Numbers and Algebraic Expressions
Numbers and Algebraic Expressions is the fluency domain: the Tokyo Metropolitan examination uses it to check whether you can handle integers, square roots, and symbolic manipulation quickly and accurately before the longer problems later in the paper.
Core rules
- Expand with the distributive law: a(b+c)=ab+ac. Learn the identities (a+b)²=a²+2ab+b², (a-b)²=a²-2ab+b², and (a+b)(a-b)=a²-b².
- Factorise by taking out a common factor first, then recognise a difference of squares or x²+(a+b)x+ab=(x+a)(x+b).
- For surds, use √(ab)=√a×√b and rationalise denominators by multiplying by a suitable root or conjugate. Only like radicals can be added: 2√3+5√3=7√3, but √2+√3 cannot be combined.
- Watch signs when subtracting a bracket: every term inside changes sign.
Worked example
Simplify √50 − √18 + 2/√2. First simplify each surd: √50=√(25×2)=5√2, and √18=3√2. For the fraction, multiply numerator and denominator by √2: 2/√2=(2√2)/2=√2. Then 5√2−3√2+√2=3√2. The key is to convert everything to like radicals before adding.
Traps
- Writing (x+3)² as x²+9, missing the 6x term.
- Cancelling terms rather than factors in a fraction.
- Losing a negative sign when expanding −(x−2).
- Leaving a radical in the denominator when a simplified answer is required.