Order of Operations - ISEE Upper Level Mathematics Achievement

1. What is $18 \div (3 \times 2)$?

   $3$. Parentheses are evaluated first, computing $3 \times 2$ inside, then dividing 18 by the result.

2. What is $-3^2$ (no parentheses around $-3$)?

   $-9$. The negative sign applies after the exponent, so compute $3^2$ first, then negate it.

3. What is $7 - \{3 + [2 \times (4 - 1)]\}$?

   $-2$. Innermost parentheses and brackets are evaluated step-by-step, starting with $(4-1)$, then multiplication, addition, and finally subtraction.

4. What is $(8 - 3) \times 2$?

   $10$. Parentheses take precedence, so subtract inside first, then multiply by 2.

5. What is $\frac{3}{4} + \frac{1}{2}$?

   $\frac{5}{4}$. Addition of fractions requires a common denominator, so convert $\frac{1}{2}$ to $\frac{2}{4}$ before adding.

6. State the order of operations for evaluating expressions (use the standard PEMDAS order).

   Parentheses, Exponents, Multiply/Divide (L to R), Add/Subtract (L to R). This sequence follows the PEMDAS rule, ensuring operations are performed in the correct precedence to evaluate expressions accurately.

7. What is $24 \div (6 + 2) \times 5$?

   $15$. Parentheses are evaluated first, then division and multiplication left to right.

8. What is $5 - (2 - 7)$?

   $10$. Parentheses are evaluated first, and subtracting a negative is equivalent to addition.

9. What does it mean to evaluate addition and subtraction from left to right in an expression?

   Compute $+$ and $-$ in the order they appear, left to right. Addition and subtraction share the same precedence level and are evaluated sequentially from left to right after multiplication and division.

10. What is $8 - 3 \times 2$?

    $2$. Multiplication precedes subtraction, so compute $3 \times 2$ first, then subtract from 8.

11. What is $8 \div \frac{1}{2} - 3$?

    $13$. Division precedes subtraction, so compute $8 \div \frac{1}{2}$ first, then subtract 3.

12. What is $18 \div 3 \times 2$?

    $12$. Division and multiplication are performed left to right, so divide 18 by 3 first, then multiply by 2.

13. What is $6(2) - 3^2$?

    $3$. Implied multiplication and exponent are computed before subtraction, with $3^2$ taking precedence over multiplication.

14. What is $3[2 + (5 - 1)]$?

    $18$. Innermost parentheses are evaluated first, then brackets imply multiplication by 3.

15. What does it mean to evaluate multiplication and division from left to right in an expression?

    Compute $\times$ and $\div$ in the order they appear, left to right. Multiplication and division share the same precedence level and are evaluated sequentially from left to right before addition and subtraction.

16. What is $2^3 + 1$?

    $9$. Exponents precede addition, so compute $2^3$ first, then add 1.

17. What is $4 + 2(3 + 1)$?

    $12$. Parentheses are evaluated first, then implied multiplication, followed by addition.

18. What is $\frac{2^3 \times 3}{4}$?

    $6$. Exponents precede multiplication, which precedes division, so compute $2^3$ first, then multiply by 3, and divide by 4.

19. What is $(-3)^2$?

    $9$. Parentheses include the negative sign with the base, so square -3 directly.

20. What is $\frac{1}{2} \times 8 + 3$?

    $7$. Multiplication precedes addition, so compute $\frac{1}{2} \times 8$ first, then add 3.

21. What is $\left(\frac{2}{3}\right)^2 \times 9$?

    $4$. Exponents are evaluated first inside parentheses, then multiplication by 9.

22. What is $24 \div 6 + 2 \times 5$?

    $14$. Division and multiplication are performed left to right before addition, treating the expression as $(24 \div 6) + (2 \times 5)$.

23. What is $2 + 3 \times (4 - 6)$?

    $-4$. Parentheses are evaluated first, then multiplication before addition.

24. What is $\left(\frac{3}{4} + \frac{1}{2}\right) \times 4$?

    $5$. Parentheses are evaluated first by adding the fractions, then multiplying the sum by 4.

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