Number and Algebra - Laws of exponents and radicals

The Base and Exponent: In the expression $a^n$, $a$ is the base and $n$ is the exponent (or index). Visually, the exponent is written as a small superscript to the right of the base, indicating how many times the base is multiplied by itself. For example, $2^3$ means $2 \times 2 \times 2$.

Negative Exponents and Reciprocals: A negative exponent signifies the reciprocal of the base raised to the positive power, expressed as $a^{-n} = \frac{1}{a^n}$. Visually, as the exponent $x$ in a function like $y = 2^x$ becomes negative, the graph's height decreases, approaching the x-axis (a horizontal asymptote) without ever touching it.

The Zero Exponent Rule: Any non-zero base raised to the power of zero is defined as $1$, so $a^0 = 1$. This can be visualized through the Quotient Law: $\frac{a^n}{a^n}$ must equal $1$ because any number divided by itself is $1$, and according to exponent laws, $\frac{a^n}{a^n} = a^{n-n} = a^0$.

Fractional Exponents as Radicals: Exponents written as fractions represent roots. In $a^{\frac{m}{n}}$, the denominator $n$ indicates the 'root' or index of the radical, while the numerator $m$ indicates the power. Visually, $a^{\frac{1}{2}}$ is equivalent to the square root symbol $\sqrt{a}$, and $a^{\frac{1}{3}}$ is equivalent to the cube root symbol $\sqrt[3]{a}$.

Combining Bases: The Product and Quotient Laws only apply when the bases are identical. If you have $(2^3 \times 3^2)$, you cannot add the exponents because the bases (2 and 3) are different. Visually, terms must be grouped by like-bases before indices are simplified.

Simplifying Surds: A radical (or surd) is simplified by identifying the largest perfect square factor of the radicand. For example, $\sqrt{50}$ can be visualized as $\sqrt{25 \times 2}$. Since $25$ is a perfect square, it is moved outside the radical as its square root, resulting in $5\sqrt{2}$.

Rationalizing the Denominator: This is the process of removing a radical from the bottom of a fraction. If a fraction is $\frac{1}{\sqrt{a}}$, you multiply both the numerator and denominator by $\sqrt{a}$ to get $\frac{\sqrt{a}}{a}$. This changes the visual form of the expression without changing its numerical value.