Middle School Math Quiz: Evaluate Expressions With Exponents

20 questions · exam conditions

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Evaluate Expressions With ExponentsQuestion 1 of 20

If $x = 2$ and $y = 3$ , what is the value of $x^y + y^x$ ?

17

13

25

36

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Middle School Math Quiz

Middle School Math Quiz: Evaluate Expressions With Exponents

Practice Evaluate Expressions With Exponents in Middle School Math with focused quiz questions that help you check what you know, review explanations, and build confidence with test-style prompts.

What this quiz covers

This quiz focuses on Evaluate Expressions With Exponents, giving you a quick way to practice the rules, question types, and explanations that matter most for Middle School Math.

How to use this quiz

Try each quiz question before looking at the correct answer. Use the explanations to review missed ideas, then come back to similar questions until the pattern feels familiar.

All questions

Question 1

If $x = 2$ and $y = 3$ , what is the value of $x^y + y^x$ ?

$17$ (correct answer)
$13$
$25$
$36$

Explanation: Substitute the values:

x^y + y^x = 2^3 + 3^2 = 8 + 9 = 17

. Choice B results from calculating

2^2 + 3^2 = 4 + 9 = 13

. Choice C comes from incorrectly computing

(x + y)^2 = 5^2 = 25

. Choice D represents

x^y \times y^x = 8 \times 9 = 72

, but then making an arithmetic error.

Question 2

Sarah needs to evaluate $(2^3)^2 - 2^3 \times 2$ . What is the correct result?

$48$ (correct answer)
$52$
$56$
$64$

Explanation: First,

(2^3)^2 = 2^{3 \times 2} = 2^6 = 64

. Then,

2^3 \times 2 = 8 \times 2 = 16

. Finally,

64 - 16 = 48

. Choice B results from calculating

2^6 - 2^3 = 64 - 8 = 56

but making an arithmetic error. Choice C comes from

64 - 8 = 56

(forgetting the multiplication by 2). Choice D is just

2^6

without subtracting.

Question 3

A game uses the points rule $6 \times 2^2 - 10$ . How many points is that?

14 (correct answer)
6
38
24

Explanation: This question is all about order of operations — the special order we follow so everyone gets the same answer. A helpful rule is to handle exponents first, then multiply and divide, and finally add or subtract. Let's work through

6 \times 2^2 - 10

one step at a time: First, the exponent.

2^2

means

2 \times 2 = 4

. Now we have

6 \times 4 - 10

. Next, multiply:

6 \times 4 = 24

.
Now we have

24 - 10

. Finally, subtract:

24 - 10 = 14

. So the game gives you

14

points! Think of it like getting dressed: socks go on before shoes, no matter what. Math has an order too, and powers come early in the routine. Try this at home: Make up your own expression like

5 \times 3^2 - 4

and solve it step by step. Powers first!

Question 4

A science club records a value using the expression $10^2 \div 5^2$ . What is the value?

100
4 (correct answer)
400
20

Explanation: This question tests evaluating numerical expressions with whole-number exponents like

10^2

and

5^2

using order of operations (PEMDAS: exponents before multiplication/addition, parentheses first). Exponent notation means the base is multiplied by itself the number of times indicated by the exponent, so

5^2 = 5 \times 5 = 25

, not

5 \times 2 = 10

; follow PEMDAS by handling parentheses first, then exponents, multiplication/division left to right, and addition/subtraction left to right, for example, in

4^2 \div 2^2

, compute exponents to

16 \div 4 = 4

. For example, to evaluate

2^3 \div 2^2

, step 1:

2^3 = 8

, step 2:

2^2 = 4

, step 3:

8 \div 4 = 2

; or for

3 \times 2^2

, step 1:

2^2 = 4

, step 2:

3 \times 4 = 12

; or for

(2 + 3)^2

, step 1:

2 + 3 = 5

, step 2:

