Use order of operations to evaluate expressions - TACHS Math

1. Evaluate $\{12 - 2[3 + 1(4 - 2)]\} \times 0$.
   Answer: 0 When you encounter complex expressions with multiple operations, the order of operations (PEMDAS/BODMAS) is crucial, but there's also a powerful shortcut to recognize here. Let's work through this step by step. First, tackle the innermost parentheses: $(4 - 2) = 2$. Next, multiply: $1(2) = 2$. Then evaluate the brackets: $3 + 2 = 5$, so $2[5] = 10$. Moving to the braces: $12 - 10 = 2$. Finally, we have $2 \times 0 = 0$. However, here's the key insight: whenever you see any expression multiplied by zero, the result is always zero, regardless of how complex the expression inside the brackets might be. You could have spent time carefully working through all those nested operations, but recognizing that anything times zero equals zero immediately gives you the answer. Choice (A) 4 might tempt students who incorrectly evaluate the expression inside the braces as 4, forgetting about the multiplication by zero. Choice (C) -4 could catch students who make sign errors during the nested operations and again ignore the zero multiplication. Choice (D) 8 might result from computational errors in the bracket evaluations while missing the zero factor. Strategy tip: On the TACHS, when you see an expression multiplied or divided by zero, stop and apply the zero property immediately. Multiplication by zero always yields zero, saving you time and reducing calculation errors. This pattern appears frequently on standardized tests as both a time-saver and a trap for students who dive into complex calculations unnecessarily.
2. Evaluate $7 + 3 \times (12 - 8)^2 \div 6$.
   Answer: 15 When you encounter an expression with multiple operations, you must follow the order of operations (PEMDAS): Parentheses, Exponents, Multiplication and Division (left to right), Addition and Subtraction (left to right). Let's work through $7 + 3 \times (12 - 8)^2 \div 6$ step by step: First, handle the parentheses: $(12 - 8) = 4$ Next, apply the exponent: $4^2 = 16$ Now our expression becomes: $7 + 3 \times 16 \div 6$ For multiplication and division, work left to right: $3 \times 16 = 48$, then $48 \div 6 = 8$ Finally, add: $7 + 8 = 15$ The answer is A) 15. Let's see where the wrong answers come from. B) 31 likely results from adding before multiplying and dividing, giving $10 \times 4^2 \div 6 = 160 \div 6 ≈ 27$, though this doesn't quite match. C) 39 probably comes from calculating $7 + 3 \times 4^2 = 7 + 48 = 55$ and making an error, or from $7 + 32 = 39$ by using $3 \times 4^2 = 3 \times 8$ instead of $3 \times 16$. D) 63 results from ignoring order of operations entirely and working left to right: $(7 + 3) \times (12 - 8)^2 \div 6 = 10 \times 16 \div 6$. Remember: PEMDAS errors are common traps on standardized tests. Always work through operations in the correct order, and double-check your arithmetic at each step.
3. Evaluate $\dfrac{2\left[5^{2} - (4 + 1)\right]}{3^{2}}$.
   Answer: $\dfrac{40}{9}$ This problem tests your ability to follow the order of operations (PEMDAS) when evaluating complex expressions with fractions, exponents, and grouping symbols. Let's work through this step by step. First, evaluate the expression inside the brackets: $5^{2} - (4 + 1)$. Start with the parentheses: $4 + 1 = 5$. Then calculate the exponent: $5^{2} = 25$. So we have $25 - 5 = 20$. Next, multiply by 2: $2[20] = 40$. Finally, divide by $3^{2}$: $3^{2} = 9$, giving us $\dfrac{40}{9}$, which is answer choice D. Let's examine why the other answers are incorrect. Choice A ($\dfrac{35}{9}$) likely comes from miscalculating $5^{2} - (4 + 1)$ as $25 - 4 - 1 = 20$, then somehow getting 35 in the numerator—perhaps by adding instead of subtracting somewhere. Choice B ($\dfrac{20}{9}$) results from forgetting to multiply by 2 after finding $5^{2} - (4 + 1) = 20$. Choice C ($\dfrac{25}{3}$) comes from multiple errors: not subtracting 5 from 25 (keeping just $5^{2} = 25$), then incorrectly calculating $3^{2}$ as 3 instead of 9. Remember to work systematically through order of operations: handle what's inside grouping symbols first, then exponents, then multiplication and division from left to right. Double-check each step, especially with exponents, as these are common sources of calculation errors.
