When you encounter an expression with multiple operations like this, you need to carefully apply the order of operations (PEMDAS): Parentheses, Exponents, Multiplication and Division (left to right), Addition and Subtraction (left to right).

Let's work through $$-3^2 + 2(-4 + 7) - 5 \times 2$$ step by step:

First, handle parentheses: $$(-4 + 7) = 3$$

Next, exponents: $$-3^2 = -9$$ (careful here - the negative sign isn't being squared, just the 3)

Then multiplication from left to right: $$2(3) = 6$$ and $$5 \times 2 = 10$$

Now you have: $$-9 + 6 - 10$$

Finally, addition and subtraction from left to right: $$-9 + 6 = -3$$, then $$-3 - 10 = -13$$

The answer is A) -13.

Looking at the wrong answers: B) -15 likely comes from incorrectly calculating $$(-3)^2 = 9$$ instead of $$-3^2 = -9$$, giving you $$9 + 6 - 10 = 5$$, then making an error in the final arithmetic. C) -11 probably results from an addition/subtraction error in the final step. D) -17 might come from mishandling the parentheses or making multiple arithmetic mistakes.

The key trap here is understanding that $$-3^2 = -9$$, not $$+9$$. The exponent applies only to the 3, then you apply the negative sign. Always write out each step of PEMDAS clearly to avoid order-of-operations errors on the ISEE.

When you encounter a complex fraction like this, the key is applying the order of operations (PEMDAS) systematically to both the numerator and denominator before dividing.

Let's work through the numerator first: $$6^2 - 4 \times 3$$. Following order of operations, we handle exponents before multiplication: $$6^2 = 36$$, then $$4 \times 3 = 12$$, giving us $$36 - 12 = 24$$.

For the denominator: $$2^3 \div 4 + 1$$. Again, exponents first: $$2^3 = 8$$, then division: $$8 \div 4 = 2$$, and finally addition: $$2 + 1 = 3$$.

Now we have $$\frac{24}{3} = 8$$, which is answer choice A.

The wrong answers likely come from order of operations mistakes. Choice B (6) might result from incorrectly calculating the numerator as $$6^2 - 4 \times 3 = 6 \times 2 \times 3 = 18$$, then getting the denominator wrong. Choice C (4) could come from errors in both numerator and denominator calculations, perhaps treating $$6^2$$ as $$6 \times 2 = 12$$. Choice D (12) might result from correctly finding the numerator as 24 but making an error in the denominator, possibly calculating it as 2 instead of 3.

Remember: when simplifying complex expressions, work methodically through PEMDAS in each part separately before combining results. Write out each step to avoid rushing through calculations and making arithmetic errors.

This problem tests your mastery of order of operations (PEMDAS), which is crucial for complex algebraic expressions. When you see nested operations like this, work systematically through each step.

Let's evaluate $$\frac{6 + 2 \times 3^2}{4^2 - 3 \times 4} - 1$$ using PEMDAS. Start with the numerator: $$6 + 2 \times 3^2$$. First handle the exponent: $$3^2 = 9$$. Then multiply: $$2 \times 9 = 18$$. Finally add: $$6 + 18 = 24$$.

For the denominator: $$4^2 - 3 \times 4$$. Calculate the exponent: $$4^2 = 16$$. Then multiply: $$3 \times 4 = 12$$. Subtract: $$16 - 12 = 4$$.

Now you have $$\frac{24}{4} - 1 = 6 - 1 = 5$$. The answer is A.

Here's why the other choices represent common mistakes: Choice B (4) results from forgetting to subtract the final 1, stopping at $$\frac{24}{4} = 6$$. Choice C (3) comes from incorrectly calculating the denominator as 8 instead of 4, giving $$\frac{24}{8} - 1 = 3 - 1 = 2$$, or from other order-of-operations errors. Choice D (6) happens when you completely ignore the "$$-1$$" at the end.

Strategy tip: With complex expressions, write out each PEMDAS step separately rather than trying to do multiple operations mentally. This prevents the careless errors that create most wrong answer choices on these problems. Always double-check that you've addressed every part of the expression.

