Number Sense and Operations
This domain tests how well you handle the real number system, exponent properties, and scientific notation. Expect questions that ask you to classify numbers, approximate irrational values, simplify expressions with integer exponents, and compute with very large or very small quantities written in scientific notation.
Real numbers and approximation
A rational number can be written as a fraction of two integers; its decimal form terminates or repeats. An irrational number cannot. Square roots of non-perfect squares (√2, √50) and π are irrational. To estimate √50, find consecutive perfect squares: 49 < 50 < 64, so 7 < √50 < 8; since 50 is closer to 49, √50 ≈ 7.1. Use squaring to check an estimate.
Exponent rules
For nonzero base a: a^m·a^n = a^(m+n); a^m ÷ a^n = a^(m−n); (a^m)^n = a^(mn); a^0 = 1; a^(−n) = 1/a^n. These rules apply to numeric and variable bases, but not to sums: (a+b)^2 ≠ a^2+b^2.
Scientific notation
Scientific notation is a × 10^n with 1 ≤ |a| < 10. To multiply, multiply coefficients and add exponents; to divide, divide coefficients and subtract exponents. Adjust the coefficient if it falls outside [1,10). To add or subtract, rewrite both numbers with the same power of 10, then combine coefficients.
Worked example
Simplify (2.5 × 10^4)(4 × 10^−7). Multiply coefficients: 2.5 × 4 = 10. Add exponents: 10^4 × 10^−7 = 10^−3. So the product is 10 × 10^−3. Because 10 is not less than 10, rewrite as 1.0 × 10^1 × 10^−3 = 1.0 × 10^−2. Check: 25,000 × 0.0000004 = 0.01, which agrees.
Traps that cost marks
- Adding exponents when bases are added, or multiplying exponents when bases are multiplied.
- Reporting 12.8 × 10^8 instead of 1.28 × 10^9; the coefficient must be at least 1 and less than 10.
- Evaluating −3^2 as 9; exponent applies before the negative sign, so −3^2 = −9, while (−3)^2 = 9.
- Calling √36 irrational because the symbol is a radical; √36 = 6 is rational.
- Rounding an irrational estimate too early and losing the comparison.