4. Evaluate $\left( \dfrac{3}{4} + \dfrac{1}{2} \right) \times 8$.
   Answer: 10 This problem tests your ability to follow the order of operations when working with fractions and whole numbers. When you see parentheses in an expression, always handle what's inside them first before moving to other operations. Start by adding the fractions inside the parentheses: $\frac{3}{4} + \frac{1}{2}$. To add fractions, you need a common denominator. Since 4 is a multiple of 2, convert $\frac{1}{2}$ to fourths: $\frac{1}{2} = \frac{2}{4}$. Now you can add: $\frac{3}{4} + \frac{2}{4} = \frac{5}{4}$. Next, multiply this result by 8: $\frac{5}{4} \times 8 = \frac{5 \times 8}{4} = \frac{40}{4} = 10$. Choice A (12) likely comes from incorrectly adding $\frac{3}{4} + \frac{1}{2} = \frac{4}{6}$ (adding numerators and denominators separately) and then multiplying by 8. Choice B (6) might result from converting to decimals incorrectly: thinking $\frac{3}{4} = 0.3$ instead of 0.75, giving 0.8 × 8 = 6.4, rounded down. Choice C (8) could come from mishandling the order of operations—perhaps multiplying 8 by just one of the fractions first. The key strategy here is to always work inside parentheses first and find common denominators when adding fractions. Converting mixed operations with fractions to improper fractions often makes multiplication easier than working with decimals.
5. Evaluate $(18 \div 3) - (2^{3} - 5) \times 2$.
   Answer: 0 When you encounter complex expressions with multiple operations, the order of operations (PEMDAS/BODMAS) is crucial. You must work through parentheses first, then exponents, then multiplication and division from left to right, and finally addition and subtraction from left to right. Let's evaluate $(18 \div 3) - (2^{3} - 5) \times 2$ step by step. First, handle the parentheses: $18 \div 3 = 6$ and $2^{3} - 5 = 8 - 5 = 3$. This gives us $6 - 3 \times 2$. Next, perform the multiplication: $3 \times 2 = 6$. Finally, subtract: $6 - 6 = 0$. Looking at the wrong answers: Choice B (4) likely comes from incorrectly calculating $18 \div 3 - 2$ by mishandling the second parenthetical expression. Choice C (6) results from either calculating just $18 \div 3$ and stopping, or from working left to right without following order of operations: $(6 - 3) \times 2 = 6$. Choice D (12) probably comes from adding instead of subtracting: $6 + 3 \times 2 = 6 + 6 = 12$. The correct answer is A) 0. Strategy tip: When you see complex expressions, write out each step clearly and double-check that you're following PEMDAS strictly. The most common errors on order of operations questions come from rushing through the steps or working purely left to right. Take your time with parentheses and exponents first.
6. Evaluate $[3(2+5) - 4]^{2} \div 17$.
   Answer: 17 This question tests your ability to follow the order of operations (PEMDAS/BODMAS) when evaluating complex expressions with grouping symbols, exponents, and division. Let's work through this step-by-step. Start with the innermost parentheses: $2 + 5 = 7$. Now the expression becomes $[3(7) - 4]^{2} \div 17$. Next, multiply inside the brackets: $3 \times 7 = 21$, giving us $[21 - 4]^{2} \div 17$. Subtract within the brackets: $21 - 4 = 17$, so we have $[17]^{2} \div 17$. Apply the exponent: $17^{2} = 289$, leaving us with $289 \div 17$. Finally, divide: $289 \div 17 = 17$. Looking at the wrong answers: Answer B (34) might result from forgetting to square the 17 and just multiplying $17 \times 2 = 34$. Answer C (51) could come from incorrectly calculating $17 \times 3 = 51$, perhaps from a computational error earlier in the process. Answer D (289) is what you'd get if you stopped after computing $17^{2}$ and forgot the final division step. The key strategy here is to work methodically through the order of operations without skipping steps. Write out each intermediate result to avoid losing track of where you are in the calculation. Many students rush through these problems and make careless errors in the final steps, so double-check your arithmetic at each stage.
