PEMDAS Order of Operations Practice

Order of Operations Framework (PEMDAS)

• The order of operations defines the standardized procedural hierarchy for evaluating mathematical expressions, abbreviated as PEMDAS:

  • P (Parentheses and Grouping Symbols): Operations enclosed in grouping symbols must be evaluated first from the innermost set outward. Grouping symbols include:

  • Standard parentheses: $( \, )$

  • Square brackets: $[ \, ]$

  • Absolute value bars: $| \, |$, which also require taking the non-negative magnitude of the evaluated interior expression.

  • Fraction bars: A horizontal fraction bar acts as an implicit grouping symbol, requiring complete evaluation of the entire numerator and the entire denominator before performing division.

  • E (Exponents): Powers, exponents, and roots are evaluated immediately after all operations inside grouping symbols are completed.

  • Base sign distinction: $(-a)^n$ includes the negative sign in the exponentiation, whereas $-a^n$ evaluates the power first and then applies the negative sign.

  • Even powers of negative bases yield positive values: $(-a)^{2k} > 0$.

  • Odd powers of negative bases yield negative values: $(-a)^{2k+1} < 0$.

  • Zero exponent rule: Any non-zero real number raised to the zero power equals one ($x^0 = 1$ for $x \neq 0$).

• MD (Multiplication and Division): Evaluated from left to right in the order they appear. Neither multiplication nor division takes inherent priority over the other.

• AS (Addition and Subtraction): Evaluated from left to right in the order they appear. Subtraction is mathematically equivalent to adding the opposite (additive inverse): $a - b = a + (-b)$.

Rules for Signed Integer Operations

• Integer Addition:

  • Summing integers with the same sign: Add absolute values and preserve the common sign ($-a + (-b) = -(a + b)$).

  • Summing integers with opposite signs: Subtract the smaller absolute value from the larger absolute value and retain the sign of the number with the larger absolute value.

• Integer Subtraction:

  • Transform subtraction into addition of the opposite: $a - (-b) = a + b$ and $a - b = a + (-b)$.

• Integer Multiplication and Division:

  • Product or quotient of two numbers with identical signs is positive: $(+) \times (+) = (+)$ and $(-) \times (-) = (+)$.

  • Product or quotient of two numbers with opposing signs is negative: $(+) \times (-) = (-)$ and $(-) \times (+) = (-)$.

  • When multiplying more than two integers, sign parity depends solely on the number of negative factors:

  • An even count of negative factors yields a positive product.

  • An odd count of negative factors yields a negative product.

Multi-Step Algebraic Evaluations

• Evaluation of $-2(-15 + 8)^2$:

  • Step 1 (Parentheses): Evaluate the expression inside the parentheses:     $-15 + 8 = -7$

  • Step 2: Substitute the result back into the expression:     $-2(-7)^2$

  • Step 3 (Exponents): Evaluate the squared negative term:     $(-7)^2 = 49$

  • Step 4 (Multiplication): Multiply the leading coefficient by the squared value:     $-2(49) = -98$

  • Final value: $-98$

• Evaluation of $-12 - 3 \cdot 8 + (-1)$ (Problem j):

  • Step 1 (Multiplication): Perform multiplication prior to addition or subtraction:     $3 \cdot 8 = 24$

  • Step 2: Rewrite the expression as additions of signed terms:     $-12 - 24 + (-1) = -12 + (-24) + (-1)$

  • Step 3 (Addition Left-to-Right): Combine the first two negative terms:     $-12 + (-24) = -36$

  • Step 4 (Final Addition): Add the remaining term:     $-36 + (-1) = -37$

  • Final value: $-37$

• Evaluation of $-10 \cdot (-5)^2 - (-15)$ (Problem l):

  • Step 1 (Exponents): Compute the exponent value:     $(-5)^2 = 25$

  • Step 2 (Subtraction of Negative): Convert $- (-15)$ to addition of a positive integer:     $- (-15) = + 15$

  • Step 3 (Multiplication): Compute the product of the coefficient and the squared base:     $-10 \times 25 = -250$

  • Step 4 (Addition): Combine the terms:     $-250 + 15 = -235$

  • Final value: $-235$

• Evaluation of $-78 - 19 - 32 + (-17)$ (Problem m):

  • Step 1: Convert all subtractions into addition of negative integers:     $-78 + (-19) + (-32) + (-17)$

  • Step 2: Group and sum terms pairwise:     $-78 + (-19) = -97$     $-32 + (-17) = -49$

