Mathematics Fundamentals: Fractions, Exponents, and Order of Operations

Fundamentals of Signed Numbers and Multiplication Rules

  • Number Line Movement and Double Negatives:

    • Movement on a number line can be modeled by directional shifts: moving backward twice or moving forward three times.

    • Taking the negative of a negative number yields a positive value: −(−x)=x-(-x) = x.

    • Example evaluation: −(−3)+2=2+3=5-(-3) + 2 = 2 + 3 = 5.

  • Rules for Multiplication of Signed Numbers:

    • minus×minus=plus\text{minus} \times \text{minus} = \text{plus} (Equivalent to (−1)×(−1)=+1(-1) \times (-1) = +1).

    • minus×plus=minus\text{minus} \times \text{plus} = \text{minus}.

    • plus×minus=minus\text{plus} \times \text{minus} = \text{minus}.

    • plus×plus=plus\text{plus} \times \text{plus} = \text{plus}.

  • Concrete Examples of Signed Multiplication:

    • (−2)×(−3)=+6(-2) \times (-3) = +6 (Using properties of negatives: −(−(2×3))=+6-(- (2 \times 3)) = +6).

    • (−2)×3=−6(-2) \times 3 = -6 (By the multiplicative associative property: −(2×3)=−6-(2 \times 3) = -6).

    • 3×(−4)=−123 \times (-4) = -12 (By rewrite: −(3×4)=−12-(3 \times 4) = -12).

    • 3 \times 4 = 12$.\n\n# Rational Numbers and Basic Fraction Operations\n\n- **Definition of Rational Numbers:**\n - A rational number is any number of the form rac{p}{q},where, wherepandandqareintegers(are integers (p, q ext{ are integers})and) andq eq 0.\n - Examples of rational numbers include rac{3}{5},, rac{4}{3},and, and rac{2}{6}.\n\n- **Addition and Subtraction of Fractions with Common Denominators:**\n - When adding fractions with equal denominators, add the numerators directly over the common denominator:\n     rac{a}{c} + rac{b}{c} = rac{a + b}{c}\n - Example: rac{3}{5} + rac{4}{5} = rac{3 + 4}{5} = rac{7}{5}.\n - **Geometric Interpretation:** Divide the unit segment from 0toto1intointo5equalpartitions(boxes).Usingequal partitions (boxes). Using3partitionunits(partition units ( rac{3}{5})andadding) and adding4morepartitionunitsyieldsmore partition units yields7partitionunitsintotal.Sincepartition units in total. Since1 = rac{5}{5},thevalue, the value rac{7}{5}representsrepresents1fullunitplusfull unit plus2 extra partition units.\n - Subtraction formula: rac{a}{c} - rac{b}{c} = rac{a - b}{c}.\n - Example: rac{2}{3} - rac{1}{3} = rac{2 + (-1)}{3} = rac{2 - 1}{3} = rac{1}{3}.\n\n- **Multiplication of Fractions:**\n - Formula for multiplying two fractions: \n     rac{a}{c} imes rac{b}{d} = rac{a imes b}{c imes d}\n - Example: rac{3}{5} imes rac{-2}{3} = rac{3 imes (-2)}{5 imes 3} = - rac{6}{15}.\n\n# Least Common Denominator (LCD) and Multiples\n\n- **Definition of the Least Common Denominator (LCD):**\n - The Least Common Denominator is the smallest positive number that is evenly divisible by all denominators in a set of rational numbers.\n\n- **Procedure for Finding the LCD:**\n - List the positive multiples of each denominator.\n - Select the smallest multiple common to all lists.\n\n- **Example 1: Finding LCD between denominators 5andand3:**\n - Multiples of 5::5, 10, 15, 20, imes\n - Multiples of 3::3, 6, 9, 12, 15, imes\n - The smallest common value is 15.Thus,. Thus, ext{LCD} = 15.\n - Note: In this case, 3 imes 5 = 15, but the LCD is not always equal to the simple product of the denominators.\n\n- **Example 2: Finding LCD between denominators 8andand12:**\n - Direct product calculation: 8 imes 12 = 96 (This is a common denominator, but not the least common denominator).\n - Multiples of 8::8, 16, 24, 32, imes\n - Multiples of 12::12, 24, 36, 48, imes\n - The smallest common value is 24.Thus,. Thus, ext{LCD} = 24$.

