Order of Operations - SHSAT Math

1. Compute $\dfrac{(4+2)^{2}}{3}$.
   Answer: 12 This question tests your understanding of order of operations (PEMDAS/BODMAS) and careful arithmetic computation. When evaluating $\dfrac{(4+2)^{2}}{3}$, you must follow the correct sequence: parentheses first, then exponents, then division. Start by computing what's inside the parentheses: $(4+2) = 6$. Next, apply the exponent: $6^{2} = 36$. Finally, divide by 3: $\dfrac{36}{3} = 12$. Looking at the wrong answers reveals common computational errors. Choice B (24) likely comes from incorrectly computing $6^{2} = 24$ instead of 36, or from making an arithmetic mistake in the final division. Choice C (36) represents stopping after finding $6^{2} = 36$ but forgetting to complete the division by 3. This is a classic "incomplete calculation" trap. Choice D (6) suggests either computing just the parentheses $(4+2) = 6$ and stopping there, or making multiple errors that coincidentally yield 6. The correct answer is A (12), which follows from the complete and accurate calculation: $\dfrac{(4+2)^{2}}{3} = \dfrac{6^{2}}{3} = \dfrac{36}{3} = 12$. Strategy tip: On order of operations problems, write out each step clearly and don't skip ahead mentally. Many students make errors by trying to do too much at once. Also, when you see answer choices that represent intermediate steps in your calculation (like 6 and 36 here), that's a signal to double-check that you've completed the entire problem.
2. What is the value of $\dfrac{9-3}{2}+4(1.5)$ ?
   Answer: 9 This problem tests your ability to apply the order of operations (PEMDAS) correctly when evaluating expressions with multiple operations. Let's work through $\dfrac{9-3}{2}+4(1.5)$ step by step. First, handle what's inside the parentheses and fraction bar: $9-3=6$, so we have $\dfrac{6}{2}+4(1.5)$. Next, perform division and multiplication from left to right: $\dfrac{6}{2}=3$ and $4(1.5)=6$. Finally, add the results: $3+6=9$. Looking at the wrong answers: Choice B (6) likely comes from only calculating one part of the expression, such as just $\dfrac{9-3}{2}=3$ and $4(1.5)=6$, then choosing the larger value instead of adding them. Choice C (4.5) might result from incorrectly calculating $4(1.5)$ as $4+1.5$ instead of $4 \times 1.5$, giving $3+1.5=4.5$. Choice D (12) could come from multiplying the two parts instead of adding them: $3 \times 4=12$, or from incorrectly computing $\dfrac{9-3}{2}+4(1.5)$ as $\dfrac{9-3}{2+4}(1.5)$. The correct answer is A (9). Study tip: When evaluating expressions with multiple operations, write out each step clearly and double-check that you're following PEMDAS correctly. The most common errors on these problems come from rushing through the order of operations or making small arithmetic mistakes.
3. What is the value of $6 + 3 \times 4 - 5$ ?
   Answer: 13 When you encounter an expression with multiple operations like this one, you need to follow the order of operations (PEMDAS/BODMAS). This means you perform multiplication and division before addition and subtraction, working from left to right. Let's work through $6 + 3 \times 4 - 5$ step by step. First, identify that you have addition, multiplication, and subtraction. According to order of operations, you must do the multiplication first: $3 \times 4 = 12$. Now your expression becomes $6 + 12 - 5$. Next, work from left to right with the remaining operations: $6 + 12 = 18$, then $18 - 5 = 13$. The answer is 13, which is choice A. Let's see where the wrong answers come from. Choice B (17) likely results from incorrectly doing $6 + 3 = 9$, then $9 \times 4 = 36$, then $36 - 5 = 31$ — wait, that's not 17 either. Actually, B probably comes from $6 + 3 \times 4 = 6 + 12 = 18$, then forgetting to subtract 5, giving you $18 - 1 = 17$ through a calculation error. Choice C (23) results from working strictly left to right without following order of operations: $6 + 3 = 9$, then $9 \times 4 = 36$, then $36 - 5 = 31$ — actually, this suggests 23 might come from $6 \times 4 - 1 = 23$ through misreading. Choice D (9) comes from $6 + 4 - 5 = 5$ if you somehow skip the 3 entirely. Remember: multiplication and division always come before addition and subtraction. When you see mixed operations, circle the multiplication and division first, solve those, then work left to right with what remains.