5^2 = 25

. For

10^2 \div 5^2

, first compute exponents:

10^2 = 100

and

5^2 = 25

, then divide

100 \div 25 = 4

. A common error is treating exponents as multiplication, like

10^2 = 20

leading to

20 \div 10 = 2

, or violating order by dividing bases first as

(10 \div 5)^2 = 2^2 = 4

(which coincidentally matches but is wrong process), or arithmetic errors like

100 \div 25 = 5

. The strategy is to (1) scan for parentheses and compute inside first, (2) evaluate all exponents like

10^2

and

5^2

, (3) multiply/divide left to right, (4) add/subtract if present left to right, and (5) verify the result is reasonable, such as

100 \div 25

equaling 4. Common exponents to know include

2^2 = 4

5^2 = 25

10^2 = 100

; mistakes often involve confusing division of powers with other operations.

Question 5

A student is checking homework and wants to verify the value of $4^2 + 3\times 2^3$ . What is the correct value?

40 (correct answer)
48
112
88

Explanation: This question tests evaluating numerical expressions with whole-number exponents like 4² and 2³ using order of operations (PEMDAS: exponents before multiplication/addition, parentheses first). Exponent notation means the base is multiplied by itself the number of times indicated by the exponent, so 4² = 4 × 4 = 16, not 4 × 2 = 8; follow PEMDAS by handling parentheses first, then exponents, multiplication/division left to right, and addition/subtraction left to right, for example, in 3 × 2² + 4, compute 2² = 4 first, then 3 × 4 = 12, then 12 + 4 = 16, and with multiple exponents like 2³ + 4², evaluate each separately to get 8 + 16 = 24. For example, to evaluate 2³ + 4², step 1: 2³ = 8, step 2: 4² = 16, step 3: 8 + 16 = 24; or for 3 × 2², step 1: 2² = 4, step 2: 3 × 4 = 12; or for (2 + 3)², step 1: 2 + 3 = 5, step 2: 5² = 25. For 4² + 3 × 2³, first compute exponents: 4² = 16 and 2³ = 8, then multiply 3 × 8 = 24, and add 16 + 24 = 40. A common error is violating order by multiplying bases first like (4 + 3 × 2)³ which is incorrect, or treating exponents as multiplication like 2³ = 6 leading to 16 + 3 × 6 = 16 + 18 = 34, or adding before multiplying. The strategy is to (1) scan for parentheses and compute inside first, (2) evaluate all exponents like 4² and 2³, (3) multiply/divide left to right, (4) add/subtract left to right, and (5) verify the result is reasonable, such as 16 + 24 equaling 40. Common exponents to know include 2³ = 8, 3² = 9, 4² = 16; mistakes often involve incorrect order like (4 + 3) × 2³ = 7 × 8 = 56.

Question 6

Evaluate $8^2 \div 4^2 + 3$ using the order of operations.

