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Apply Order Of Operations - TEAS Mathematics

  1. Simplify the expression: −2[3−(4−8)÷22]2 -2[3 - (4-8) \div 2^2]^2
    Answer: -32 We apply the order of operations (PEMDAS/BODMAS), working from the inside out. 1. Innermost Parentheses: (4−8)=−4(4-8) = -4. 2. Exponent inside brackets: 22=42^2 = 4. 3. Division inside brackets: The expression inside the brackets is now [3−(−4)÷4][3 - (-4) \div 4]. Perform the division: −4÷4=−1-4 \div 4 = -1. 4. Subtraction inside brackets: The brackets become [3−(−1)][3 - (-1)], which simplifies to 3+1=43 + 1 = 4. 5. Exponent outside brackets: The expression is now −2[4]2-2[4]^2. We apply the exponent before multiplying: 42=164^2 = 16. 6. Final Multiplication: −2×16=−32-2 \times 16 = -32.
  2. Calculate the value of the following expression: [(23−5)2×(4−1)]÷(−32) [ (2^3 - 5)^2 \times (4-1) ] \div (-3^2)
    Answer: -3 We must follow the order of operations (PEMDAS/BODMAS), paying close attention to grouping symbols and signs. 1. Innermost Parentheses: Inside the brackets, we have (23−5)(2^3 - 5) and (4−1)(4-1). We evaluate these first. - For (23−5)(2^3 - 5): exponent first, 23=82^3 = 8. Then subtraction, 8−5=38 - 5 = 3. - For (4−1)(4-1): 4−1=34 - 1 = 3. 2. Operations within Brackets: The expression inside the brackets is now [(3)2×3] [ (3)^2 \times 3 ] . - Exponent: 32=93^2 = 9. - Multiplication: 9×3=279 \times 3 = 27. 3. Exponent outside Brackets: We evaluate −32-3^2. The exponent applies only to the 3, not the negative sign. So, −32=−(3×3)=−9-3^2 = -(3 \times 3) = -9. 4. Final Division: The expression is now 27÷(−9)27 \div (-9), which equals −3-3.
  3. What is the value of 4×6−24÷4×2+62÷3−5 4 \times 6 - 24 \div 4 \times 2 + 6^2 \div 3 - 5 ?
    Answer: 19 We follow the order of operations (PEMDAS/BODMAS). 1. Exponent: First, evaluate 62=366^2 = 36. 2. Multiplication and Division: These have equal precedence, so we work from left to right. - 4×6=244 \times 6 = 24 - 24÷4=624 \div 4 = 6 - 6×2=126 \times 2 = 12 - 36÷3=1236 \div 3 = 12 The expression becomes 24−12+12−524 - 12 + 12 - 5. 3. Addition and Subtraction: These also have equal precedence, so we work from left to right. - 24−12=1224 - 12 = 12 - 12+12=2412 + 12 = 24 - 24−5=1924 - 5 = 19.
  4. Simplify the expression: −42+(−3)3−[5−(9−2)0]×2 -4^2 + (-3)^3 - [5 - (9-2)^0] \times 2
    Answer: -51 Following the order of operations (PEMDAS/BODMAS): 1. Parentheses/Brackets: Inside the brackets, we have parentheses and an exponent. First, 9−2=79-2 = 7. Then, the zero exponent: 70=17^0 = 1. Any non-zero number raised to the power of 0 is 1. The brackets become [5−1]=4[5 - 1] = 4. 2. Exponents: Evaluate the other exponents. Note that −42=−(4×4)=−16-4^2 = -(4 \times 4) = -16, as the exponent applies only to the 4. (−3)3=−27(-3)^3 = -27, as the exponent applies to -3. 3. Multiplication: The expression is now −16+(−27)−[4]×2-16 + (-27) - [4] \times 2. Perform the multiplication: 4×2=84 \times 2 = 8. 4. Addition and Subtraction (from left to right): The expression is −16−27−8-16 - 27 - 8. −16−27=−43-16 - 27 = -43. −43−8=−51-43 - 8 = -51.