4. What is the value of $2^{3} + (4 - 1)^{2}$ ?
   Answer: 17 This question tests your ability to apply the order of operations (PEMDAS) while evaluating expressions with exponents and parentheses. When you see an expression like this, work systematically through each operation in the correct order: Parentheses, Exponents, Multiplication/Division (left to right), Addition/Subtraction (left to right). Let's evaluate $2^{3} + (4 - 1)^{2}$ step by step: First, handle the parentheses: $(4 - 1) = 3$ Next, evaluate the exponents: $2^{3} = 8$ and $3^{2} = 9$ Finally, add: $8 + 9 = 17$ Looking at the wrong answers: Choice A (11) likely comes from calculating $2^{3} + (4 - 1) = 8 + 3 = 11$ — forgetting to square the result in parentheses. Choice C (23) might result from incorrectly calculating $2^{3} = 8$ but then doing $8 + (4 - 1)^{3} = 8 + 27 = 35$ or making other computational errors. Choice D (27) could come from calculating $(4 - 1)^{3} = 27$ instead of $(4 - 1)^{2} = 9$, then subtracting $2^{3}$ instead of adding it. The correct answer is B (17). Study tip: When evaluating expressions with multiple operations, write out each step clearly. Don't try to do everything in your head — small errors in order of operations or arithmetic can easily lead you to a tempting wrong answer that appears among the choices.
5. Refer to the table to answer the question. What is the value of $2x^{2}-yz$ when the given values are used?
   Answer: 28 From the table, $x=3,\;y=-2,\;z=5$. First square $x$: $x^{2}=9$. Multiply: $2\times9=18$. Next, $yz=(-2)(5)=-10$. Subtract: $18-(-10)=18+10=28$. B: Stops after calculating $2x^2$ without subtracting $yz$. C: Incorrectly computes $18-10=8$ instead of $18-(-10)$. D: Only calculates $yz=-10$ without the rest of the expression.
6. What is the value of $2^3 \cdot 3^2 - 4(5 + 2^2) + 3 \cdot 2^4 \div 4$?
   Answer: $48$ Following order of operations: First evaluate exponents: $2^3 = 8$, $3^2 = 9$, $2^2 = 4$, $2^4 = 16$. Then parentheses: $(5 + 4) = 9$. Now we have $8 \cdot 9 - 4(9) + 3 \cdot 16 \div 4$. Next, multiplication and division from left to right: $8 \cdot 9 = 72$, $4(9) = 36$, $3 \cdot 16 = 48$, $48 \div 4 = 12$. Finally, addition and subtraction from left to right: $72 - 36 + 12 = 48$. Choice A results from calculation errors in the exponents. Choice B comes from forgetting to add the final term $12$. Choice D results from treating division with lower precedence than addition.
7. Find the value of $3\{4+2[5-(3+1)]\}$.
   Answer: 18 When you encounter expressions with multiple grouping symbols like brackets, braces, and parentheses, you must work from the innermost groupings outward, following the order of operations (PEMDAS/BODMAS). Let's solve $3\{4+2[5-(3+1)]\}$ step by step: Start with the innermost parentheses: $(3+1) = 4$ Now the expression becomes: $3\{4+2[5-4]\}$ Next, work inside the brackets: $5-4 = 1$ This gives us: $3\{4+2[1]\}$ or $3\{4+2 \times 1\}$ Multiply inside the braces: $2 \times 1 = 2$ Now we have: $3\{4+2\} = 3\{6\}$ Finally: $3 \times 6 = 18$ The answer is A) 18. Let's examine why the other choices are incorrect: B) 30 likely results from multiplying all the numbers without properly following order of operations: $3 \times 2 \times 5 = 30$. C) 6 comes from forgetting to multiply by the outer 3, stopping at $4+2 = 6$. D) 12 occurs if you incorrectly calculate the inner operations, perhaps getting $3 \times 4 = 12$ instead of working through all the groupings properly. Study tip: When facing nested grouping symbols, always work inside-out. Draw or rewrite the expression after each step to avoid losing track. The SHSAT often tests whether you can carefully follow order of operations through multiple levels of groupings.
