Number & Quantity
Number & Quantity on the Regents Algebra I exam tests reasoning with real numbers, radicals, exponents, and units. Questions often ask you to simplify a radical, rewrite a rational exponent as a radical, classify a sum or product as rational or irrational, or use units to interpret a rate. The exam supplies a reference sheet and permits a graphing calculator, so the score comes from choosing the right structure and showing the reasoning, not from arithmetic alone.
Core rules
- Simplify radicals by removing perfect-square factors: √72 = √(36·2) = 6√2. Add or subtract only like radicals.
- Rational exponents connect to roots: a^(m/n) = ⁿ√(a^m). A negative exponent means reciprocal: a^(−n)=1/a^n.
- A nonzero rational plus an irrational is irrational; a nonzero rational times an irrational is irrational. Two irrationals can multiply to a rational, as √2·√2=2.
- Treat units as algebra. In a rate, divide both the numbers and the units, and label the result.
Worked example
Simplify (√50 + √18)/√2. First factor: √50=√(25·2)=5√2 and √18=√(9·2)=3√2. The numerator is 8√2. Divide: 8√2/√2=8. The reasoning is to simplify each radical before combining, then cancel the common radical factor.
Traps
Do not split √(a+b) into √a+√b; the rule works for products, not sums. Do not drop the denominator when rationalizing. Remember that a negative exponent does not make the value negative. When a context gives units, an answer without units or with inverted units loses credit.