multiples, unit conversions, absolute value, and a few advanced topics like complex numbers, vectors and matrices. Most questions are short — if you know the rule, you can answer in under a minute.

How it shows up

  • The Math test is 45 questions in 50 minutes — a little over a minute each. Every question has four answer choices, a calculator is allowed on the whole section, there is no penalty for guessing, and no formula sheet is given.
  • Number & Quantity is one of the smaller Math categories, but its questions are often quick wins.
  • Expect things like “Which expression is equivalent to…”, simplifying radicals, a unit conversion inside a word problem, or “What is (3 − 2i)(4 + i)?”.
  • Matrix and vector questions are rare, but they’re easy points if you know the basic operations.

Key rules & concepts

The content you actually need to know.

Exponent rules

  • Multiply same base → add exponents: x3 · x4 = x7.
  • Divide same base → subtract exponents: x9 ÷ x2 = x7.
  • Power of a power → multiply exponents: (x3)2 = x6.
  • Power of a product: (3x)2 = 9x2 — the exponent hits every factor.
  • x0 = 1 (x ≠ 0) and x−n = 1 / xn. Example: 2−3 = 1/8.
  • Fractional exponents are roots: x1/2 = √x, and xm/n = (ⁿ√x)m. Example: 82/3 = (∛8)2 = 22 = 4.

Radicals

  • Simplify by pulling out perfect squares: √72 = √(36 · 2) = 6√2.
  • Only like radicals add: 3√5 + 4√5 = 7√5, but √2 + √3 does NOT equal √5.
  • √a · √b = √(ab). Example: √3 · √12 = √36 = 6.
  • Rationalize a denominator by multiplying top and bottom by the root: 6/√3 = 6√3/3 = 2√3.

Scientific notation

  • Form: a × 10n with 1 ≤ a < 10.
  • Multiply: multiply the fronts and add the exponents. (3 × 104)(5 × 106) = 15 × 1010 = 1.5 × 1011.
  • Divide: divide the fronts and subtract the exponents.

Number properties

  • A prime has exactly two factors (1 and itself). 2 is the only even prime; 1 is not prime.
  • GCF = largest number that divides both. LCM = smallest number both divide into. “When do two repeating events happen together again?” → LCM.
  • Even ± even = even, odd ± odd = even, even ± odd = odd; any integer × even = even.
  • A number is divisible by 3 (or 9) if its digit sum is divisible by 3 (or 9).
  • Absolute value is distance from zero: |−7| = 7. |x − 3| = 5 means x − 3 = 5 or x − 3 = −5, so x = 8 or x = −2.

Units and conversions

  • Multiply by conversion fractions that equal 1 and cancel units: 90 min × (1 h / 60 min) = 1.5 h.
  • Square units use the square of the conversion: 1 yd = 3 ft, so 1 yd2 = 9 ft2. 1 ft2 = 144 in2.
  • Cubic units use the cube: 1 yd3 = 27 ft3.

Complex numbers

  • i = √(−1), so i2 = −1.
  • Powers of i repeat every 4: i, −1, −i, 1. To find iⁿ, divide n by 4 and use the remainder (remainder 0 → 1, 1 → i, 2 → −1, 3 → −i).
  • Add/subtract like terms: (2 + 3i) + (4 − i) = 6 + 2i.
  • Multiply with FOIL, then replace i2 with −1.
  • The conjugate of a + bi is a − bi, and (a + bi)(a − bi) = a2 + b2 — a real number.

Matrices and vectors

  • Add or subtract matrices of the same size entry by entry; a scalar multiplies every entry.
  • Matrix multiplication (m × n)(n × p) gives an m × p matrix. The inside numbers must match.
  • Determinant of a 2 × 2 matrix [a b; c d] is ad − bc.
  • Vectors add component by component: ⟨a, b⟩ + ⟨c, d⟩ = ⟨a + c, b + d⟩. The magnitude of ⟨a, b⟩ is √(a2 + b2).

Strategy & traps

Step by step

  1. 1Name the topic first: exponents? radicals? units? complex numbers?
  2. 2Rewrite into the simplest form — same base, simplified radical, i2 → −1.
  3. 3If variables make it confusing, pick an easy number (like x = 2) and test each answer choice.
  4. 4Before choosing, double-check signs and units.

Adding exponents when you should multiply

(x3)2 = x6, not x5. Power of a power → multiply.

Distributing an exponent over addition

(a + b)2 is a2 + 2ab + b2, never just a2 + b2.

Square units with a linear factor

Square yards → square feet multiplies by 9, not 3.

Board Buddy
Board Buddy tip: Stuck on an “equivalent expression”? Plug in x = 2 to the question and every choice — the one that matches wins.

Worked examples

Try each one first, then reveal the solution.

Example 1

Which of the following is equivalent to (2x3y)2 · (3xy4)?

  • A6x6y6
  • B12x6y8
  • C36x7y6
  • D12x7y6

Example 2

For i = √(−1), what is the product (3 − 2i)(4 + i)?

  • A10 − 5i
  • B14 − 5i
  • C12 − 2i
  • D14 + 11i

Example 3

A rectangular garden plot has an area of 18 square yards. What is its area in square feet? (1 yard = 3 feet)

  • A6
  • B54
  • C162
  • D486

Practice

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Question 1

What is the value of 163/4?

Question 2

Which of the following is equivalent to √50 + √18?

Question 3

Bus A leaves the station every 12 minutes, and Bus B leaves every 18 minutes. Both buses leave together at 7:00 a.m. What is the next time they will leave together?

Question 4

For i = √(−1), what is the value of i23?

Question 5

Vectors u = ⟨3, −4⟩ and v = ⟨−1, 2⟩. What is the magnitude of 2u + v?