8. What is the value of $4 + 3[2^2 + 5(3 - 1)] - 6$?
   Answer: $40$ When you encounter a complex expression with multiple operations, the order of operations (PEMDAS) is your roadmap. Work systematically: Parentheses first, then Exponents, then Multiplication and Division from left to right, finally Addition and Subtraction from left to right. Let's break down $4 + 3[2^2 + 5(3 - 1)] - 6$ step by step: First, handle the innermost parentheses: $(3 - 1) = 2$ Now we have: $4 + 3[2^2 + 5(2)] - 6$ Next, the exponent: $2^2 = 4$ This gives us: $4 + 3[4 + 5(2)] - 6$ Continue with multiplication inside the brackets: $5(2) = 10$ So: $4 + 3[4 + 10] - 6$ Simplify inside the brackets: $4 + 10 = 14$ Now: $4 + 3[14] - 6$ Multiply: $3[14] = 42$ Finally: $4 + 42 - 6 = 40$ The answer is D) $40$. Here's why the other choices are wrong: A) $38$ likely comes from incorrectly calculating $4 + 42 - 8$ (perhaps misreading the final 6). B) $52$ probably results from adding instead of subtracting at the end: $4 + 42 + 6$. C) $46$ comes from forgetting to subtract 6 entirely: $4 + 42$. Strategy tip: Write out each step when working with order of operations. Don't try to skip steps mentally—small arithmetic errors in complex expressions are easy to make but costly on the SHSAT.
9. What is the value of $\dfrac{3(4+5)}{2}-1$ ?
   Answer: 12.5 This question tests your understanding of the order of operations (PEMDAS/BODMAS) - the sequence you must follow when evaluating mathematical expressions with multiple operations. When you see an expression like this, work systematically through the order of operations: Parentheses, Exponents, Multiplication/Division (left to right), Addition/Subtraction (left to right). Let's evaluate $\dfrac{3(4+5)}{2}-1$ step by step: First, handle the parentheses: $(4+5) = 9$ Next, perform the multiplication and division from left to right: $3 \times 9 = 27$, then $\dfrac{27}{2} = 13.5$ Finally, subtract: $13.5 - 1 = 12.5$ Looking at the wrong answers: Choice B (13.5) represents stopping after the division step and forgetting to subtract 1 - a common error when students lose track of the final operation. Choice C (26) likely comes from incorrectly calculating $3(4+5) = 3 \times 9 = 27$, then subtracting 1 to get 26, but forgetting to divide by 2. Choice D (35) appears to result from adding instead of following proper order: $3 + 4 + 5 + 2 - 1 = 13$, or some other computational error. The correct answer is A (12.5). Strategy tip: Always write out each step of the order of operations, especially on standardized tests. Many wrong answer choices are specifically designed to catch students who skip steps or work too quickly through the calculation sequence.
10. Compute $\dfrac{2+4}{3}\times(7-2^{2})$.
    Answer: 6 When you encounter an expression with multiple operations like this, success depends on following the correct order of operations (PEMDAS): Parentheses, Exponents, Multiplication/Division (left to right), Addition/Subtraction (left to right). Let's work through $\dfrac{2+4}{3}\times(7-2^{2})$ step by step. First, handle what's inside parentheses and the fraction's numerator. In the numerator: $2+4=6$. In the second parentheses, you need the exponent first: $2^{2}=4$, then $7-4=3$. So now you have $\dfrac{6}{3}\times 3$. Next, perform the division: $\dfrac{6}{3}=2$. Finally, multiply: $2\times 3=6$. Looking at the answer choices, (A) 6 is correct. The wrong answers likely come from order of operations errors. (B) 4 might result from calculating $2^{2}=4$ and stopping there, or from incorrectly combining terms. (C) 3 could come from evaluating just the second parentheses $(7-2^{2})=3$ or making an arithmetic mistake in the final steps. (D) 9 might result from adding instead of multiplying the final terms, giving $2+3+4=9$, or from other calculation errors. The key strategy here is to slow down and carefully apply PEMDAS. Many students rush through order of operations problems and make careless mistakes. Write out each step clearly, and double-check that you're handling exponents before subtraction and completing all operations inside parentheses before moving to multiplication.
