Comprehensive Study Note: Laws of Exponents

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Rob (Host): Hi. This is Rob. Welcome to Math Antics. In this video, we're gonna learn about the laws of exponents. If you look up the laws of exponents online or in the math book, you'll probably see a long list of equations that looks something like this. Wow. That's kind of overwhelming when you see them all at once. But don't worry. We'll take them one step at a time, and you'll see that they're not that complicated after all.

Previous prerequisite knowledge: It is highly recommended to watch the previous videos regarding the basics of exponents and "Exponents in Algebra" before proceeding.
The laws are essentially the same as "rules."
Anything raised to the first power is itself:
- $x^1 = x$
Anything raised to the zero power is equal to one:
- $x^0 = 1$

Standard exponents represent repeated multiplication for positive integer values ( $x^2$ , $x^3$ ).
Integers can also be negative. Exponents such as $x^{-1}$ , $x^{-2}$ , or $x^{-3}$ represent repeated division rather than repeated multiplication.
The Negative Exponent Law:
- $x^{-n} = \frac{1}{x^{n}}$
Interpretation:
- A negative number is the inverse of its positive counterpart, and division is the inverse operation of multiplication.
- $x^{-1} = \frac{1}{x}$
- $x^{-2} = 1 / x / x$
- $x^{-3} = 1 / x / x / x$
Fractional Representation: Mathematicians prefer to express negative exponents as a fraction where $1$ is divided by the base raised to the positive power (e.g., $x$ 's multiplied together on the bottom).

Method 1: Repeated Division Pattern
- Start with $1$ and divide by the base ( $2$ ) three times.
- $1 / 2 / 2 / 2 = 0.125$
Method 2: Fraction Form (Inverse Law)
- $2^{-3} = \frac{1}{2^3}$
- $\frac{1}{2 \times 2 \times 2} = \frac{1}{8}$
- $\frac{1}{8} = 0.125$
General Pattern:
- $x^{-1} = \frac{1}{x^1}$ or just $\frac{1}{x}$
- $x^{-2} = \frac{1}{x^2}$
- $x^{-3} = \frac{1}{x^3}$

This rule applies to a base raised to a power that is then nested within another power (like Russian nesting dolls).
The Law:
- $(x^m)^n = x^{m \times n}$
Example: Simplifying $(x^2)^3$
- The rule state we multiply exponents: $2 \times 3 = 6$ , resulting in $x^6$ .
- Verification: $(x^2)^3$ means multiplying three $x^2$ terms together: $x^2 \times x^2 \times x^2$ .
- Each $x^2$ is $x \times x$ , so: $(x \times x) \times (x \times x) \times (x \times x) = x^6$ .
Example with Negative Exponents:
- $(x^2)^{-3} = x^{2 \times -3} = x^{-6}$
- Using the negative exponent law: $x^{-6} = \frac{1}{x^6}$ .
- Verification: Rewrite $(x^2)^{-3}$ as $\frac{1}{(x^2)^3}$ , which simplifies to $\frac{1}{x^2 \times x^2 \times x^2}$ , which further simplifies to $\frac{1}{x^6}$ .

Identical Bases Multiplication Law:
- $x^m \times x^n = x^{m+n}$
- The bases must be the same, but the exponents can be different or the same.
- Example: $2^3 \times 2^4$
  - Expanding: $(2 \times 2 \times 2) \times (2 \times 2 \times 2 \times 2) = 2^7$ .
  - Shortcut: $3 + 4 = 7$ , so $2^7$ .
- Example: Combining 10 $x$ 's multiplied together can be grouped as $x^4 \times x^6 = x^{4+6} = x^{10}$ .
Identical Bases Division Law:
- $\frac{x^m}{x^n} = x^{m-n}$
- Rule: Subtract the exponent on the bottom from the exponent on the top.
- Example: $\frac{5^3}{5^2}$
  - Expanding: $\frac{5 \times 5 \times 5}{5 \times 5}$ .
  - Simplifying fractions: Cancel out two pairs of 5s ( $\frac{5}{5} = 1$ ), leaving $5 / 1$ or $5$ .
  - Shortcut: $3 - 2 = 1$ , so $5^1 = 5$ .
- Example with Negative Result: $\frac{x^4}{x^6}$
  - Shortcut: $4 - 6 = -2$ , so $x^{-2}$ .
  - Expanding: Four $x$ factors on top, six on bottom. After canceling four pairs, we are left with $1$ on top and $x \times x$ on bottom, which is $\frac{1}{x^2}$ .
  - The two results ( $x^{-2}$ and $\frac{1}{x^2}$ ) are identical according to the law of negative exponents.

In these laws, the bases are different, but the exponents are the same. This is the opposite of the previous section.
These laws involve distributing or "undistributing" common exponents.
Power of a Product Law:
- $(x \times y)^m = x^m \times y^m$
- Verification: $(x \times y)^2 = (x \times y) \times (x \times y)$ . By commutative property: $x \times x \times y \times y = x^2 \times y^2$ .
Power of a Quotient Law:
- $(\frac{x}{y})^n = \frac{x^n}{y^n}$
- Verification: $(\frac{x}{y})^2 = (\frac{x}{y}) \times (\frac{x}{y})$ . To multiply fractions, multiply tops and bottoms: $\frac{x \times x}{y \times y} = \frac{x^2}{y^2}$ .
Reversibility: These laws work in reverse (undistributing).
- $x^2 \times y^2 = (xy)^2$
- $\frac{x^2}{y^2} = (\frac{x}{y})^2$

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