7 (correct answer)
19
4
256

Explanation: This question tests evaluating numerical expressions with whole-number exponents like 8² and 4² using the order of operations, often remembered as PEMDAS, where exponents are evaluated before division and addition, and parentheses come first. Exponent notation means the base is multiplied by itself as many times as the exponent indicates, so 2³ = 2 × 2 × 2 = 8, not 2 × 3 = 6; the order of operations is: (1) parentheses first, for example in (2 + 3)² you compute 2 + 3 = 5 first, (2) then exponents like 5² = 25, (3) multiply or divide from left to right, and (4) add or subtract from left to right; for instance, in 3 × 2² + 4, you do the exponent 2² = 4 first, then multiply 3 × 4 = 12, then add 12 + 4 = 16; when there are multiple exponents, evaluate each one independently, such as in 2³ + 4² where 2³ = 8 and 4² = 16, then add to get 24. For example, to evaluate 10² ÷ 5², step 1: 10² = 100, step 2: 5² = 25, step 3: 100 ÷ 25 = 4; or for 2³ + 4², step 1: 2³ = 8, step 2: 4² = 16, step 3: 8 + 16 = 24; or for (2 + 3)², step 1: 2 + 3 = 5, step 2: 5² = 25. For the expression 8² ÷ 4² + 3, first evaluate the exponents: 8² = 64 and 4² = 16, then divide 64 ÷ 16 = 4, and finally add 4 + 3 = 7. A common error is adding before dividing like 4² + 3 = 16 + 3 = 19 then 8² ÷ 19 which is wrong, or violating order by doing division after addition, or treating exponent as multiplication such as 4² = 4 × 2 = 8 then 64 ÷ 8 + 3 = 8 + 3 = 11. To solve these, use this strategy: (1) scan for parentheses and compute inside them first, (2) identify and evaluate all exponents like 8² or 4², (3) multiply or divide from left to right if present, (4) add or subtract from left to right, and (5) verify if the result is reasonable, such as 64 ÷ 16 = 4 plus 3 is 7. Remember common exponents: 2² = 4, 2³ = 8, 2⁴ = 16, 3² = 9, 3³ = 27, 4² = 16, 5² = 25, 10² = 100, 10³ = 1000; avoid mistakes like treating exponents as multiplication, violating order by adding first, ignoring parentheses, or adding before exponentiating.

Question 7

Evaluate $\left(8-3\right)^2 + 2^3$ .

13
17
64
33 (correct answer)

Explanation: This question tests evaluating numerical expressions with whole-number exponents like 5² and 2³ using order of operations (PEMDAS: exponents before multiplication/addition, parentheses first). Exponent notation means the base is multiplied by itself the number of times indicated by the exponent, so 2³ = 2 × 2 × 2 = 8, not 2 × 3 = 6; follow PEMDAS by handling parentheses first, then exponents, multiplication/division left to right, and addition/subtraction left to right, for example, in (3 + 2)², compute inside to 5, then 5² = 25. For example, to evaluate (2 + 3)², step 1: 2 + 3 = 5, step 2: 5² = 25; or for 2³ + 3², step 1: 2³ = 8, step 2: 3² = 9, step 3: 8 + 9 = 17; or for 3 × 2², step 1: 2² = 4, step 2: 3 × 4 = 12. For (8 - 3)² + 2³, first compute inside parentheses: 8 - 3 = 5, then 5² = 25, next 2³ = 8, and add 25 + 8 = 33. A common error is ignoring parentheses like 8 - 3² + 2³ = 8 - 9 + 8 = 7, or distributing exponent wrongly as 8² - 3² + 2³ = 64 - 9 + 8 = 63, or arithmetic like 25 + 8 = 32. The strategy is to (1) scan for parentheses and compute inside first, (2) evaluate all exponents like 5² and 2³, (3) multiply/divide if present, (4) add/subtract left to right, and (5) verify reasonable, such as 25 + 8 equaling 33. Common exponents to know include 2³ = 8, 3² = 9, 5² = 25; mistakes often involve computing exponents before parentheses.

Question 8

Evaluate the expression $10^2 \div 5^2$ .