  5. Simplify the expression: −52+12−2(3)42−2×5 -5^2 + \frac{12 - 2(3)}{4^2 - 2 \times 5}
    Answer: -24 We follow the order of operations (PEMDAS/BODMAS). The fraction bar acts as a grouping symbol, so we simplify the numerator and denominator first. 1. Evaluate the first term: −52-5^2. The exponent applies only to the 5, not the negative sign. So, −52=−(5×5)=−25-5^2 = -(5 \times 5) = -25. 2. Simplify the fraction's numerator: 12−2(3)12 - 2(3). Multiplication first: 2(3)=62(3) = 6. Then subtraction: 12−6=612 - 6 = 6. 3. Simplify the fraction's denominator: 42−2×54^2 - 2 \times 5. Exponent first: 42=164^2 = 16. Then multiplication: 2×5=102 \times 5 = 10. Then subtraction: 16−10=616 - 10 = 6. 4. Divide the fraction: 66=1\frac{6}{6} = 1. 5. Perform the final addition: The expression is now −25+1-25 + 1, which equals −24-24.
  6. What is the value of the expression 33−∣4−2(3+1)2∣÷4 3^3 - |4 - 2(3+1)^2| \div 4 ?
    Answer: 20 Following the order of operations (PEMDAS/BODMAS), we first address the operations inside the absolute value bars, which act as a grouping symbol. 1. Innermost Parentheses: 3+1=43+1 = 4. 2. Exponent within absolute value: 42=164^2 = 16. 3. Multiplication within absolute value: 2(16)=322(16) = 32. 4. Subtraction within absolute value: 4−32=−284 - 32 = -28. 5. Absolute Value: ∣−28∣=28|-28| = 28. Now the expression is 33−28÷43^3 - 28 \div 4. 6. Exponent: 33=273^3 = 27. 7. Division: 28÷4=728 \div 4 = 7. 8. Subtraction: 27−7=2027 - 7 = 20.
  7. What is the value of 3+4×223+4\times2^2?
    Answer: 1919 When you encounter an expression with multiple operations like this one, you need to follow the order of operations (PEMDAS/BODMAS). This means you must perform operations in a specific sequence: Parentheses, Exponents, Multiplication and Division (left to right), then Addition and Subtraction (left to right). Let's work through 3+4×223+4\times2^2 step by step. First, handle the exponent: 22=42^2 = 4. Now you have 3+4×43+4\times4. Next, perform the multiplication: 4×4=164\times4 = 16. Finally, add: 3+16=193+16 = 19. Looking at the wrong answers reveals common order of operations mistakes. Choice A (1111) results from incorrectly doing addition first: 3+4=73+4 = 7, then 7×22=7×4=287\times2^2 = 7\times4 = 28... this doesn't match, so this likely comes from a different error path. Choice C (1414) comes from forgetting to apply the exponent properly, treating 222^2 as just 22: 3+4×2=3+8=113+4\times2 = 3+8 = 11... again, this suggests multiple error combinations. Choice D (3232) results from working left to right without following proper order: (3+4)×22=7×4=28(3+4)\times2^2 = 7\times4 = 28, though the exact path to 32 might involve treating the entire expression as (3+4)×2×2(3+4)\times2\times2. The correct answer is B (1919). Remember this key strategy: When you see mixed operations, always write out PEMDAS and work through each step methodically. The TEAS frequently tests whether you'll rush and work left-to-right instead of following proper mathematical order.
  8. What is the value of 12÷(23)2−34×(13+16) \frac{1}{2} \div (\frac{2}{3})^2 - \frac{3}{4} \times (\frac{1}{3} + \frac{1}{6}) ?
    Answer: 34\frac{3}{4} We follow the order of operations (PEMDAS/BODMAS), handling each part of the expression separately before the final subtraction. First term: 12÷(23)2\frac{1}{2} \div (\frac{2}{3})^2 1. Parentheses/Exponent: (23)2=2232=49(\frac{2}{3})^2 = \frac{2^2}{3^2} = \frac{4}{9}. 2. Division: 12÷49\frac{1}{2} \div \frac{4}{9}. To divide by a fraction, multiply by its reciprocal: 12×94=98\frac{1}{2} \times \frac{9}{4} = \frac{9}{8}. Second term: 34×(13+16)\frac{3}{4} \times (\frac{1}{3} + \frac{1}{6}) 1. Parentheses/Addition: To add 13+16\frac{1}{3} + \frac{1}{6}, find a common denominator (6): 26+16=36\frac{2}{6} + \frac{1}{6} = \frac{3}{6}, which simplifies to 12\frac{1}{2}. 2. Multiplication: 34×12=38\frac{3}{4} \times \frac{1}{2} = \frac{3}{8}. Final Subtraction: Combine the results: 98−38=68\frac{9}{8} - \frac{3}{8} = \frac{6}{8}, which simplifies to 34\frac{3}{4}.
