Numbers and Expressions
What the exam tests
Numbers and Expressions rewards exact manipulation: factoring polynomials, handling radicals and absolute values, and simplifying rational expressions without changing the domain. Kyoto entrance problems often combine these skills inside a short proof or a parameter question.
Core rules
- a^3 ± b^3 = (a ± b)(a^2 ∓ ab + b^2); a^2 - b^2 = (a - b)(a + b).
- √(x^2) = |x|, not x. For real x, √x√y = √(xy) only when x, y ≥ 0.
- Rationalize denominators by multiplying by the conjugate; state any excluded values.
- When a parameter appears, compare coefficients only after both sides are polynomials in the same variable.
Worked example
Simplify (√5 + √3)/(√5 - √3). Multiply numerator and denominator by √5 + √3: the denominator becomes 5 - 3 = 2, and the numerator becomes (√5 + √3)^2 = 8 + 2√15. Hence the expression equals 4 + √15. The denominator is nonzero because √5 ≠ √3, so no further restriction is needed.
Traps
Do not write √(a+b) = √a + √b. Do not cancel terms across a plus sign. If a square root is squared away, check the sign or domain. In coefficient comparison, first expand and collect like powers; a missing term has coefficient zero. Also check whether a simplification changes the domain.