ACT Math Number and Quantity: exponents, roots, scientific notation, complex numbers, ratios, percentages, vectors and matrices

What the Number and Quantity area covers

Number and Quantity, about 7 to 10 percent of the ACT Math test, is the foundational arithmetic of higher math: exponents and roots, the real and complex number systems, ratios and rates, percentages, and basic vectors and matrices. These skills also power the Integrating Essential Skills questions, so they pay off well beyond their share. This guide ties together the dot points: exponents, roots and scientific notation, the real and complex number systems, ratios, proportions and rates, percentages and percent change, and vectors and matrices.

Exponents, roots and scientific notation

The exponent laws are $x^{a} x^{b} = x^{a+b}$, $\frac{x^{a}}{x^{b}} = x^{a-b}$, $(x^{a})^{b} = x^{ab}$, $x^{0} = 1$, $x^{-a} = \frac{1}{x^{a}}$, and $x^{m/n} = \sqrt[n]{x^{m}}$. Simplify a root by pulling out perfect-power factors ($\sqrt{72} = 6\sqrt{2}$). Scientific notation writes $a \times 10^{n}$ with $1 \le a < 10$: multiply by multiplying the $a$ values and adding exponents, divide by dividing the $a$ values and subtracting exponents.

The real and complex number systems

The reals nest as naturals, integers, rationals (terminating or repeating decimals) and irrationals (non-repeating, like $\sqrt{2}$ and $\pi$). Absolute value is distance from zero, always non-negative. A complex number $a + bi$ uses $i^{2} = -1$: add and subtract by parts, multiply by distributing and converting $i^{2}$ to $-1$, and use the four-step cycle $i, -1, -i, 1$ for powers of $i$.

Ratios, proportions and rates

A proportion $\frac{a}{b} = \frac{c}{d}$ solves by cross-multiplying, keeping like quantities aligned. A rate compares different units; reduce to a unit rate and scale. Direct variation ($y = kx$) holds the ratio constant; inverse variation ($y = \frac{k}{x}$) holds the product constant. Convert units by multiplying by a fraction that cancels the unwanted unit.

Percentages and percent change

Treat a percent as a multiplier: $p\%$ of $N$ is $\frac{p}{100} N$; an increase multiplies by $1 + \frac{p}{100}$, a decrease by $1 - \frac{p}{100}$. Successive percentages multiply their factors. A reverse percentage divides by the multiplier. Percent change is $\frac{\text{new} - \text{old}}{\text{old}} \times 100\%$.

Vectors and matrices

Operate component by component (vectors) or entry by entry (matrices) for sums and scalar multiples. A vector's magnitude is $\sqrt{a^{2} + b^{2}}$. Matrix multiplication uses "row times column", needs matching inner dimensions, and is not commutative.

Check your knowledge

Try these, then read the solutions.

1. Simplify $\frac{x^{9}}{x^{4}}$. [1 point]
2. Compute $(2 + i)(3 - i)$. [2 points]
3. If 4 pens cost \5.20$, what do 9 pens cost at the same rate? [2 points]
4. A price rises 25% to \50$. What was the original? [2 points]
5. Find the magnitude of $\langle 9, 12 \rangle$. [1 point]