When you encounter a complex expression with multiple operations, the order of operations (PEMDAS/BODMAS) is your roadmap. 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 work through $$4^2 \div 2 + 3 \times(7 - 5)^2 - 4$$ step by step:

First, handle the parentheses: $(7 - 5) = 2$

Next, calculate all exponents: $4^2 = 16$ and $(2)^2 = 4$

Now the expression becomes: $$16 \div 2 + 3 \times 4 - 4$$

Then perform multiplication and division from left to right: $16 \div 2 = 8$ and $3 \times 4 = 12$

This gives us: $$8 + 12 - 4$$

Finally, addition and subtraction from left to right: $8 + 12 = 20$, then $20 - 4 = 16$

The answer is A) 16.

Common mistakes lead to the other choices. Choice B) 18 likely comes from forgetting to subtract the final 4. Choice C) 14 might result from incorrectly calculating $4^2 \div 2$ as 2 instead of 8. Choice D) 20 happens when you stop before the final subtraction step.

Strategy tip: Write out each step of PEMDAS when working with complex expressions. Don't try to do multiple operations in your head simultaneously—the ISEE rewards careful, methodical work over speed, and order of operations questions are designed to catch rushed calculations.

This question tests ISEE Upper Level Mathematics Achievement by evaluating expressions using the order of operations. The order of operations is a set of rules to determine which operations to perform first in a mathematical expression, typically remembered by PEMDAS: Parentheses, Exponents, Multiplication and Division (left-to-right), Addition and Subtraction (left-to-right). In the given expression $(9 + 15) \div 3 + 2^3 \times 4 - 10$, we first calculate within parentheses: $9 + 15 = 24$, then the exponent: $2^3 = 8$, divide: $24 \div 3 = 8$, multiply: $8 \times 4 = 32$, add: $8 + 32 = 40$, and finally subtract: $40 - 10 = 30$. The correct answer, C (30), results from applying these rules correctly, ensuring that each operation respects the hierarchy. A common mistake, such as choosing B (18), might occur if students misapply the order of operations or make arithmetic errors. To help students master this skill, practice identifying the highest priority operations first and using the PEMDAS acronym to guide the sequence. Encourage students to double-check each step, especially when dealing with complex expressions, to avoid intermediate errors.

When you see a complex fraction with multiple operations, you must follow the order of operations (PEMDAS) carefully in both the numerator and denominator before dividing.

Let's work through this step by step. In the numerator $$24 - 3 \times 2^3$$, you first handle the exponent: $$2^3 = 8$$. Then multiply: $$3 \times 8 = 24$$. Finally subtract: $$24 - 24 = 0$$. So the numerator equals 0.

In the denominator $$4 + 2 \times 3$$, you multiply first: $$2 \times 3 = 6$$. Then add: $$4 + 6 = 10$$. So the denominator equals 10.

Therefore: $$\frac{24 - 3 \times 2^3}{4 + 2 \times 3} = \frac{0}{10} = 0$$

Choice A (0) is correct because any fraction with 0 in the numerator equals 0, regardless of the denominator value.

Choice B (1) likely comes from incorrectly getting the same value for both numerator and denominator, perhaps by making order of operations errors that coincidentally produce equal results.

Choice C (2) might result from calculating $$24 \div 12$$ if you incorrectly simplified the original expression or made computational errors.

Choice D (3) could come from various order of operations mistakes, such as working left to right instead of following PEMDAS.

Strategy tip: Always work out the numerator and denominator completely and separately before dividing. Double-check your exponents and multiplication before moving to addition and subtraction. Remember that any fraction with 0 on top equals 0.

When you encounter complex expressions like this, the key is applying the order of operations (PEMDAS) systematically: Parentheses, Exponents, Multiplication/Division (left to right), Addition/Subtraction (left to right).

Let's work through $$\frac{2^4 + 3 \times 4}{8 - 2^2} + 7$$ step by step. First, handle the exponents: $$2^4 = 16$$ and $$2^2 = 4$$. Next, perform the multiplication: $$3 \times 4 = 12$$. Now the expression becomes $$\frac{16 + 12}{8 - 4} + 7$$.