4 (correct answer)
100
16
20

Explanation: This question tests evaluating numerical expressions with whole-number exponents like 10² and 5² using the order of operations, often remembered as PEMDAS, where exponents are evaluated before division, and parentheses come first. Exponent notation means the base is multiplied by itself as many times as the exponent indicates, so 2³ = 2 × 2 × 2 = 8, not 2 × 3 = 6; the order of operations is: (1) parentheses first, for example in (2 + 3)² you compute 2 + 3 = 5 first, (2) then exponents like 5² = 25, (3) multiply or divide from left to right, and (4) add or subtract from left to right; for instance, in 3 × 2² + 4, you do the exponent 2² = 4 first, then multiply 3 × 4 = 12, then add 12 + 4 = 16; when there are multiple exponents, evaluate each one independently, such as in 2³ + 4² where 2³ = 8 and 4² = 16, then add to get 24. For example, to evaluate 2³ + 4², step 1: compute 2³ = 2 × 2 × 2 = 8, step 2: compute 4² = 4 × 4 = 16, step 3: add 8 + 16 = 24; or for 10² ÷ 5, step 1: 10² = 100, step 2: 100 ÷ 5 = 20; or for (2 + 3)², step 1: 2 + 3 = 5, step 2: 5² = 25. For the expression 10² ÷ 5², evaluate the exponents: 10² = 100 and 5² = 25, then divide 100 ÷ 25 = 4. A common error is dividing before exponents like 10 ÷ 5 = 2 then 2² ÷ 2² which is wrong, or treating exponent as multiplication such as 5² = 5 × 2 = 10 then 100 ÷ 10 = 10, or arithmetic error like 100 ÷ 25 = 5. To solve these, use this strategy: (1) scan for parentheses and compute inside them first, (2) identify and evaluate all exponents like 10² or 5², (3) multiply or divide from left to right if present, (4) add or subtract from left to right, and (5) verify if the result is reasonable, such as 100 ÷ 25 = 4. Remember common exponents: 2² = 4, 2³ = 8, 2⁴ = 16, 3² = 9, 3³ = 27, 4² = 16, 5² = 25, 10² = 100, 10³ = 1000; avoid mistakes like treating exponents as multiplication, violating order by dividing first, ignoring parentheses, or adding before exponentiating.

Question 9

Evaluate $2^3\times\left(3+1\right) - 6$ using order of operations.

58
20
10
26 (correct answer)

Explanation: This question tests evaluating numerical expressions with whole-number exponents like 2³ using order of operations (PEMDAS: exponents before multiplication/addition, parentheses first). Exponent notation means the base is multiplied by itself the number of times indicated by the exponent, so 2³ = 2 × 2 × 2 = 8, not 2 × 3 = 6; follow PEMDAS by handling parentheses first, then exponents, multiplication/division left to right, and addition/subtraction left to right, for example, in 2 × (3 + 1) - 4, compute parentheses to 4, then multiply 2 × 4 = 8, then 8 - 4 = 4. For example, to evaluate 2³ + 4², step 1: 2³ = 8, step 2: 4² = 16, step 3: 8 + 16 = 24; or for (2 + 3)², step 1: 2 + 3 = 5, step 2: 5² = 25; or for 3 × 2², step 1: 2² = 4, step 2: 3 × 4 = 12. For 2³ × (3 + 1) - 6, first compute parentheses: 3 + 1 = 4, then exponent 2³ = 8, multiply 8 × 4 = 32, and subtract 32 - 6 = 26. A common error is ignoring parentheses like 2³ × 3 + 1 - 6 = 8 × 3 + 1 - 6 = 24 + 1 - 6 = 19, or violating order by subtracting first, or treating exponent as multiplication like 2³ = 6 leading to 6 × 4 - 6 = 24 - 6 = 18. The strategy is to (1) scan for parentheses and compute inside first, (2) evaluate all exponents like 2³, (3) multiply/divide left to right, (4) add/subtract left to right, and (5) verify reasonable, such as 8 × 4 = 32 minus 6 equaling 26. Common exponents to know include 2² = 4, 2³ = 8, 3² = 9; mistakes often involve multiplying after subtracting.

Question 10

Evaluate $6 + 2^4 \div 4$ using order of operations.

