Number and Quantity
Core ideas
Number and Quantity questions ask you to move confidently among fractions, decimals, percents, exponents, radicals, and scientific notation. Know the real-number hierarchy: natural numbers, whole numbers, integers, rationals, and irrationals. A rational number can be written as a fraction of two integers and its decimal terminates or repeats; an irrational number such as √2 or π has a nonrepeating, nonterminating decimal. Use exponent laws fluently: a^m · a^n = a^(m+n), a^m / a^n = a^(m−n), (a^m)^n = a^(mn), and a^(−n) = 1/a^n. A fractional exponent means a^(m/n) is the nth root of a^m. Scientific notation writes a number as c × 10^n with 1 ≤ |c| < 10.
Traps
Do not distribute a radical or exponent over addition: √(9+16) is 5, not 7. A negative exponent signals a reciprocal, not a negative value. When squaring a binomial, (a+b)^2 = a^2+2ab+b^2, not a^2+b^2. Converting a small decimal to scientific notation requires a negative power of ten, and the coefficient must stay between 1 and 10. Before comparing values, rewrite them in the same form.
Worked example
Simplify 27^(2/3). Rewrite the exponent: 27^(2/3) = (the cube root of 27)^2. Since the cube root of 27 is 3, the result is 3^2 = 9. Rule of thumb: when a radical or fractional exponent looks awkward, convert it to a fractional exponent and apply exponent laws step by step.