  9. Simplify the following expression: −(−2)4−[3×(5−8)]÷(−3)2 -(-2)^4 - [3 \times (5 - 8)] \div (-3)^2
    Answer: -15 To simplify the expression, we follow the order of operations (PEMDAS/BODMAS). 1. Parentheses/Brackets: First, evaluate the expression inside the parentheses: 5−8=−35 - 8 = -3. The expression becomes −(−2)4−[3×(−3)]÷(−3)2 -(-2)^4 - [3 \times (-3)] \div (-3)^2 . Then evaluate inside the brackets: 3×(−3)=−93 \times (-3) = -9. 2. Exponents: Evaluate all exponents. (−2)4=16(-2)^4 = 16 and (−3)2=9(-3)^2 = 9. The expression becomes −(16)−[−9]÷9 -(16) - [-9] \div 9 . 3. Multiplication and Division (from left to right): Perform the division: −9÷9=−1-9 \div 9 = -1. The expression is now −16−(−1)-16 - (-1). 4. Addition and Subtraction (from left to right): Perform the subtraction: −16−(−1)=−16+1=−15-16 - (-1) = -16 + 1 = -15.
  10. Evaluate 52−324 .\dfrac{5^2-3^2}{4}\,.
    Answer: 44 This problem tests your ability to follow the order of operations (PEMDAS/BODMAS) when evaluating expressions with exponents and fractions. To solve 52−324\dfrac{5^2-3^2}{4}, you must work systematically. First, evaluate the exponents in the numerator: 52=255^2 = 25 and 32=93^2 = 9. Next, perform the subtraction: 25−9=1625 - 9 = 16. Finally, divide by the denominator: 164=4\dfrac{16}{4} = 4. Looking at the answer choices, A) 44 is correct because it follows the proper order of operations throughout. B) 55 likely results from incorrectly calculating 52−325^2 - 3^2 as (5−3)2=22=4(5-3)^2 = 2^2 = 4, then somehow getting 44=1\dfrac{4}{4} = 1 and making another error, or from misreading the original expression entirely. C) 77 might come from calculating 52−32=25−9=165^2 - 3^2 = 25 - 9 = 16, then forgetting to divide by 4, and instead subtracting: 25−9−9=725 - 9 - 9 = 7. This represents confusion about the fraction bar's role as a division symbol. D) 1111 could result from adding instead of subtracting in the numerator: 52+32=25+9=345^2 + 3^2 = 25 + 9 = 34, then making an arithmetic error when dividing by 4. Remember that the fraction bar acts as both a grouping symbol and division operator. Always complete all operations in the numerator first, then divide by the denominator. Double-check your exponent calculations, as these are common sources of error on the TEAS.
  11. What is the value of the expression 48÷(−2)×3−10+52×2 48 \div (-2) \times 3 - 10 + 5^2 \times 2 ?
    Answer: -32 According to the order of operations (PEMDAS/BODMAS): 1. Exponents: First, calculate 52=255^2 = 25. 2. Multiplication and Division: These operations have the same precedence and must be performed from left to right. 48÷(−2)=−2448 \div (-2) = -24. Then, −24×3=−72-24 \times 3 = -72. On the right side, 25×2=5025 \times 2 = 50. The expression simplifies to −72−10+50-72 - 10 + 50. 3. Addition and Subtraction: These operations also have the same precedence and must be performed from left to right. −72−10=−82-72 - 10 = -82. Then, −82+50=−32-82 + 50 = -32.
  12. Evaluate the expression: 33−5×32(4−1)−22−1 \frac{3^3 - 5 \times 3}{2(4-1) - 2^2} - 1
    Answer: 5 The fraction bar acts as a grouping symbol, so we must simplify the numerator and the denominator completely before dividing. 1. Simplify the Numerator: 33−5×33^3 - 5 \times 3. First, the exponent: 33=273^3 = 27. Then multiplication: 5×3=155 \times 3 = 15. Finally, subtraction: 27−15=1227 - 15 = 12. 2. Simplify the Denominator: 2(4−1)−222(4-1) - 2^2. First, the parentheses: 4−1=34-1 = 3. Then the exponent: 22=42^2 = 4. The denominator becomes 2(3)−42(3) - 4. Next, multiplication: 2(3)=62(3) = 6. Finally, subtraction: 6−4=26 - 4 = 2. 3. Perform the Division: The fraction is now 122=6\frac{12}{2} = 6. 4. Final Subtraction: Perform the last operation: 6−1=56 - 1 = 5.