10 (correct answer)
7
8
12

Explanation: This question tests evaluating numerical expressions with whole-number exponents like 2⁴ using order of operations (PEMDAS: exponents before multiplication/addition, parentheses first). Exponent notation means the base is multiplied by itself the number of times indicated by the exponent, so 2⁴ = 2 × 2 × 2 × 2 = 16, not 2 × 4 = 8; follow PEMDAS by handling parentheses first, then exponents, multiplication/division left to right, and addition/subtraction left to right, for example, in 3 + 2³ ÷ 4, compute 2³ = 8, then 8 ÷ 4 = 2, then 3 + 2 = 5. For example, to evaluate 2⁴ ÷ 4, step 1: 2⁴ = 16, step 2: 16 ÷ 4 = 4; or for 3 × 2², step 1: 2² = 4, step 2: 3 × 4 = 12; or for (2 + 3)², step 1: 2 + 3 = 5, step 2: 5² = 25. For 6 + 2⁴ ÷ 4, first compute exponent 2⁴ = 16, then divide 16 ÷ 4 = 4, and add 6 + 4 = 10. A common error is violating order by adding first like (6 + 2)⁴ ÷ 4 = 8⁴ ÷ 4 which is huge, or treating exponent as multiplication like 2⁴ = 8 leading to 6 + 8 ÷ 4 = 6 + 2 = 8, or arithmetic errors like 16 ÷ 4 = 3. The strategy is to (1) scan for parentheses and compute inside first, (2) evaluate all exponents like 2⁴, (3) multiply/divide left to right, (4) add/subtract left to right, and (5) verify the result is reasonable, such as 16 ÷ 4 = 4 plus 6 equaling 10. Common exponents to know include 2² = 4, 2³ = 8, 2⁴ = 16, 3² = 9; mistakes often involve adding before dividing.

Question 11

Evaluate the expression: $4^2+2^4\div 2^2$ .

8
20 (correct answer)
12
32

Explanation: This question tests evaluating numerical expressions with whole-number exponents like 4², 2⁴, and 2² using the order of operations (PEMDAS: parentheses first, then exponents, multiplication and division left to right, addition and subtraction left to right). Exponent notation means the base is multiplied by itself the number of times indicated by the exponent, so 2³ = 2 × 2 × 2 = 8, not 2 × 3 = 6; follow PEMDAS strictly, for example, in 3 × 2² + 4, compute 2² = 4 first, then 3 × 4 = 12, then 12 + 4 = 16; with multiple exponents, evaluate each one separately before other operations. For example, to evaluate 2³ + 4², step 1: 2³ = 2 × 2 × 2 = 8, step 2: 4² = 4 × 4 = 16, step 3: 8 + 16 = 24; or for 3 × 2², step 1: 2² = 4, step 2: 3 × 4 = 12; or for (2 + 3)², step 1: 2 + 3 = 5, step 2: 5² = 25. For 4² + 2⁴ ÷ 2², compute exponents: 4² = 16, 2⁴ = 16, 2² = 4, then divide 16 ÷ 4 = 4, finally add 16 + 4 = 20. A common error is adding before dividing, like 2⁴ + 2² = 16 + 4 = 20, then 4² + 20 = 16 + 20 = 36, or treating division as lower priority incorrectly. The strategy is to (1) scan for parentheses (none), (2) evaluate all exponents (4² = 16, 2⁴ = 16, 2² = 4), (3) divide left to right (16 ÷ 4 = 4), (4) add (16 + 4 = 20), and (5) verify reasonableness, like 16 + 4 = 20. Common exponents: 2⁴ = 16, 4² = 16; mistakes include confusing 2⁴ ÷ 2² as (2 ÷ 2)^{4-2} = 1² = 1 instead of 16 ÷ 4 = 4.

Question 12

Which of the following expressions has the greatest value?

