Powers & Roots

What Are Exponents? Powers, Bases and Repeated Multiplication

An exponent (or power) says how many times to multiply a number, the base, by itself: $2^5 = 2 \times 2 \times 2 \times 2 \times 2 = 32$.

The Laws of Exponents: Product, Quotient and Power Rules

The exponent laws are shortcuts for combining powers of the same base: multiply powers by adding exponents, divide by subtracting them, and raise a power to a power by multiplying them.

Zero and Negative Exponents: Why a⁰ = 1 and a⁻ⁿ = 1/aⁿ

Any non-zero number to the power zero is $1$, and a negative exponent means a reciprocal: $a^{-n} = \frac{1}{a^n}$. So $2^{-3} = \frac{1}{8}$.

Square Roots and Surds: Simplifying Radicals Step by Step

The square root $\sqrt{n}$ is the non-negative number that squares to $n$: $\sqrt{49} = 7$. A root that cannot be written as a whole number or fraction, like $\sqrt{2}$, is called a surd (or radical).

Scientific Notation (Standard Form) Explained with Examples

Scientific notation, also called standard form, writes a number as $a \times 10^n$ where $1 \le a < 10$ and $n$ is a whole number, such as $1.5 \times 10^8$ or $4.2 \times 10^{-4}$.

Fractional Exponents: Roots Written as Powers

A fractional exponent is a root written as a power: $a^{1/n} = \sqrt[n]{a}$, and $a^{m/n} = \left(\sqrt[n]{a}\right)^m$. The bottom of the fraction is the root; the top is the power.