7. Evaluate $\sqrt{36} + 4^{2} \div 2^{3}$.
   Answer: 8 When you encounter expressions with multiple operations like this one, you need to apply the order of operations (PEMDAS) systematically: Parentheses, Exponents, Multiplication/Division (left to right), Addition/Subtraction (left to right). Let's work through $\sqrt{36} + 4^{2} \div 2^{3}$ step by step. First, handle the exponents and square root: $\sqrt{36} = 6$, $4^{2} = 16$, and $2^{3} = 8$. This gives us $6 + 16 \div 8$. Next, perform the division before addition: $16 \div 8 = 2$. Finally, add: $6 + 2 = 8$. Looking at the wrong answers: Choice B (10) likely comes from adding $6 + 4$ and ignoring the division entirely. Choice C (14) results from incorrectly adding $6 + 16$ without performing the division by $2^{3}$. Choice D (20) might occur if someone incorrectly calculated $4^{2} \div 2^{3}$ as $4 \div 2 = 2$, then multiplied by something, or made multiple computational errors. The correct answer is A) 8. Study tip: On order of operations problems, work methodically through PEMDAS and write out each step. The most common mistakes happen when students rush through the sequence or forget that division comes before addition. Always double-check your exponent calculations, as $2^{3} = 8$, not 6, and $4^{2} = 16$, not 8.
8. Evaluate $-4^{2} + 6 \times (3 - 1)$.
   Answer: -4 When you encounter expressions with multiple operations, order of operations (PEMDAS) is crucial. You must carefully evaluate exponents, handle parentheses, and watch for negative sign placement. Let's work through $-4^{2} + 6 \times (3 - 1)$ step by step. First, note that $-4^2$ means "the negative of 4 squared," not "negative 4, quantity squared." So $-4^2 = -(4^2) = -16$. Next, evaluate the parentheses: $(3-1) = 2$. Then perform the multiplication: $6 \times 2 = 12$. Finally, add: $-16 + 12 = -4$. Choice A (-28) likely comes from incorrectly treating $-4^2$ as $(-4)^2 = 16$, then somehow getting the operations wrong to reach a large negative number. Choice B (20) suggests the student calculated $(-4)^2 = 16$, then added $16 + 6 \times 2 = 16 + 12 = 28$, but made a sign error. Choice D (4) probably results from the same $(-4)^2 = 16$ mistake, giving $16 + 6 \times 2 = 28$, then somehow subtracting to get 4, or calculating $16 - 12 = 4$. The correct answer is C (-4). Key strategy: Pay close attention to negative signs with exponents. $-a^2$ always means "negative of (a squared)," while $(-a)^2$ means "negative a, quantity squared." This distinction frequently appears on standardized tests and is a common source of errors.
9. Evaluate $\dfrac{2^{3}\,\cdot\,5\; -\; (7 - 3)}{4 + 1}$.
   Answer: 7.2 This problem tests your ability to apply the order of operations (PEMDAS) correctly when evaluating complex expressions with multiple operations. Let's work through this step-by-step using PEMDAS: Parentheses, Exponents, Multiplication/Division (left to right), Addition/Subtraction (left to right). Starting with $\frac{2^{3}\,\cdot\,5\; -\; (7 - 3)}{4 + 1}$: First, handle parentheses and exponents: $2^3 = 8$ and $(7-3) = 4$. The denominator becomes $4 + 1 = 5$. Next, perform multiplication: $8 \cdot 5 = 40$. Then subtraction in the numerator: $40 - 4 = 36$. Finally, divide: $\frac{36}{5} = 7.2$. Choice A (8.0) likely comes from incorrectly calculating just $2^3$ and ignoring the rest of the operations. Choice B (7.8) might result from computational errors in the multiplication or division steps. Choice C (6.2) could occur if you incorrectly subtract before multiplying, getting $(8-4) \cdot 5 = 20$ in the numerator, then $\frac{20}{5} + 1.2 = 6.2$ through faulty reasoning. The correct answer is D (7.2). Strategy tip: When you see complex expressions like this, write out each step of PEMDAS clearly. Don't try to do multiple operations mentally at once—this leads to the kinds of errors that create the wrong answer choices. Always double-check your arithmetic, especially with exponents and multi-step calculations.