$6^2$
$4^3$
$2^6$
$3^4$ (correct answer)

Explanation: When comparing expressions with exponents, you need to calculate the actual value of each expression rather than just looking at the base and exponent separately. Let's work through each option systematically. For choice A,

6^2

means

6 \times 6 = 36

. For choice B,

4^3

means

4 \times 4 \times 4 = 64

. For choice C,

2^6

means

2 \times 2 \times 2 \times 2 \times 2 \times 2 = 64

. Finally, for choice D,

3^4

means

3 \times 3 \times 3 \times 3 = 81

. Since 81 is the largest value, choice D is correct. Choice A gives us only 36, which is significantly smaller than the others. Choice B equals 64, which might seem large because of the higher base number (4), but it's still less than 81. Choice C also equals 64 – this one can be tricky because the large exponent (6) might make it seem like it should be the biggest, but the small base (2) keeps the final value lower than choice D. The key insight here is that you can't determine which exponential expression is largest just by comparing bases or exponents alone. A smaller base with a larger exponent might give you a smaller result than a larger base with a smaller exponent, depending on the specific numbers involved. Remember: when comparing exponential expressions, always calculate the actual values. Don't assume that bigger exponents or bigger bases automatically mean bigger results.

Question 13

Evaluate the expression: $4^2+2^4\div 2^2$ .

12
8
20 (correct answer)
32

Question 14

A student writes the expression $3\times 2^3 + 4^2$ on the board. What is the value of the expression?

48
40 (correct answer)
112
28

Explanation: This question tests evaluating numerical expressions with whole-number exponents like 2³ and 4² using the order of operations, often remembered as PEMDAS, where exponents are evaluated before multiplication and addition, and parentheses come first. Exponent notation means the base is multiplied by itself as many times as the exponent indicates, so 2³ = 2 × 2 × 2 = 8, not 2 × 3 = 6; the order of operations is: (1) parentheses first, for example in (2 + 3)² you compute 2 + 3 = 5 first, (2) then exponents like 5² = 25, (3) multiply or divide from left to right, and (4) add or subtract from left to right; for instance, in 3 × 2² + 4, you do the exponent 2² = 4 first, then multiply 3 × 4 = 12, then add 12 + 4 = 16; when there are multiple exponents, evaluate each one independently, such as in 2³ + 4² where 2³ = 8 and 4² = 16, then add to get 24. For example, to evaluate 2³ + 4², step 1: compute 2³ = 2 × 2 × 2 = 8, step 2: compute 4² = 4 × 4 = 16, step 3: add 8 + 16 = 24; or for 3 × 2², step 1: 2² = 4, step 2: 3 × 4 = 12; or for (2 + 3)², step 1: 2 + 3 = 5, step 2: 5² = 25. For the expression 3 × 2³ + 4², first evaluate the exponents: 2³ = 8 and 4² = 16, then multiply 3 × 8 = 24, and finally add 24 + 16 = 40. A common error is treating an exponent as multiplication, like thinking 2³ = 2 × 3 = 6, which would lead to 3 × 6 + 16 = 18 + 16 = 34, or violating order by multiplying before exponents, such as treating it as 3 × 2 = 6 then 6³ + 4² which is wrong, or an arithmetic error like 24 + 16 = 30. To solve these, use this strategy: (1) scan for parentheses and compute inside them first, (2) identify and evaluate all exponents like 2³ or 4², (3) multiply or divide from left to right if present, (4) add or subtract from left to right, and (5) verify if the result is reasonable, such as 2³ around 8, 4² = 16, so with multiplication by 3 and addition, about 40 makes sense. Remember common exponents: 2² = 4, 2³ = 8, 2⁴ = 16, 3² = 9, 3³ = 27, 4² = 16, 5² = 25, 10² = 100, 10³ = 1000; avoid mistakes like treating exponents as multiplication, violating order by multiplying first, ignoring parentheses, or adding before exponentiating.

Question 15

Evaluate $6 + 2^3\times 3$ using the order of operations.