11. Find the value of $|5-2\times4|-3$.
    Answer: 0 This question tests your understanding of order of operations and absolute value. When you see expressions with multiple operations and absolute value bars, you must carefully follow the correct sequence of calculations. Let's work through $|5-2\times4|-3$ step by step. First, handle what's inside the absolute value bars using order of operations (PEMDAS). Within $|5-2\times4|$, multiplication comes before subtraction, so: $2\times4 = 8$, giving us $|5-8| = |-3|$. The absolute value of $-3$ is $3$. Finally, subtract: $3-3 = 0$. Now let's examine why each answer choice might be tempting. Choice A (0) is correct based on our calculation above. Choice B (3) represents stopping after finding $|5-2\times4| = 3$ but forgetting to subtract the final 3. Choice C (5) likely comes from incorrectly doing subtraction before multiplication inside the absolute value bars, calculating $|5-2|\times4-3 = 3\times4-3 = 9$, though this doesn't directly give 5. Choice D (-3) results from forgetting to apply the absolute value, calculating $5-2\times4-3 = 5-8-3 = -6$, or making other order of operations errors. The key strategy here is to work methodically: handle parentheses and absolute value bars as grouping symbols, apply order of operations within them, then continue from left to right. Always double-check that you've applied absolute value correctly—it always produces a non-negative result.
12. What is $\left(\dfrac{6}{3}\right)^{2}+4\times2$ ?
    Answer: 12 This question tests your ability to apply the order of operations (PEMDAS) correctly when evaluating mathematical expressions. When you see an expression with multiple operations, you must follow the proper sequence: Parentheses, Exponents, Multiplication and Division (left to right), then Addition and Subtraction (left to right). Let's work through $\left(\dfrac{6}{3}\right)^{2}+4\times2$ step by step: First, handle the parentheses: $\dfrac{6}{3} = 2$ Next, apply the exponent: $2^2 = 4$ Then perform the multiplication: $4 \times 2 = 8$ Finally, add: $4 + 8 = 12$ Looking at the wrong answers: Choice B (16) likely comes from incorrectly calculating $2^2 + 4 \times 2$ as $2 + 4 = 6$, then $6^2 + 8 = 36 + 8$, but this shows confusion with order of operations. Choice C (20) might result from adding first: $(2 + 4) \times 2 + 8$, ignoring that exponents come before addition. Choice D (24) could come from treating the entire expression as $(2 + 4) \times 4$, completely misapplying the order of operations. The correct answer is A (12). Strategy tip: When you see complex expressions on the SHSAT, always write out each step of PEMDAS clearly. Don't try to do multiple operations in your head at once—this leads to the exact mistakes that create the wrong answer choices. Work methodically through each operation in the correct order.
13. Evaluate $12-3\left[2^{2}-(1-3)\right]$.
    Answer: -6 This problem tests your ability to apply the order of operations (PEMDAS) correctly when evaluating complex expressions with nested operations. Let's work through $12-3\left[2^{2}-(1-3)\right]$ step by step. Start with the innermost parentheses: $(1-3) = -2$. Next, evaluate the exponent: $2^2 = 4$. Now the expression becomes $12-3[4-(-2)]$. Inside the brackets: $4-(-2) = 4+2 = 6$. So we have $12-3[6] = 12-18 = -6$. Looking at the answer choices, choice A gives us $-6$, which matches our calculation. Choice B ($6$) likely comes from forgetting the negative sign when subtracting $18$ from $12$. Choice C ($18$) occurs if you calculate $3 \times 6 = 18$ but forget to subtract it from $12$. Choice D ($-18$) results from computing $3 \times 6 = 18$, correctly getting the negative result of $12-18$, but then incorrectly applying another negative sign. The key trap in this problem is the double negative when evaluating $4-(-2)$. Many students rush through this step and miss that subtracting a negative number means adding the positive. Strategy tip: When working with order of operations, write out each step completely and double-check your work with parentheses and negative signs. These are the most common sources of error on SHSAT algebra problems. Always verify your final answer by substituting it back or checking the reasonableness of your result.
