Number - Laws of exponents and radicals

Understanding Exponents: An exponent (or index) indicates how many times a base number is multiplied by itself. In the expression $a^n$, $a$ is the base and $n$ is the exponent. Visually, you can think of $a^2$ as the area of a square with side length $a$, and $a^3$ as the volume of a cube with side length $a$.

The Multiplication and Division Laws: When multiplying two powers with the same base, you add the exponents: $a^m \cdot a^n = a^{m+n}$. When dividing, you subtract the exponents: $\frac{a^m}{a^n} = a^{m-n}$. Imagine a string of repeated multiplications being combined or cancelled out to visualize why these additions and subtractions occur.

Power of a Power: When a power is raised to another power, such as $(a^m)^n$, the exponents are multiplied to get $a^{m \cdot n}$. This represents $n$ groups of $a$ multiplied $m$ times. If you visualize a grid where each cell contains $a^m$, the total number of factors of $a$ is the product of the dimensions.

Zero and Negative Exponents: Any non-zero base raised to the power of zero is $1$ ($a^0 = 1$). A negative exponent indicates the reciprocal of the base raised to the positive version of that power ($a^{-n} = \frac{1}{a^n}$). Visually, moving from $a^1$ to $a^0$ and then $a^{-1}$ involves repeatedly dividing by the base $a$.

Rational (Fractional) Exponents: An exponent in the form of a fraction $\frac{m}{n}$ represents both a power and a root. The denominator $n$ indicates the $n$-th root, and the numerator $m$ indicates the power. For example, $a^{\frac{1}{2}}$ is the square root $\sqrt{a}$, which is the side length of a square with area $a$.

Radicals and Surds: A radical expression uses the symbol $\sqrt[n]{x}$, where $n$ is the index and $x$ is the radicand. If a root cannot be simplified to a rational number (like $\sqrt{2}$ or $\sqrt{3}$), it is called a surd. To simplify radicals, look for the largest perfect $n$-th power factor of the radicand and 'pull it out' of the radical symbol.

Scientific Notation: This is a way of writing very large or very small numbers using powers of $10$ in the form $a \times 10^k$, where $1 \le |a| < 10$. Moving the decimal point to the left increases the exponent (positive $k$ for large numbers), while moving it to the right decreases the exponent (negative $k$ for decimals).