72
42
30 (correct answer)
24

Explanation: This question tests evaluating numerical expressions with whole-number exponents like 2³ using the order of operations, often remembered as PEMDAS, where exponents are evaluated before multiplication and addition, and parentheses come first. Exponent notation means the base is multiplied by itself as many times as the exponent indicates, so 2³ = 2 × 2 × 2 = 8, not 2 × 3 = 6; the order of operations is: (1) parentheses first, for example in (2 + 3)² you compute 2 + 3 = 5 first, (2) then exponents like 5² = 25, (3) multiply or divide from left to right, and (4) add or subtract from left to right; for instance, in 3 × 2² + 4, you do the exponent 2² = 4 first, then multiply 3 × 4 = 12, then add 12 + 4 = 16; when there are multiple exponents, evaluate each one independently, such as in 2³ + 4² where 2³ = 8 and 4² = 16, then add to get 24. For example, to evaluate 3 × 2², step 1: compute 2² = 4, step 2: multiply 3 × 4 = 12; or for 2³ + 4², step 1: 2³ = 8, step 2: 4² = 16, step 3: 8 + 16 = 24; or for (2 + 3)², step 1: 2 + 3 = 5, step 2: 5² = 25. For the expression 6 + 2³ × 3, first evaluate the exponent: 2³ = 8, then multiply 8 × 3 = 24, and finally add 6 + 24 = 30. A common error is adding before multiplying, like 6 + 2³ = 8 then 8 × 3 = 24, or treating exponent as multiplication such as 2³ = 6 then 6 + 6 × 3 = 6 + 18 = 24, or violating order by doing 2³ × 3 = 24 but then adding wrongly. To solve these, use this strategy: (1) scan for parentheses and compute inside them first, (2) identify and evaluate all exponents like 2³, (3) multiply or divide from left to right if present, (4) add or subtract from left to right, and (5) verify if the result is reasonable, such as 8 × 3 = 24 plus 6 is 30. Remember common exponents: 2² = 4, 2³ = 8, 2⁴ = 16, 3² = 9, 3³ = 27, 4² = 16, 5² = 25, 10² = 100, 10³ = 1000; avoid mistakes like treating exponents as multiplication, violating order by adding first, ignoring parentheses, or adding before exponentiating.

Question 16

Marcus is calculating $3^4 + 2^5$ and claims the answer is $5^9$ . What is the actual value of $3^4 + 2^5$ ?

$113$
$81$
$97$ (correct answer)
$243$

Explanation: First calculate each term:

3^4 = 3 \times 3 \times 3 \times 3 = 81

and

2^5 = 2 \times 2 \times 2 \times 2 \times 2 = 32

. Then add:

81 + 32 = 97

. Choice A represents

3^4 + 2^5

calculated as

81 + 32

but with an arithmetic error. Choice B is just

3^4

without adding

2^5

. Choice D is

3^5

instead of

3^4

Question 17

A student claims that $3\times 2^2 = 36$ . What is the correct value of $3\times 2^2$ ?

18
12 (correct answer)
36
24

Explanation: This question tests evaluating numerical expressions with whole-number exponents like 2² using order of operations (PEMDAS: exponents before multiplication/addition, parentheses first). Exponent notation means the base is multiplied by itself the number of times indicated by the exponent, so 2² = 2 × 2 = 4, not 2 × 2 = 4 wait, but common mistake is thinking it's base times exponent directly; follow PEMDAS by handling parentheses first, then exponents, multiplication/division left to right, and addition/subtraction left to right, for example, in 3 × 2², compute 2² = 4 first, then 3 × 4 = 12. For example, to evaluate 3 × 2², step 1: 2² = 4, step 2: 3 × 4 = 12; or for (3 × 2)², step 1: 3 × 2 = 6 inside implied, but actually parentheses would make it 36; or for 2³, step 1: 2 × 2 × 2 = 8. The student claims 3 × 2² = 36, but correctly it's first 2² = 4, then 3 × 4 = 12. A common error is violating order by multiplying first then exponentiating as (3 × 2)² = 6² = 36, which is what the student did, or treating exponent as addition like 3 × (2 + 2) = 12, or other arithmetic. The strategy is to (1) scan for parentheses, (2) evaluate exponents like 2² first, (3) multiply/divide left to right, (4) add/subtract if present, and (5) verify by knowing 2² = 4, times 3 is 12. Common exponents to know include 2² = 4, 2³ = 8; mistakes like this often come from confusing multiplication and exponentiation order.