14. Evaluate $\dfrac{(2+3)(4-1)}{2^{2}}$.
    Answer: 3.75 When you see a complex fraction with multiple operations, always follow the order of operations (PEMDAS): Parentheses, Exponents, Multiplication/Division, Addition/Subtraction. Start by evaluating what's inside the parentheses: $(2+3) = 5$ and $(4-1) = 3$. Next, handle the exponent in the denominator: $2^2 = 4$. Now your expression becomes $\frac{5 \times 3}{4} = \frac{15}{4}$. To convert this improper fraction to a decimal, divide: $15 ÷ 4 = 3.75$. So the answer is A) 3.75. Let's see where the wrong answers come from. Choice B) 6 likely results from incorrectly calculating the denominator as 2 instead of $2^2 = 4$, giving $\frac{15}{2} = 7.5$, then making an arithmetic error. Choice C) 7.5 comes from that same denominator mistake: treating $2^2$ as just 2, so $\frac{15}{2} = 7.5$. Choice D) 12 probably comes from multiplying the numerator results without dividing by the denominator at all: $5 \times 3 = 15$, then making a calculation error, or possibly adding instead of multiplying: $(2+3) + (4-1) + 2^2 = 5 + 3 + 4 = 12$. Remember: when you see exponents mixed with other operations, don't rush. Exponents come before multiplication and division in the order of operations, and $2^2$ means $2 \times 2 = 4$, not just 2. Double-check your fraction-to-decimal conversion by multiplying back.
15. Compute $9-4^{2}\div8$.
    Answer: 7 When you encounter expressions with multiple operations, order of operations (PEMDAS/BODMAS) is crucial. You must perform operations in the correct sequence: Parentheses, Exponents, Multiplication and Division (left to right), then Addition and Subtraction (left to right). Let's work through $9-4^{2}\div8$ step by step. First, handle the exponent: $4^2 = 16$. The expression becomes $9-16\div8$. Next, perform the division: $16\div8 = 2$. Finally, subtract: $9-2 = 7$. Looking at the answer choices, (A) 7 is correct because we followed the proper order of operations. Choice (B) 2 likely comes from incorrectly doing $9-4 = 5$, then $5^2 = 25$, then $25\div8 = 3.125$, which doesn't match, or from some other order mix-up. Choice (C) 1 might result from calculating $4^2\div8 = 16\div8 = 2$, then mistakenly doing $9\div2 - 7 = 1$ or similar errors. Choice (D) 25 comes from the common mistake of ignoring order of operations entirely and working left to right: $9-4 = 5$, then $5^2 = 25$, completely forgetting about the division by 8. Strategy tip: Always write out each step when working with order of operations. Don't try to do multiple steps mentally—exponents and division/multiplication before addition/subtraction. This systematic approach prevents the costly mistakes that create these attractive wrong answers on the SHSAT.
16. What is the value of $-2^{3}+3^{2}$ ?
    Answer: 1 This question tests your understanding of order of operations and how negative signs interact with exponents—a common source of confusion on standardized tests. Let's work through $-2^{3}+3^{2}$ step by step. The key insight is recognizing that $-2^{3}$ means "the negative of $2^{3}$," not "negative 2, cubed." According to order of operations, exponents come before multiplication (and applying a negative sign is multiplication by -1). First, calculate the exponents: $2^{3} = 8$ and $3^{2} = 9$. Then apply the negative sign: $-2^{3} = -8$. Finally, add: $-8 + 9 = 1$. Looking at the wrong answers: Choice B (-1) likely comes from calculating $-8 + 9$ incorrectly or from misunderstanding the order of operations. Choice C (17) suggests someone calculated $(-2)^{3} + 3^{2} = -8 + 9$ but then made an error, possibly getting $8 + 9 = 17$ by dropping the negative. Choice D (-17) indicates the student correctly found $8 + 9 = 17$ but incorrectly applied a negative sign to the final result. The correct answer is A) 1. Study tip: When you see a negative sign before an exponent without parentheses, remember that the exponent applies only to the number, then you apply the negative. If the problem meant "negative 2, cubed," it would be written as $(-2)^{3}$. This distinction frequently appears on the SHSAT, so practice identifying it.