10. Evaluate $20 - [3(4 - 2)]^{2} \div 9$.
    Answer: 16 When you encounter complex expressions with multiple operations, the order of operations (PEMDAS/BODMAS) is your roadmap. This problem tests whether you can systematically work through parentheses, exponents, multiplication/division, and addition/subtraction in the correct sequence. Start with the innermost parentheses: $4 - 2 = 2$. Now the expression becomes $20 - [3(2)]^{2} \div 9$. Next, handle the multiplication inside the brackets: $3(2) = 6$, giving you $20 - [6]^{2} \div 9$. Apply the exponent: $6^{2} = 36$, so now you have $20 - 36 \div 9$. Perform the division: $36 \div 9 = 4$. Finally, subtract: $20 - 4 = 16$. Choice A (4) would result from incorrectly calculating just the final division $36 \div 9 = 4$ without completing the subtraction. Choice C (8) might come from miscalculating $20 - 36 \div 9$ as $20 - 9 - 4$ or making an error with the exponent. Choice D (12) could result from computing $(20 - 36) \div 9$ incorrectly or mishandling the order of operations by grouping operations that shouldn't be grouped together. The correct answer is B (16). Remember: when facing multi-step expressions, work systematically through PEMDAS. Write out each step clearly to avoid rushed mistakes, especially with nested parentheses and exponents. The TACHS often includes answer choices that represent common calculation errors, so double-check your order of operations.
11. Evaluate $5^{2} - 4 \times 3 + 6 \div 2$.
    Answer: 16 When you encounter an expression with multiple operations like this one, you need to follow the order of operations (PEMDAS/BODMAS): Parentheses, Exponents, Multiplication and Division (left to right), then Addition and Subtraction (left to right). Let's work through $5^{2} - 4 \times 3 + 6 \div 2$ step by step: First, handle the exponent: $5^{2} = 25$ Next, perform multiplication and division from left to right: $4 \times 3 = 12$ and $6 \div 2 = 3$ Now your expression becomes: $25 - 12 + 3$ Finally, work left to right with addition and subtraction: $25 - 12 = 13$, then $13 + 3 = 16$ The answer is C) 16. Let's see where the wrong answers come from. Choice A) 17 likely results from incorrectly calculating $5^{2} - 4 \times 3 + 6 \div 2$ as $25 - 12 + 4 = 17$, perhaps from miscomputing $6 \div 2$. Choice B) 6 probably comes from working strictly left to right without following order of operations: $(5^{2} - 4) \times 3 + 6 \div 2 = 21 \times 3 + 3$, then making calculation errors. Choice D) 28 might result from incorrectly grouping operations or computational mistakes. Remember: Order of operations questions are common on standardized tests. Always write out each step clearly and double-check your arithmetic. The most frequent error is ignoring the proper sequence, so memorize PEMDAS and practice applying it systematically.
12. Evaluate $\dfrac{9 + 3 \times 2}{5 - 2^{2}}$.
    Answer: 15 When you see a complex fraction like this, you're being tested on the order of operations (PEMDAS/BODMAS). The key is working systematically through each part before attempting the final division. Start with the numerator: $9 + 3 \times 2$. Following order of operations, multiplication comes before addition, so calculate $3 \times 2 = 6$ first, then $9 + 6 = 15$. For the denominator: $5 - 2^2$. Exponents come before subtraction, so calculate $2^2 = 4$ first, then $5 - 4 = 1$. Now you have $\frac{15}{1} = 15$, which is answer choice D. The wrong answers represent common order of operations mistakes. Choice A (-15) likely comes from incorrectly calculating the denominator as $5 - 2^2 = (5-2)^2 = 9$, then getting $\frac{15}{9}$ and somehow arriving at a negative result. Choice B (3) might result from working left to right in the numerator without following order of operations: $(9 + 3) \times 2 = 24$, then making errors in the denominator. Choice C (5) could come from calculating $\frac{9 + 3 \times 2}{5 - 2^2}$ as $\frac{15}{3}$ by miscomputing the denominator as 3 instead of 1. Remember: when evaluating expressions with multiple operations, always follow PEMDAS strictly. Work through parentheses first, then exponents, then multiplication and division from left to right, and finally addition and subtraction from left to right. Take your time with each step to avoid careless errors.

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