Number and Quantity
Number and Quantity on NYS high-school mathematics exams checks whether you can move flexibly among fractions, exponents, radicals, complex numbers, and measured quantities. Items often ask for an equivalent expression or a justified unit conversion rather than a routine calculation.
Core rules and concepts
- For a > 0, a^(m/n) = (a^(1/n))^m = (a^m)^(1/n). A negative exponent means reciprocal: a^(-p)=1/a^p.
- Radicals: sqrt(ab)=sqrt(a)sqrt(b) when a,b >=0; sqrt(x^2)=|x|, not automatically x. Rationalize denominators by multiplying by a form of 1.
- Complex numbers use i^2=-1. Add/subtract real and imaginary parts; multiply by distributing; divide by multiplying numerator and denominator by the conjugate. Write results as a+bi.
- Unit rates and dimensional analysis: multiply by conversion factors equal to 1, canceling units; check that the final unit matches the question.
Worked example
Simplify 27^(-2/3). Since 27=3^3, write (3^3)^(-2/3)=3^(3 * -2/3)=3^(-2)=1/9. Reasoning: the negative sign takes the reciprocal, and the denominator 3 takes the cube root before the numerator squares. A common alternate route is (cube root of 27)^(-2)=3^(-2)=1/9.
Traps that cost marks
- Treating a negative base with an even root as real; (-8)^(2/3)=4, but (-8)^(1/2) is not real.
- Writing sqrt(x^2)=x when x may be negative.
- Adding radicals such as sqrt(2)+sqrt(3) as sqrt(5).
- Inverting a conversion factor, which changes the unit but not the number correctly.