17. Find the value of $5+2^{3}\div2$.
    Answer: 9 When you encounter an expression with multiple operations like this one, you need to follow the order of operations (PEMDAS/BODMAS). This means handling Parentheses, Exponents, Multiplication and Division (left to right), then Addition and Subtraction (left to right). Let's work through $5+2^{3}\div2$ step by step. First, calculate the exponent: $2^3 = 2 \times 2 \times 2 = 8$. Now the expression becomes $5 + 8 \div 2$. Next, perform the division: $8 \div 2 = 4$. Finally, add: $5 + 4 = 9$. Looking at the wrong answers reveals common order of operations mistakes. Choice B (8) likely comes from calculating $2^3 = 8$ and then incorrectly thinking this is the final answer, or from doing $5 + 2 = 7$, then $7^3 \div 2$, which shows misunderstanding of how exponents work in the order of operations. Choice C (11) probably results from adding first: $5 + 2 = 7$, then cubing to get $7^3 = 343$, then dividing by 2 to get approximately 171.5, or more likely from doing $5 + 2^3 = 5 + 8 = 13$, then subtracting 2 instead of dividing. Choice D (13) comes from ignoring the division entirely and just calculating $5 + 2^3 = 5 + 8 = 13$. Remember: exponents come before addition, and division comes before addition. When you see mixed operations, always write out PEMDAS and work through each step methodically. Don't rush—order of operations errors are among the most common mistakes on standardized math tests.
18. If $x = 3$ and $y = 2$, what is the value of $2x^2 - 3(x + y)^2 + 4xy$?
    Answer: $-33$ Substitute $x = 3$ and $y = 2$ and evaluate each term separately. First term: $2x^2 = 2(9) = 18$. Second term: $x + y = 5$, so $3(x + y)^2 = 3(25) = 75$. Third term: $4xy = 4(3)(2) = 24$. Combine: $18 - 75 + 24 = -33$. The value $-57$ comes from stopping after $18 - 75$ and forgetting to add the $4xy$ term. The value $-39$ comes from treating $x^2$ as $2x$, so the first term becomes $2(6) = 12$ and $12 - 75 + 24 = -39$. Squaring the sum before multiplying by 3 and keeping track of all three terms gives $-33$.
19. If $x = 4$ and $y = -3$, what is the value of $\frac{x^2 - y^2}{x + y} + 2xy$?
    Answer: $-17$ First, substitute values: $x^2 = 16$, $y^2 = 9$, $x + y = 1$, $xy = -12$. The expression becomes $\frac{16 - 9}{1} + 2(-12) = \frac{7}{1} - 24 = 7 - 24 = -17$. Note that $\frac{x^2 - y^2}{x + y} = \frac{(x+y)(x-y)}{x+y} = x - y = 7$ when $x + y \neq 0$. Choice B results from computing $2xy$ as $2(-3)(4) = -24$ but then adding instead of the correct calculation. Choice C comes from finding only $\frac{x^2 - y^2}{x + y} = 7$ and forgetting the $2xy$ term. Choice D results from sign errors in computing $y^2$ and $xy$.
20. What is the value of $\dfrac12\,(8-3)^{2}-7$ ?
    Answer: 5.5 This problem tests your ability to follow the order of operations (PEMDAS/BODMAS) when evaluating expressions with parentheses, exponents, multiplication, and subtraction. Let's work through $\frac{1}{2}(8-3)^2-7$ step by step. First, handle what's inside the parentheses: $8-3=5$. Next, apply the exponent: $5^2=25$. Then multiply by $\frac{1}{2}$: $\frac{1}{2} \times 25 = 12.5$. Finally, subtract 7: $12.5-7=5.5$. Looking at the wrong answers reveals common mistakes. Choice B (12.5) represents stopping too early—you've correctly calculated $\frac{1}{2}(8-3)^2$ but forgot to subtract the 7 at the end. Choice C (18.5) likely comes from incorrectly adding 7 instead of subtracting it: $12.5+7=18.5$. Choice D (-5.5) could result from a sign error where you calculated $7-12.5$ instead of $12.5-7$, essentially reversing the subtraction. The correct answer is A) 5.5. Strategy tip: On order of operations problems, work methodically through PEMDAS and resist the urge to rush. Many wrong answers on the SHSAT are designed to catch students who skip steps or make sign errors. Always double-check your final arithmetic, especially when subtracting at the end of a multi-step calculation.

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