Question 18

Which value makes this statement true? $3^4-2^3=\,?$

65
73 (correct answer)
56
19

Explanation: This question tests evaluating numerical expressions with whole-number exponents like 3⁴ and 2³ using the order of operations (PEMDAS: parentheses first, then exponents, multiplication and division left to right, addition and subtraction left to right). Exponent notation means the base is multiplied by itself the number of times indicated by the exponent, so 2³ = 2 × 2 × 2 = 8, not 2 × 3 = 6; follow PEMDAS strictly, for example, in 3 × 2² + 4, compute 2² = 4 first, then 3 × 4 = 12, then 12 + 4 = 16; with multiple exponents, evaluate each one separately before other operations. For example, to evaluate 2³ + 4², step 1: 2³ = 2 × 2 × 2 = 8, step 2: 4² = 4 × 4 = 16, step 3: 8 + 16 = 24; or for 3 × 2², step 1: 2² = 4, step 2: 3 × 4 = 12; or for (2 + 3)², step 1: 2 + 3 = 5, step 2: 5² = 25. For 3⁴ - 2³, compute exponents: 3⁴ = 81 (3×3×3×3), 2³ = 8, then subtract 81 - 8 = 73. A common error is treating as (3 - 2)^4 ^3 or something, or computing 3^4 as 3×4=12, leading to 12 - 8 = 4. The strategy is to (1) scan for parentheses (none), (2) evaluate exponents (3⁴ = 81, 2³ = 8), (3) no multiply/divide, (4) subtract (81 - 8 = 73), and (5) verify, like 81 - 8 about 70-80. Common exponents: 3⁴ = 81, 2³ = 8; mistakes include 3⁴ as 3^2=9 ×2=18 or arithmetic 81 - 8 = 72.

Question 19

Evaluate: $(2+5)^2-3^3$ .

13
49
22 (correct answer)
76

Explanation: This question tests evaluating numerical expressions with whole-number exponents like ² and 3³ using the order of operations (PEMDAS: parentheses first, then exponents, multiplication and division left to right, addition and subtraction left to right). Exponent notation means the base is multiplied by itself the number of times indicated by the exponent, so 2³ = 2 × 2 × 2 = 8, not 2 × 3 = 6; follow PEMDAS strictly, for example, in 3 × 2² + 4, compute 2² = 4 first, then 3 × 4 = 12, then 12 + 4 = 16; with multiple exponents, evaluate each one separately before other operations. For example, to evaluate 2³ + 4², step 1: 2³ = 2 × 2 × 2 = 8, step 2: 4² = 4 × 4 = 16, step 3: 8 + 16 = 24; or for 3 × 2², step 1: 2² = 4, step 2: 3 × 4 = 12; or for (2 + 3)², step 1: 2 + 3 = 5, step 2: 5² = 25. For (2 + 5)² - 3³, first handle parentheses: 2 + 5 = 7, then exponent 7² = 49, next 3³ = 27, then subtract 49 - 27 = 22. A common error is ignoring parentheses, like 2² + 5² - 3³ = 4 + 25 - 27 = 2, or treating exponent outside as distributing, like 2² + 5² = 29, then -27 = 2. The strategy is to (1) scan for parentheses and compute inside first (2 + 5 = 7), (2) evaluate exponents (7² = 49, 3³ = 27), (3) no multiplication or division, (4) subtract (49 - 27 = 22), and (5) verify reasonableness, like 49 - 27 is about 20-25. Common exponents include 3³ = 27, 7² = 49; mistakes involve arithmetic like 49 - 27 = 32 or forgetting to cube 3 as 9.

Question 20

Which value makes this statement true? $3^4-2^3=\,?$

19
73 (correct answer)
65
56