Number and Quantity: Real Numbers, Exponents, and Precision
What this domain tests
Number and Quantity questions ask you to reason about the real number system and to compute accurately with it. You should be fluent with rational and irrational numbers, ordering and comparing values, absolute value as distance on the number line, and the rules for integer and rational exponents. Know that $a^{m/n} = \sqrt[n]{a^m}$, that $a^{-k} = 1/a^k$, and that $a^0 = 1$ for any nonzero $a$. Be comfortable converting between decimals, fractions, and percents, and with scientific notation, including multiplying and dividing numbers written as $k \times 10^n$.
Traps to watch for
A common trap is treating $\sqrt{a+b}$ as $\sqrt{a}+\sqrt{b}$; radicals do not distribute over addition. Another is confusing $-3^2$ (which equals $-9$) with $(-3)^2$ (which equals $9$). Questions about rational versus irrational numbers often hinge on recognizing that sums such as $\sqrt{2}+2$ remain irrational, while $\sqrt{2}\cdot\sqrt{2}$ is rational. Rounding and precision items may ask for the appropriate number of significant figures, so do not over-report digits. When comparing negative numbers, remember the order reverses: $-7 < -2$, and $|-7| > |-2|$.
Worked example
Simplify $\dfrac{8^{2/3}}{2^{-1}}$. First, $8^{2/3} = (\sqrt[3]{8})^2 = 2^2 = 4$. The denominator $2^{-1} = \tfrac12$. So the expression is $4 \div \tfrac12 = 8$. As a rule of thumb, rewrite every base as a power of the same small prime before combining exponents; this turns messy radical and fraction problems into simple exponent arithmetic.