Number and Quantity
The Number and Quantity domain on MCAS Grade 10 tests fluency with real and complex numbers: rational exponents, radicals, scientific notation, units, and the structure of the number system. Questions often ask for an equivalent form rather than a long decimal.
Core rules
- Rational exponents: a^(m/n) = the nth root of a^m = (the nth root of a)^m. The denominator is the root; the numerator is the power.
- Simplify radicals by removing perfect-square factors: √72 = √(36·2) = 6√2.
- Exponent laws: a^m·a^n = a^(m+n), (a^m)^n = a^(mn), a^(−n) = 1/a^n, and a^0 = 1 for a ≠ 0.
- Complex numbers: i = √(−1), so i² = −1. Add or subtract like terms; multiply by distributing and replacing i² with −1.
- Scientific notation: a × 10^n with 1 ≤ |a| < 10. Moving the decimal left increases the exponent; moving it right decreases it.
- Units: use dimensional analysis so units cancel and the result carries the requested unit.
Worked example
Evaluate 27^(−2/3). Rewrite the negative exponent as a reciprocal: 27^(−2/3) = 1/27^(2/3). The denominator 3 means cube root, and 27^(1/3) = 3. The numerator 2 means square: 3² = 9. So the value is 1/9.
Traps
- √(a+b) ≠ √a + √b; radicals distribute over multiplication, not addition.
- A negative exponent gives a reciprocal, not a negative value.
- √(x²) = |x|, not always x.
- In a^(m/n), do not reverse the root and the power.