Exhaustive Guide to Exponent Rules and Tessellations

The Product Rule ( $x^m \times x^n = x^{m+n}$ ): When multiplying two numbers that share the same base (represented here as $x$ ), the rule is to simply add the exponents together.
- Example: Given the expression $x^2 \times x^3$ , you add the exponents ( $2 + 3$ ) to find the result, which is $x^5$ .
The Quotient Rule ( $x^m \div x^n = x^{m-n}$ ): As a direct contrast to the multiplication rule, when dividing two numbers with the same base, you subtract the exponent of the denominator from the exponent of the numerator.
- Example: Given the expression $x^6 \div x^4$ , you perform the operation $6 - 4$ to arrive at the result, which is $x^2$ .
The Power to a Power Rule ( $(x^m)^n = x^{m \times n}$ ): When an expression that already contains an exponent is raised to another power, you must multiply the exponents together.
- Example and Warning: Consider the expression $(3x^2)^3$ . To solve this correctly, you must apply the power to every element inside the parentheses.
- First, calculate the coefficient: $3^3 = 27$ . Note that this is not $3 \times 3$ , but rather $3$ raised to the power of $3$ .
- Second, apply the rule to the variable: $(x^2)^3 = x^{2 \times 3}$ , which equals $x^6$ .
- The final simplified result is $27x^6$ .

Negative Exponent in the Numerator ( $x^{-n} = \frac{1}{x^n}$ ): A negative exponent indicates a reciprocal. An expression with a negative exponent in the numerator (or as a whole number) is equivalent to one over that base raised to the corresponding positive exponent.
- Place Value Context: A simpler numeric example is $10^{-1}$ , which equals $\frac{1}{10^1}$ or $\frac{1}{10}$ . In a place value table, this is precisely where we find the value $0.1$ .
- Algebraic Example: The expression $x^{-3}$ is equivalent to $\frac{1}{x^3}$ .
Negative Exponent in the Denominator ( $\frac{1}{x^{-n}} = x^n$ ): If a term has a negative exponent while residing in the denominator, it must be moved to the numerator to make the exponent positive.
- Example: The expression $\frac{1}{x^{-3}}$ is equivalent to $x^3$ (or $\frac{x^3}{1}$ ).

Multivariable Simplification Example: Consider the complex fraction $\frac{10x^{-2}y^5z^1}{5y^3z^3}$ . The simplification process involves several steps:
- Coefficients: Divide the numerical coefficients: $10 \div 5 = 2$ . The integer $2$ remains in the numerator.
- Term $x$ : The term $x^{-2}$ has a negative exponent in the numerator. According to rule four, it moves to the denominator as a positive exponent, becoming $x^2$ .
- Term $y$ : Use the Quotient Rule for $y^5 \div y^3$ . Subtract the exponents: $5 - 3 = 2$ . Since the result is positive, $y^2$ stays in the numerator.
- Term $z$ : Use the Quotient Rule for $z^1 \div z^3$ . Subtracting the exponents ( $1 - 3$ ) results in $z^{-2}$ . Because the resulting exponent is negative, it must be moved to the denominator as $z^2$ .
- Final Result: $\frac{2y^2}{x^2z^2}$ .

Definition ( $\sqrt[n]{x^m} = x^{\frac{m}{n}}$ ): This rule describes how to convert a radical (root) into a rational (fractional) exponent. The index of the root ( $n$ ) becomes the denominator of the fractional exponent, while the power the base is raised to ( $m$ ) becomes the numerator.
Example 1 (Radical to Rational): For the expression $\sqrt{x}$ , there is an understood index of $2$ (square root) and an internal exponent of $1$ for the $x$ . This converts to $x^{\frac{1}{2}}$ .
Example 2 (Rational to Radical): To convert $x^{\frac{3}{4}}$ back into radical form, move the denominator ( $4$ ) to the index position. The result is the fourth root of $x$ cubed, or $\sqrt[4]{x^3}$ .

Definition: A tessellation is a tiling pattern created when a surface is completely covered with one geometric figure or a combination of different figures in a repeating pattern.
Standard Requirements:
- There must be no overlaps between the figures.
- There must be no gaps between the figures.
Real-World Examples:
- Kitchen floor tiles.
- Bathroom floor tiles.
- Classroom floor configurations.
- Common shapes used include squares and diamonds (tilted squares).
Geometric Examples of Tessellating Shapes:
- Hexagons: These shapes successfully tessellate with no gaps.
- Parallelograms: These geometric figures can cover a surface completely in a repeating pattern.
- Squares: A common example often seen on tiled floors.

The Point-Angle Rule: A critical mathematical property of any tessellation is that the sum of the internal angles at any single vertex (point where the shapes meet) must be exactly $360^{\circ}$ .
Verification with Squares: Since every internal angle of a square is $90^{\circ}$ , four squares meeting at a point will provide $90 + 90 + 90 + 90 = 360$ , which satisfies the rule.
Verification with Hexagons: While not mathematically detailed in the transcript, it is noted that if you add up the angles where hexagons meet in a tessellation, they also sum to $360^{\circ}$ .

External Study Resources: If students find negative exponent rules or overall exponent rules confusing, they are encouraged to seek out supplementary examples online by searching for "exponent rules" or "negative exponent rules."
Immediate Next Steps:
- Students should be prepared to turn in their classroom notes.
- There will be a quiz covering this material.
- Further detailed work with tessellations will be conducted during the next class session.