LAWS OF EXPONENT

Presentation Overview

• Presenter: UPLB DOST Scholars' Society

• Year: 2020

Section 1: True or False Statements

Recall Exercise

• State whether the following statements are TRUE or FALSE:

  • A. $(x^2)(y^2) = (xy)^{2+2}$

  • B. $4^3 = 3 imes 3 imes 3 imes 3$

  • C. $-567^0 = 1$

  • D. $5a^1 = rac{1}{5a}$

  • E. $(24^2)^4 = (24)^6$

Section 2: Understanding Exponents

Definition of Exponents

• Exponent: A mathematical notation indicating the number of times a number (the base) is multiplied by itself.

Exponents Concepts

• Example of Exponents:

  • Base: 2

  • $2 imes 2 imes 2 = 8$

  • Incorrect example:

  • $3 imes 3$

  • Incorrect interpretation: $2 imes 3$

Section 3: Laws of Exponents

1. Multiplication of Powers with Common Bases

• Rule: When multiplying powers with the same base, add the exponents.

  • Formula: $(A^m)(A^n) = A^{m+n}$

  • Example:

  • $(4^3)(4^2) = 4^{3+2} = 4^5 = 1024$

  • Note: This rule is NOT applicable for exponents with different bases.

2. Multiplication of Powers with Different Bases but Same Exponents

• Rule: Multiply the bases and raise the product to the common exponent.

  • Formula: $(A^m)(B^m) = (AB)^m$

  • Example:

  • $(8^3)(5^3) = (8 imes 5)^3 = 40^3 = 64000$

  • Note: Known as the Power of a Product Rule.

3. Division of Powers with Common Bases

• Rule: When dividing powers with the same base, subtract the exponents.

  • Formula: $(A^m)/(A^n) = A^{m-n}$

  • Example:

  • $(4^5)/(4^2) = 4^{5-2} = 4^3 = 64$

  • Note: This rule is NOT applicable for exponents with different bases.

4. Power of a Fraction Rule

• Rule: Distribute the exponent to both the numerator and denominator.

  • Formula: $( rac{A}{B})^n = rac{A^n}{B^n}$

  • Example:

  • $( rac{3}{2})^2 = rac{3^2}{2^2} = rac{9}{4}$

5. Power of a Power

• Rule: Multiply the exponents when raising a power to a power.

  • Formula: $(A^m)^n = A^{mn}$

  • Example:

  • $(5^3)^2 = 5^{3 imes 2} = 5^6 = 15625$

6. Zero Exponent

• Rule: Any non-zero number raised to the power of zero is equal to one.

  • Formula: $(A^0 = 1, ext{ where } A <br>eq 0)$

7. Inverse Rule/Negative Exponent

• Rule: A negative exponent indicates the reciprocal of the base raised to the opposite positive exponent.

  • Formula: $(A^{-m} = rac{1}{A^m})$

  • Example:

  • $(6^{-5} = rac{1}{6^5}) = rac{1}{7776}$

8. Fractional Exponent

• Definition: A fractional exponent represents a root.

  • Formula: $(A^{m/n} = <br>oot{n}{A^m})$

  • Example:

  • $(5^{2/3} = <br>oot{3}{5^2})$

Section 4: True/False Revisions

• State whether the following statements are TRUE or FALSE and clarify false claims.

  • A. $(x^2)(y^2) = (xy)^{2 + 2}$ (FALSE)

  • B. $43 = 3 imes 3 imes 3 imes 3$ (FALSE)

  • C. $-567^0 = 1$ (TRUE)

  • D. $5a^1 = [1/(5a)]$ (FALSE)

  • E. $(24^2)^4 = (24)^6$ (FALSE)

Section 5: Special Products and Factoring

• Definition of a Polynomial: A polynomial of degree $n$ is a function of the form:

  • $f(x) = a<em>nx^n + a</em>{n-1}x^{n-1} + … + a<em>2x^2 + a</em>1x + a_0$

  • where the coefficients $a$ are real numbers.

