Apply integer and rational exponent laws, simplify and operate with square and higher roots, and convert and compute with numbers in scientific notation (Number and Quantity).

What this topic is asking

The ACT Number and Quantity area tests fluency with exponents, roots and scientific notation, the machinery for writing and manipulating very large, very small and repeatedly-multiplied numbers. These rules appear on their own and inside algebra, geometry and science-context questions, so making them automatic frees time for harder reasoning elsewhere.

The laws of exponents

Every exponent question reduces to these rules.

The single most-tested confusion is product versus power: multiplying same-base powers adds exponents, while raising a power to a power multiplies them.

Simplifying roots

A radical is in simplest form when no perfect-$n$th-power factor remains under it.

The same idea works for higher roots: for a cube root, pull out perfect-cube factors. To rationalise a denominator like $\dfrac{1}{\sqrt{3}}$, multiply top and bottom by $\sqrt{3}$ to get $\dfrac{\sqrt{3}}{3}$.

Rational exponents and roots together

A rational exponent is a root: $x^{1/2} = \sqrt{x}$, $x^{1/3} = \sqrt[3]{x}$, and $x^{2/3} = \sqrt[3]{x^{2}}$. This lets you switch whichever form is easier. For example $8^{2/3} = (\sqrt[3]{8})^{2} = 2^{2} = 4$: take the root first (because $8$ is a perfect cube), then apply the power. Reading $m/n$ as "the $n$th root, raised to the $m$th power" turns an intimidating expression into two simple steps.

Scientific notation

Scientific notation writes a number as $a \times 10^{n}$ with $1 \le a < 10$ and $n$ an integer.

• A positive exponent means a large number ($3 \times 10^{5} = 300{,}000$); a negative exponent means a small number ($3 \times 10^{-4} = 0.0003$).
• To multiply: multiply the $a$ values and add the exponents, then adjust so $1 \le a < 10$.
• To divide: divide the $a$ values and subtract the exponents.

Why these rules save time

On the ACT, a question buried in a science or geometry context may hand you numbers in scientific notation precisely to test whether you can compute with them quickly. Knowing to add exponents on a product and subtract on a quotient turns a daunting expression like $(4 \times 10^{6})(2 \times 10^{-9})$ into $8 \times 10^{-3}$ in one line. The exponent and root laws are the kind of automatic skill that, once solid, never costs you a wrong answer.

Try this

Q1. Simplify $(2x^{3})^{4}$. [1 point]

• Cue. Raise each factor to the 4th: $2^{4} x^{3 \cdot 4} = 16x^{12}$.

Q2. Write $\sqrt{50}$ in simplest radical form. [1 point]

• Cue. $50 = 25 \times 2$, so $\sqrt{50} = 5\sqrt{2}$.