Addition and Subtraction of Fractions with Different Denominators

  • Method 1: Cross-Multiplication (Classic Method):

    • Formula: ab+cd=a×d+b×cb×d\frac{a}{b} + \frac{c}{d} = \frac{a \times d + b \times c}{b \times d}.

    • Example: Evaluate 38+112\frac{3}{8} + \frac{1}{12}:     3×12+1×88×12=36+896=4496\frac{3 \times 12 + 1 \times 8}{8 \times 12} = \frac{36 + 8}{96} = \frac{44}{96}

    • Simplification using common factors:     4496=4×114×24=1124\frac{44}{96} = \frac{4 \times 11}{4 \times 24} = \frac{11}{24}

    • Since 1111 is a prime number, 1124\frac{11}{24} cannot be simplified further.

  • Method 2: Equivalent Fractions via the LCD:

    • Property of Equivalent Fractions: a×cb×c=ab\frac{a \times c}{b \times c} = \frac{a}{b}.

    • Critical Rule: When converting a fraction to an equivalent denominator, multiply both the numerator and the denominator by the required factor. Multiplying only the denominator alters the value of the fraction (e.g., 33×8≠38\frac{3}{3 \times 8} \neq \frac{3}{8}).

    • Step-by-step evaluation of 38+112\frac{3}{8} + \frac{1}{12} using LCD=24\text{LCD} = 24:

    • Convert 38\frac{3}{8} by multiplying top and bottom by 33:       3×38×3=924\frac{3 \times 3}{8 \times 3} = \frac{9}{24}

    • Convert 112\frac{1}{12} by multiplying top and bottom by 22:       1×212×2=224\frac{1 \times 2}{12 \times 2} = \frac{2}{24}

    • Add equivalent fractions:       924+224=9+224=1124\frac{9}{24} + \frac{2}{24} = \frac{9 + 2}{24} = \frac{11}{24}

  • Example: Evaluating −23+−512-\frac{2}{3} + \frac{-5}{12}:

    • Using LCD (LCD=12\text{LCD} = 12):

    • Convert −23-\frac{2}{3} by multiplying numerator and denominator by 44:       −2×43×4=−812\frac{-2 \times 4}{3 \times 4} = -\frac{8}{12}

    • Combine:       −812+−512=−8−512=−1312-\frac{8}{12} + \frac{-5}{12} = \frac{-8 - 5}{12} = -\frac{13}{12}

    • Using Cross-Multiplication:

    • (−2)×12+(−5)×33×12=−24−1536=−3936\frac{(-2) \times 12 + (-5) \times 3}{3 \times 12} = \frac{-24 - 15}{36} = -\frac{39}{36}

    • Simplify by dividing top and bottom by 33:       3×(−13)3×12=−1312\frac{3 \times (-13)}{3 \times 12} = -\frac{13}{12}

  • Adding Multiple Rational Numbers:

    • Problem: Evaluate −2+32+−23-2 + \frac{3}{2} + \frac{-2}{3}.

    • Approach A: Grouping via Associative Property:

    • Rewrite −2-2 as −21\frac{-2}{1}.

    • Add first two terms: −21+32\frac{-2}{1} + \frac{3}{2}. The LCD between 11 and 22 is 22.       2×(−2)2×1+32=−4+32=−12\frac{2 \times (-2)}{2 \times 1} + \frac{3}{2} = \frac{-4 + 3}{2} = -\frac{1}{2}

    • Add third term: −12+−23-\frac{1}{2} + \frac{-2}{3}. The LCD between 22 and 33 is 66.       3×(−1)3×2+2×(−2)2×3=−36+−46=−3−46=−76\frac{3 \times (-1)}{3 \times 2} + \frac{2 \times (-2)}{2 \times 3} = \frac{-3}{6} + \frac{-4}{6} = \frac{-3 - 4}{6} = -\frac{7}{6}

    • Approach B: Global LCD for All Terms Simultaneously:

    • Denominators are 11, 22, and 33. Global LCD=6\text{LCD} = 6.

    • First term: 6×(−2)6×1=−126\frac{6 \times (-2)}{6 \times 1} = -\frac{12}{6}.

    • Second term: 3×33×2=96\frac{3 \times 3}{3 \times 2} = \frac{9}{6}.

    • Third term: 2×(−2)2×3=−46\frac{2 \times (-2)}{2 \times 3} = -\frac{4}{6}.

