For real numbers $a, b$ and integers $m, n$, the following laws define how to manipulate expressions involving powers:

• Product Rule: $a^m \cdot a^n = a^{m+n}$ (Add exponents when multiplying like bases).
• Quotient Rule: $\frac{a^m}{a^n} = a^{m-n}$ (Subtract exponents when dividing like bases).
• Power Rule: $(a^m)^n = a^{mn}$ (Multiply exponents when raising a power to a power).
• Power of a Product: $(ab)^n = a^n b^n$.
• Power of a Quotient: $(\frac{a}{b})^n = \frac{a^n}{b^n}$.
• Zero Exponent: $a^0 = 1$ (for $a \neq 0$).
• Negative Exponent: $a^{-n} = \frac{1}{a^n}$ (Flip the base).

Scientific notation provides a compact way to write very large or very small numbers using powers of 10.

• Standard Form: A number is written as $a \times 10^n$, where $1 \le |a| < 10$ and $n$ is an integer.
• Large Numbers: Move the decimal point to the left. The exponent $n$ is positive (e.g., $4,500,000 = 4.5 \times 10^6$).
• Small Numbers: Move the decimal point to the right. The exponent $n$ is negative (e.g., $0.00032 = 3.2 \times 10^{-4}$).
• Multiplication/Division: Multiply/divide the coefficients ($a$) and use exponent rules for the powers of 10.

Rational (fractional) exponents link powers and roots. The denominator of the exponent becomes the index of the root, and the numerator becomes the power.

Special Cases:

• $a^{1/2} = \sqrt{a}$ (Square Root)
• $a^{1/3} = \sqrt[3]{a}$ (Cube Root)

• Product Property: $$\sqrt[n]{ab} = \sqrt[n]{a} \cdot \sqrt[n]{b}$$
• Quotient Property: $$\sqrt[n]{\frac{a}{b}} = \frac{\sqrt[n]{a}}{\sqrt[n]{b}}$$

In standard form, we do not leave radicals in the denominator of a fraction.

• Monomial Denominator: Multiply numerator and denominator by the root needed to complete a perfect power.
• Binomial Denominator: Multiply numerator and denominator by the conjugate. The conjugate of $a + \sqrt{b}$ is $a - \sqrt{b}$.

When solving equations involving radicals, squaring both sides can sometimes introduce false solutions that do not work in the original equation.