  13. Evaluate the following expression: (5+[3−2(4−6)2])2(5 + [3 - 2(4 - 6)^2])^2
    Answer: 0 We use the order of operations (PEMDAS/BODMAS), starting from the innermost grouping symbol. 1. Innermost Parentheses: (4−6)=−2(4 - 6) = -2. 2. Exponent within brackets: (−2)2=4(-2)^2 = 4. 3. Multiplication within brackets: 2(4)=82(4) = 8. 4. Subtraction within brackets: 3−8=−53 - 8 = -5. 5. Addition within outer parentheses: 5+[−5]=05 + [-5] = 0. 6. Final Exponent: 02=00^2 = 0.
  14. What is the result of 811/2×(3−1−2−2)+50 81^{1/2} \times (3^{-1} - 2^{-2}) + 5^0 ?
    Answer: 74\frac{7}{4} To evaluate this expression, follow the order of operations and rules for exponents. 1. Exponents: Evaluate each term with an exponent. 811/2=81=981^{1/2} = \sqrt{81} = 9. 3−1=133^{-1} = \frac{1}{3}. 2−2=122=142^{-2} = \frac{1}{2^2} = \frac{1}{4}. 50=15^0 = 1. 2. Parentheses: Substitute the exponent values and perform the subtraction inside the parentheses: 13−14\frac{1}{3} - \frac{1}{4}. To subtract, find a common denominator, which is 12. 412−312=112\frac{4}{12} - \frac{3}{12} = \frac{1}{12}. 3. Multiplication: The expression is now 9×112+19 \times \frac{1}{12} + 1. Multiply 9×112=9129 \times \frac{1}{12} = \frac{9}{12}, which simplifies to 34\frac{3}{4}. 4. Addition: Finally, perform the addition: 34+1=34+44=74\frac{3}{4} + 1 = \frac{3}{4} + \frac{4}{4} = \frac{7}{4}.
  15. Simplify the following expression: 13(10−4)2−5(18÷32×2) \frac{1}{3}(10 - 4)^2 - 5(18 \div 3^2 \times 2)
    Answer: -8 We follow the order of operations (PEMDAS/BODMAS) for each term. First term: 13(10−4)2\frac{1}{3}(10 - 4)^2 1. Parentheses: 10−4=610 - 4 = 6. 2. Exponent: 62=366^2 = 36. 3. Multiplication: 13(36)=12\frac{1}{3}(36) = 12. Second term: 5(18÷32×2)5(18 \div 3^2 \times 2) 1. Parentheses (evaluate inside first): a. Exponent: 32=93^2 = 9. b. Division and Multiplication (left to right): 18÷9=218 \div 9 = 2, then 2×2=42 \times 2 = 4. 2. Multiplication: 5(4)=205(4) = 20. Final Step: Combine the terms. 12−20=−812 - 20 = -8.
  16. Evaluate the expression: [(5−2)3÷32]2−(10−2×3) [ (5-2)^3 \div 3^2 ]^2 - (10 - 2 \times 3)
    Answer: 5 This problem requires careful application of the order of operations (PEMDAS/BODMAS) for two separate complex terms before the final subtraction. First term: [(5−2)3÷32]2 [ (5-2)^3 \div 3^2 ]^2 1. Innermost Parentheses: 5−2=35-2 = 3. 2. Exponents inside brackets: 33=273^3 = 27 and 32=93^2 = 9. 3. Division inside brackets: 27÷9=327 \div 9 = 3. 4. Exponent outside brackets: 32=93^2 = 9. Second term: (10−2×3)(10 - 2 \times 3) 1. Multiplication inside parentheses: 2×3=62 \times 3 = 6. 2. Subtraction inside parentheses: 10−6=410 - 6 = 4. Final Step: Subtract the value of the second term from the first. 9−4=59 - 4 = 5.
  17. Evaluate the expression: 5−[32÷(3−1)2×2−23] 5 - [32 \div (3 - 1)^2 \times 2 - 2^3]
    Answer: -3 We simplify using the order of operations (PEMDAS/BODMAS), starting with the innermost grouping symbols. 1. Innermost Parentheses: 3−1=23 - 1 = 2. 2. Exponents: Calculate all exponents. Inside the brackets, 22=42^2 = 4. Also, 23=82^3 = 8. The expression is now 5−[32÷4×2−8] 5 - [32 \div 4 \times 2 - 8] . 3. Multiplication and Division within Brackets (from left to right): First, 32÷4=832 \div 4 = 8. Then, 8×2=168 \times 2 = 16. The brackets simplify to [16−8][16 - 8]. 4. Subtraction within Brackets: 16−8=816 - 8 = 8. 5. Final Subtraction: The expression becomes 5−85 - 8, which equals −3-3.

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