    • Combine numerators:       −12+9−46=−3−46=−76\frac{-12 + 9 - 4}{6} = \frac{-3 - 4}{6} = -\frac{7}{6}

Multiplication and Simplification of Fractions

  • Direct Multiplication Rule:

    • To multiply rational numbers, multiply the numerators directly and the denominators directly. Common denominators are not required.   

  • Simplification Example:

    • Expression: −2×33×(−4)\frac{-2 \times 3}{3 \times (-4)}

    • Step 1: Cancel the common factor 33 in numerator and denominator:     −2−4\frac{-2}{-4}

    • Step 2: Rewrite −4-4 as −1×4-1 \times 4 and −2-2 as −1×2-1 \times 2:     −1×2−1×4\frac{-1 \times 2}{-1 \times 4}

    • Step 3: Cancel the common factor −1-1:     24\frac{2}{4}

    • Step 4: Reduce 24\frac{2}{4} to lowest terms:     24=12\frac{2}{4} = \frac{1}{2}

Mixed Numbers and Improper Fractions

  • Definitions:

    • Improper Fraction: A fraction pq\frac{p}{q} where p>qp > q. Examples: 53\frac{5}{3}, 1110\frac{11}{10}, 1000747\frac{1000}{747}.

    • Mixed Number: A representation combining a whole number integer and a proper fraction.

  • Conversion Procedures:

    • Improper Fraction to Mixed Number:

    • Divide numerator by denominator using long division.

    • Example: 1110→11 divided by 10=1 with remainder 1→1110\frac{11}{10} \rightarrow 11 \text{ divided by } 10 = 1 \text{ with remainder } 1 \rightarrow 1\frac{1}{10}.

    • Example: 425→42 divided by 5=8 with remainder 2→825\frac{42}{5} \rightarrow 42 \text{ divided by } 5 = 8 \text{ with remainder } 2 \rightarrow 8\frac{2}{5}.

    • Mixed Number to Improper Fraction:

    • Multiply whole number by denominator, add remainder, place result over original denominator.

    • Formula: Improper Fraction=Whole×Denominator+RemainderDenominator\text{Improper Fraction} = \frac{\text{Whole} \times \text{Denominator} + \text{Remainder}}{\text{Denominator}}.

    • Example: 825=8×5+25=4258\frac{2}{5} = \frac{8 \times 5 + 2}{5} = \frac{42}{5}.

  • Operations using Mixed Numbers:

    • Subtraction Example 1: Evaluate 315−2233\frac{1}{5} - 2\frac{2}{3}.

    • Method via Improper Fractions:

      • Convert 315=3×5+15=1653\frac{1}{5} = \frac{3 \times 5 + 1}{5} = \frac{16}{5}.

      • Convert 223=2×3+23=832\frac{2}{3} = \frac{2 \times 3 + 2}{3} = \frac{8}{3}.

      • Subtract using cross-multiplication:         165−83=16×3−8×515=48−4015=815\frac{16}{5} - \frac{8}{3} = \frac{16 \times 3 - 8 \times 5}{15} = \frac{48 - 40}{15} = \frac{8}{15}

    • Subtraction Example 2: Evaluate 713−4257\frac{1}{3} - 4\frac{2}{5}.

    • Method via Improper Fractions:

      • Convert 713=7×3+13=2237\frac{1}{3} = \frac{7 \times 3 + 1}{3} = \frac{22}{3}.

      • Convert 425=4×5+25=2254\frac{2}{5} = \frac{4 \times 5 + 2}{5} = \frac{22}{5}.

      • Subtract:         223−225=22×5−22×315=110−6615=4415\frac{22}{3} - \frac{22}{5} = \frac{22 \times 5 - 22 \times 3}{15} = \frac{110 - 66}{15} = \frac{44}{15}

      • Convert back to mixed number: 44 divided by 15=2 with remainder 14→2141544 \text{ divided by } 15 = 2 \text{ with remainder } 14 \rightarrow 2\frac{14}{15}.

Properties of Exponents and Powers

  • Definition of Exponential Notation:

    • For a base aa and a positive integer exponent nn:     an=a multiplied by itself n timesa^n = \text{a multiplied by itself } n \text{ times}

    • Example: 7^4 = 7 \times 7 \times 7 \times 7$.\n\n- **Sign Rules with Parentheses:**\n - -a^n:Thenegativesignisoutsidethepoweroperation.: The negative sign is outside the power operation.-7^4 = -(7 imes 7 imes 7 imes 7).\n - (-a)^n:Thenegativebaseisinsideparentheses.: The negative base is inside parentheses.(-7)^4 = (-7) imes (-7) imes (-7) imes (-7).\n - **Even Exponent Rule:** When base aisnegativeandexponentis negative and exponentniseven,is even,(-a)^n is ALWAYS POSITIVE.\n - Example: (-2)^4 = (-2) imes (-2) imes (-2) imes (-2) = 16$.