  • Step 3: Combine intermediate sums:     $-97 + (-49) = -146$

  • Final value: $-146$

• Evaluation with Absolute Value: $2 \cdot |8(-2) + (-4)|$ (Problem d):

  • Step 1 (Interior Multiplication): Multiply within the absolute value bars:     $8(-2) = -16$

  • Step 2 (Interior Addition): Add the terms inside the absolute value:     $-16 + (-4) = -20$

  • Step 3 (Absolute Value Evaluation): Compute the magnitude:     $|-20| = 20$

  • Step 4 (Exterior Multiplication): Multiply by the external factor:     $2 \cdot 20 = 40$

  • Final value: $40$

• Evaluation with Nested Brackets: $[(-15 + 12)^3 + (-11)] \div (-2)$ (Problem c):

  • Step 1 (Innermost Parentheses): Evaluate the addition within the parentheses:     $-15 + 12 = -3$

  • Step 2 (Exponent within Brackets): Raise the result to the third power:     $(-3)^3 = -27$

  • Step 3 (Bracket Evaluation): Add the remaining term inside the brackets:     $[-27 + (-11)] = -38$

  • Step 4 (Division): Divide the bracketed result by $-2$:     $-38 \div (-2) = 19$

  • Final value: $19$

Multi-Factor Integer Products

• Evaluation of $(-5)(-2)(2)(-1)$ (Problem e):

  • Negative factor tally: 3 negative factors (odd count), indicating a negative final product.

  • Step 1: Multiply the first pair of integers:     $(-5)(-2) = 10$

  • Step 2: Multiply the second pair of integers:     $(2)(-1) = -2$

  • Step 3: Multiply intermediate products:     $10 \times (-2) = -20$

  • Final value: $-20$

• Evaluation of $(-3)(-4)(-2)(6)$ (Problem f):

  • Negative factor tally: 3 negative factors (odd count), indicating a negative final product.

  • Step 1: Multiply the first two factors:     $(-3)(-4) = 12$

  • Step 2: Multiply the last two factors:     $(-2)(6) = -12$

  • Step 3: Multiply intermediate products:     $12 \cdot (-12) = -144$

  • Final value: $-144$

Rational Expressions with Multi-Term Numerators and Denominators

• Evaluation of $\frac{20 - 3(-2)}{-2 - 1 + 2^0}$ (Problem k):

  • Numerator Analysis (Num):

  • Expression: $20 - 3(-2)$

  • Convert to addition: $20 + (-3)(-2)$

  • Perform multiplication: $(-3)(-2) = 6$

  • Add terms: $20 + 6 = 26$

  • Denominator Analysis (Denom):

  • Expression: $-2 - 1 + 2^0$

  • Apply zero exponent rule: $2^0 = 1$

  • Convert to addition: $-2 + (-1) + 1$

  • Add from left to right: $-2 + (-1) = -3$, then $-3 + 1 = -2$

  • Division of Numerator by Denominator:     $\frac{26}{-2} = -13$

  • Final value: $-13$

• Evaluation of $\frac{(-2)(16)(-1)}{-4}$ (Problem g):

  • Step 1 (Numerator Product): Compute numerator terms:     $(-2)(16) = -32$     $-32(-1) = 32$

  • Step 2 (Denominator): Denominator is $-4$.

  • Step 3 (Division): Compute quotient of positive numerator and negative denominator:     $\frac{32}{-4} = -8$

  • Final value: $-8$

• Evaluation of $\frac{(-2)(9)}{-3(-1)}$ (Problem h):

  • Step 1 (Numerator): Multiply terms in numerator:     $(-2)(9) = -18$

  • Step 2 (Denominator): Multiply terms in denominator:     $-3(-1) = 3$

  • Step 3 (Division): Divide numerator by denominator:     $\frac{-18}{3} = -6$

  • Final value: $-6$

• Evaluation of $\frac{(-8)^2 - (-12) + (-1)^5}{-5}$ (Problem b):

  • Step 1 (Numerator Exponents):     $(-8)^2 = 64$     $(-1)^5 = -1$

  • Step 2 (Numerator Signs and Addition):

  • Subtraction of negative twelve: $- (-12) = + 12$

  • Numerator expression: $64 + 12 + (-1)$

  • Left-to-right addition: $64 + 12 = 76$, then $76 + (-1) = 75$

  • Step 3 (Division by Denominator):     $\frac{75}{-5} = -15$

  • Final value: $-15$