Evaluate Expressions With Exponents - 6th Grade Math

Evaluate: $(2+5)^2-3^3$ .
Answer: 22 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.
Evaluate $(9-4)^2 + 2^3$ .
Answer: 33 This question tests evaluating numerical expressions with whole-number exponents like (9 - 4)² and 2³ using the order of operations, often remembered as PEMDAS, where parentheses are handled first, then exponents before addition. 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 (2 + 3)², step 1: compute 2 + 3 = 5, step 2: 5² = 25; or for 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. For the expression (9 - 4)² + 2³, first evaluate inside parentheses: 9 - 4 = 5, then exponents: 5² = 25 and 2³ = 8, finally add 25 + 8 = 33. A common error is ignoring parentheses and computing 9² - 4² + 2³ = 81 - 16 + 8 = 73, or distributing exponent wrongly like (9)² - (4)² + 2³ = 81 - 16 + 8 = 73, or arithmetic error like 25 + 8 = 35. To solve these, use this strategy: (1) scan for parentheses and compute inside them first, (2) identify and evaluate all exponents like the parenthetical squared and 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 25 + 8 = 33. 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, ignoring parentheses by squaring each term separately, or adding before exponentiating.
A student claims that $3\times 2^2 = 36$ . What is the correct value of $3\times 2^2$ ?
Answer: 12 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.
A game score is calculated with the expression $5^2 - 2\times 3^2$ . What is the score?
Answer: 7 This question tests evaluating numerical expressions with whole-number exponents like 5² and 3² 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 3² = 3 × 3 = 9, not 3 × 2 = 6; follow PEMDAS by handling parentheses first, then exponents, multiplication/division left to right, and addition/subtraction left to right, for example, in 4² - 2 × 3, compute 4² = 16 first, then 2 × 3 = 6, then 16 - 6 = 10. For example, to evaluate 3² - 2², step 1: 3² = 9, step 2: 2² = 4, step 3: 9 - 4 = 5; or for 2 × 3², step 1: 3² = 9, step 2: 2 × 9 = 18; or for (2 + 3)², step 1: 2 + 3 = 5, step 2: 5² = 25. For 5² - 2 × 3², first compute exponents: 5² = 25 and 3² = 9, then multiply 2 × 9 = 18, and subtract 25 - 18 = 7. A common error is violating order by subtracting first like 5² - 2 = 23 then × 3² = 23 × 9 = 207, or treating exponents as multiplication like 3² = 6 leading to 25 - 2 × 6 = 25 - 12 = 13, or multiplying wrongly. The strategy is to (1) scan for parentheses and compute inside first, (2) evaluate all exponents like 5² and 3², (3) multiply/divide left to right, (4) add/subtract left to right, and (5) verify reasonable, such as 25 - 18 equaling 7. Common exponents to know include 2² = 4, 3² = 9, 5² = 25; mistakes often involve subtracting before multiplying.
A game uses the points rule $6\times 2^2-10$ . How many points is that?
Answer: 14 This question tests evaluating numerical expressions with whole-number exponents like 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 6 × 2² - 10, compute exponent 2² = 4, then multiply 6 × 4 = 24, finally subtract 24 - 10 = 14. A common error is multiplying before exponent, like (6 × 2)² - 10 = 12² - 10 = 144 - 10 = 134, or treating exponent as multiplication 6 × 2 × 2 - 10 = 24 - 10 = 14 (coincidentally correct but wrong reason). The strategy is to (1) scan for parentheses (none), (2) evaluate exponents (2² = 4), (3) multiply (6 × 4 = 24), (4) subtract (24 - 10 = 14), and (5) check if about 20-10=10ish, but precisely 14. Common exponents: 2² = 4; mistakes include subtracting first 2² - 10 = -6, then 6 × -6 = -36, or arithmetic 24 - 10 = 12.
Evaluate $\left(8-3\right)^2 + 2^3$ .
Answer: 33 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.
Which value makes this statement true? $3^4-2^3=\,?$
Answer: 73 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.
Evaluate the expression $10^2 \div 5^2$ .
Answer: 4 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.
The expression $2^3 + 3^2 \times 4$ equals which of the following?