    • Contrast: -2^4 = -(2 \times 2 \times 2 \times 2) = -16$.\n - **Odd Exponent Rule:** When base aisnegativeandexponentis negative and exponentnisodd,is odd,(-a)^n is ALWAYS NEGATIVE.\n - Example: (-2)^3 = (-2) imes (-2) imes (-2) = -8$.

  • Product Rule for Exponents:

    • am×an=am+na^m \times a^n = a^{m+n}

    • Example: 7^3 \times 7^2 = (7 \times 7 \times 7) \times (7 \times 7) = 7^{3+2} = 7^5$.\n\n- **Zero Exponent Rule:**\n - For any non-zero real number a eq 0,,a^0 = 1$.

    • Proof: 7^1 = 7^{1+0} = 7^1 \times 7^0 \rightarrow 7 = 7 \times 7^0 \rightarrow 7^0 = 1$.\n - Note: 0^0 is an indeterminate form in standard arithmetic.\n\n- **Negative Exponent Rule:**\n - a^{-n} = rac{1}{a^n}andand rac{1}{a^{-n}} = a^n\n - Proof via quotient: \n     rac{7^3}{7^3} = 1 ext{ and } rac{7^3}{7^3} = 7^3 imes 7^{-3} = 7^{3 + (-3)} = 7^0 = 1\n - **Important Note on Addition:** Exponent rules apply strictly to multiplication and division. When adding exponential expressions with identical bases and exponents, do not add exponents:\n    3^2 + 3^2 = 9 + 9 = 18 = 2 imes 3^2\n\n- **Power of a Power Rule:**\n - (a^m)^n = a^{m imes n}\n - Example: (5^2)^3 = 5^2 imes 5^2 imes 5^2 = 5^{2+2+2} = 5^6$.

    • Example: (2^3)^2 = 2^3 \times 2^3 = 2^{3+3} = 2^6 = 64$.\n - Example with negative exponents: ((2)^{-2})^{-3} = 2^{(-2) imes (-3)} = 2^6 = 64$.

    • Expansion check: ((2)^{-2})^{-3} = \frac{1}{((2)^{-2})^3} = \frac{1}{2^{-6}} = 2^6 = 64$.\n\n- **Power of a Product / Quotient Rules:**\n - (a imes b)^m = a^m imes b^m.Example:. Example:(3 imes 4)^5 = 3^5 imes 4^5$.

    • (a^m \times b^n)^p = a^{m \times p} \times b^{n \times p}$.\n - Quotient exponent rule: rac{a^m}{a^n} = a^{m-n}.Example:. Example: rac{7^3}{7^1} = 7^{3-1} = 7^2 = 49$.

Order of Operations (PEMDAS) and Evaluated Examples

  • Hierarchical Rules for Order of Operations:

    1. Parentheses: Evaluate expressions inside grouping symbols first.

    2. Exponents: Evaluate exponential expressions.

    3. Multiplication and Division: Perform in order from left to right.

    4. Addition and Subtraction: Perform in order from left to right.

  • Example 1: Evaluate 4+(3−8)24 + (3 - 8)^2

    • Step 1 (Parentheses): 3−8=−53 - 8 = -5. Expression becomes 4 + (-5)^2$.\n - Step 2 (Exponents): (-5)^2 = 25.Expressionbecomes. Expression becomes4 + 25$.

    • Step 3 (Addition): 4 + 25 = 29$.\n\n- **Example 2:** Evaluate 7 + 2 imes 3 - 4^2 imes 8\n - Step 1 (Exponents): Evaluate 4^2 = 16.Becausethenegativesignisoutsideparentheses,. Because the negative sign is outside parentheses,-4^2 = -16.Expressionbecomes. Expression becomes7 + 2 imes 3 - 16 imes 8$.

    • Step 2 (Multiplication & Division, Left to Right):

    • Multiply: 2 \times 3 = 6$.\n - Divide: 16 imes 8 = 2$.

    • Expression becomes 7 + 6 - 2$.\n - Step 3 (Addition & Subtraction, Left to Right):\n - Add: 7 + 6 = 13$.

    • Subtract: $$13 - 2 = 11$.

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