Working inside the fraction, calculate the numerator: $$16 + 12 = 28$$, and the denominator: $$8 - 4 = 4$$. This gives us $$\frac{28}{4} = 7$$. Finally, add: $$7 + 7 = 14$$.

Choice A (14) is correct. Choice B (13) likely results from a calculation error, perhaps getting 27 instead of 28 in the numerator. Choice C (15) might occur if you incorrectly calculated the denominator as 3 instead of 4, giving $$\frac{28}{3} ≈ 9.33$$, then rounding poorly. Choice D (12) could happen if you forgot to add the final 7, stopping at just the fraction result.

Remember that order of operations errors are common traps on standardized tests. Always work methodically: handle exponents first, then multiplication and division, and save addition and subtraction for last. When fractions are involved, completely simplify the numerator and denominator separately before dividing.

When you encounter an expression with multiple variables and operations, the key is to substitute the given values carefully and follow the order of operations (PEMDAS/BODMAS).

Given $$p = 4$$ and $$q = 3$$, let's substitute these values into $$p^2 \div 2 + q(p - 1) - 2q + 5$$:

$$4^2 \div 2 + 3(4 - 1) - 2(3) + 5$$

Now apply order of operations step by step:

• First, handle exponents: $$4^2 = 16$$
• Next, operations in parentheses: $$(4 - 1) = 3$$
• Then multiplication and division from left to right: $$16 \div 2 = 8$$, $$3 \times 3 = 9$$, $$2 \times 3 = 6$$
• Finally, addition and subtraction from left to right: $$8 + 9 - 6 + 5 = 16$$

The answer is A) 16.

Looking at the wrong answers: B) 17 likely results from a small arithmetic error in the final addition/subtraction steps. C) 14 could come from incorrectly calculating $$p^2 \div 2$$ as $$4^2 \div 2 = 8$$ but then making errors with the remaining terms. D) 20 might result from forgetting to subtract $$2q = 6$$ or making sign errors when combining terms.

Strategy tip: When substituting multiple variables, write out each substitution clearly and work through order of operations methodically. Double-check your arithmetic at each step, especially when combining positive and negative terms at the end.

This question tests ISEE Upper Level Mathematics Achievement by evaluating expressions using the order of operations. The order of operations is a set of rules to determine which operations to perform first in a mathematical expression, typically remembered by PEMDAS: Parentheses, Exponents, Multiplication and Division (left-to-right), Addition and Subtraction (left-to-right). In the given expression $(25 - 9) \times 3 + 2^4 \div 4$, we first calculate within parentheses: $25 - 9 = 16$, then the exponent: $2^4 = 16$, multiply: $16 \times 3 = 48$, divide: $16 \div 4 = 4$, and finally add: $48 + 4 = 52$. The correct answer, C (52), results from applying these rules correctly, ensuring that each operation respects the hierarchy. A common mistake, such as choosing B (28), might occur if students forget to add the final term or miscalculate the exponent. To help students master this skill, practice identifying the highest priority operations first and using the PEMDAS acronym to guide the sequence. Encourage students to double-check each step, especially when dealing with complex expressions, to avoid intermediate errors.

This question tests ISEE Upper Level Mathematics Achievement by evaluating expressions using the order of operations. The order of operations is a set of rules to determine which operations to perform first in a mathematical expression, typically remembered by PEMDAS: Parentheses, Exponents, Multiplication and Division (left-to-right), Addition and Subtraction (left-to-right). In the given expression $40 + (18 \div 3) \times 2^2 - 9$, we first calculate within parentheses: $18 \div 3 = 6$, then the exponent: $2^2 = 4$, multiply: $6 \times 4 = 24$, add: $40 + 24 = 64$, and finally subtract: $64 - 9 = 55$. The correct answer, B (55), results from applying these rules correctly, ensuring that each operation respects the hierarchy. A common mistake, such as choosing A (47), might occur if students perform operations out of order or make arithmetic errors. To help students master this skill, practice identifying the highest priority operations first and using the PEMDAS acronym to guide the sequence. Encourage students to double-check each step, especially when dealing with complex expressions, to avoid intermediate errors.