Number Sense, Concepts, and Operations
This domain rewards fluency with the real number system and the properties that govern operations on it. You should be able to classify numbers (natural, whole, integer, rational, irrational, real, complex), convert between fractions, decimals, and percents, and reason about prime factorization, GCF, LCM, and divisibility. Order of operations, absolute value, and integer exponent rules appear constantly, along with radicals and rational exponents.
Traps to watch
- Irrational vs. rational: a radical like $\sqrt{9}$ is rational (equals 3); only non-perfect-square roots are irrational. A decimal that repeats is rational even if it looks messy.
- Negative exponents and signs: $(-2)^2 = 4$ but $-2^2 = -4$. Also $x^{-n} = 1/x^{n}$, not a negative result.
- Absolute value as distance: $|x-3|$ is distance from $x$ to 3, so $|x-3|=5$ gives two solutions.
- Percent change: "increased by 20% then decreased by 20%" does not return to the original; it nets to a 4% decrease.
Worked example / rule of thumb
Simplify $\dfrac{2^{-2}\cdot 8^{2/3}}{4^{1/2}}$. Rewrite each base as a power of 2: $2^{-2}\cdot (2^3)^{2/3} / (2^2)^{1/2} = 2^{-2}\cdot 2^{2} / 2^{1} = 2^{0}/2^{1} = 1/2$. Rule of thumb: convert every factor to a common base, then add and subtract exponents. When a problem mixes radicals and fractional exponents, convert to exponents first-it almost always simplifies the algebra.