Restrictions on Polynomials

• A polynomial cannot have:

  • A negative exponent at the variable

  • A fractional exponent at the variable

  • The variable inside a radical

  • Division by a term containing the variable

Special Products

1. Square of a Binomial

   • Formula: $(a \bm{+} b)^2 = a^2 + 2ab + b^2$

   • Example: $(x + 1)^2 = x^2 + 2x + 1$

2. Product of Sum & Difference of Two Binomials

   • Formula: $(a + b)(a - b) = a^2 - b^2$

   • Example: $(x + 1)(x - 1) = x^2 - 1$

3. Square of Trinomial

   • Formula: $(a + b + c)^2 = a^2 + b^2 + c^2 + 2ab + 2ac + 2bc$

   • Example: $(5x - 3y + 2z)^2 = 25x^2 + 9y^2 + 4z^2 - 30xy + 20xz - 12yz$

4. Sum and Difference of Two Cubes

   • Formula: $(a^3 \bm{+} b^3) = (a + b)(a^2 - ab + b^2)$

   • Example:

     • $(27x^3 + y^3) = (3x + y)(9x^2 - 3xy + y^2)$

Common Methods/Techniques in Factoring Polynomials

1. Special Product Factorization

   • Example:

     • $x^2 - y^2 = (x + y)(x - y)$

2. Greatest Common Factor

   • Definition: The greatest number (or exponential variable) that is a factor of two or more terms.

   • Example:

     • $10x + 20y = 10(x + 2y)$

3. Factoring Trinomials

   • Definition: A trinomial consists of three terms $ax^2 + bx + c$. If a trinomial has a highest degree of two, the quadratic formula can be used.

Practice Examples

1. Completely factor examples such as $5x^2 - 20y^2$, $x^3 + 8y^3$, $6x^6 + 6y^6$, $2x^2 + 2y^2$.

Section 6: Functions and Relations

Definitions

• Function: A rule that associates each element $x$ from set $A$ with exactly one element $y$ from set $B$.

• Relation: A set of ordered pairs.

Properties of Functions

• All functions are relations; however, not all relations are functions.

Operations on Functions

• Addition: $(f + g)(x) = f(x) + g(x)$

• Subtraction: $(f - g)(x) = f(x) - g(x)$

• Multiplication: $(f g)(x) = f(x) g(x)$

• Division: $(f / g)(x) = f(x) / g(x)$

• Composite Function: $(f \bm{g})(x)$ involving substitution of functions.

Evaluation of Functions

• To evaluate a function, substitute the given input value into the function. For example:

  • If $f(x) = x^2 + 1$ and $x = 3$, then:

  • $f(3) = 3^2 + 1 = 9 + 1 = 10$

Section 7: Linear Equations

Standard Form

• Standard Equation: $ax + by = c$; where $a, b, c$ are constants ($a > 0$).

Slope-Intercept Form

• Equation: $y = mx + b$

  • $m$ = slope

  • $b$ = y-intercept

Point-Slope Form

• Equation: $y - y<em>1 = m(x - x</em>1)$; where $m$ is slope and $(x<em>1, y</em>1)$ is a point on the line.

Slope Calculation

• $m = rac{y<em>2 - y</em>1}{x<em>2 - x</em>1}$; where $(x<em>1, y</em>1)$ and $(x<em>2, y</em>2)$ are points on the line.

Common Problems

Example Problem: Finding the Equation of a Line Given Two Points

• Given points (1, 2) and (4, 5):

  1. Calculate the slope: $m = rac{5 - 2}{4 - 1} = 1$

  2. Using $y = mx + b$, find $y$-intercept with one of the points:

  • For the point (1, 2): $2 = 1(1) + b <br>ightarrow b = 1$

  1. Substitute $m$ and $b$ into the slope-intercept form $y = mx + b$:

  • $y = 1x + 1$ or $y = x + 1$

Practice Problems

• Determine the equation of a line passing through points (-5, 6) and (2, -8).

Parallel and Perpendicular Lines

• Parallel Lines have the same slope; Perpendicular Lines have slopes that are negative reciprocals of each other.

Reminder on Linear Equation Calculation Techniques

1. Practice rewriting equations in slope-intercept form.

2. Identify and use patterns relevant to special products and factoring.

3. Check for mistakes, especially during subtraction of functions.

Conclusion

• The study of exponents, functions, and linear equations is essential for developing problem-solving skills in algebra.

• Understanding these foundational concepts will prepare you for more advanced mathematical challenges.

Contact Information

• For questions about the content of this guide, contact the UPLB DOST Scholars' Society at their email: uplbdostscholarssociety@gmail.com.