Answer: $44$ When you see an expression with multiple operations like this one, you need to follow the order of operations (PEMDAS): Parentheses, Exponents, Multiplication and Division (left to right), Addition and Subtraction (left to right). Let's work through $2^3 + 3^2 \times 4$ step by step. First, handle the exponents: $2^3 = 2 \times 2 \times 2 = 8$ and $3^2 = 3 \times 3 = 9$ . So now you have $8 + 9 \times 4$ . Next, perform the multiplication: $9 \times 4 = 36$ . Finally, add: $8 + 36 = 44$ . Looking at the wrong answers, choice A ( $144$ ) likely comes from multiplying everything together: $8 \times 9 \times 2 = 144$ - but this ignores the order of operations completely. Choice B ( $68$ ) might result from adding first and then multiplying: $(8 + 9) \times 4 = 17 \times 4 = 68$ - this violates the rule that multiplication comes before addition. Choice C ( $100$ ) could come from various calculation errors, such as miscalculating the exponents or mixing up the operations. The key strategy here is to always write out each step of PEMDAS clearly. Don't try to do multiple operations in your head at once - work systematically through exponents first, then multiplication, then addition. This prevents the common mistake of working left to right without considering which operations have priority.
A game score is calculated with $7^2 - (5+1)^2$ . What is the value?
Answer: 13 This question tests evaluating numerical expressions with whole-number exponents like 7² and (5 + 1)² using the order of operations, often remembered as PEMDAS, where parentheses are handled first, then exponents before subtraction. 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 (2 + 3)², step 1: compute 2 + 3 = 5, step 2: 5² = 25; or for 5² - 3², step 1: 5² = 25, step 2: 3² = 9, step 3: 25 - 9 = 16; or for 3 × 2², step 1: 2² = 4, step 2: 3 × 4 = 12. For the expression 7² - (5 + 1)², first evaluate inside parentheses: 5 + 1 = 6, then exponents: 7² = 49 and 6² = 36, finally subtract 49 - 36 = 13. A common error is ignoring parentheses and computing 7² - 5 + 1² = 49 - 5 + 1 = 45, or distributing exponent wrongly like 7² - 5² - 1² = 49 - 25 - 1 = 23, or arithmetic error like 49 - 36 = 15. To solve these, use this strategy: (1) scan for parentheses and compute inside them first, (2) identify and evaluate all exponents like 7² or the parenthetical squared, (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 49 - 36 = 13. 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, ignoring parentheses by not adding inside first, or subtracting before exponentiating.
Evaluate $8^2 \div 4^2 + 3$ using the order of operations.
Answer: 7 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.
Evaluate: $(9-4)^2+2^3$ .
Answer: 33 This question tests evaluating numerical expressions with whole-number exponents like ² 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 (9 - 4)² + 2³, first parentheses: 9 - 4 = 5, then 5² = 25, next 2³ = 8, finally add 25 + 8 = 33. A common error is ignoring parentheses, like 9² - 4² + 2³ = 81 - 16 + 8 = 73, or distributing exponent wrongly as 9² + (-4)² = 81 + 16 = 97 + 8. The strategy is to (1) compute inside parentheses first (9 - 4 = 5), (2) evaluate exponents (5² = 25, 2³ = 8), (3) no multiply/divide, (4) add (25 + 8 = 33), and (5) verify, like 25 + 8 is about 30-35. Common exponents: 5² = 25, 2³ = 8; mistakes include arithmetic 25 + 8 = 23 or forgetting to square after subtracting.
Marcus is calculating $3^4 + 2^5$ and claims the answer is $5^9$ . What is the actual value of $3^4 + 2^5$ ?
Answer: $97$ 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 $3^4$ .
Sarah needs to evaluate $(2^3)^2 - 2^3 \times 2$ . What is the correct result?
Answer: $48$ 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.
A rectangular garden has length $2^4$ feet and width $2^2$ feet. If the garden is divided into square plots, each with side length $2$ feet, how many plots are there?
Answer: $16$ plots This problem combines exponents, area calculation, and division - key skills you'll use throughout geometry and algebra. First, let's find the garden's dimensions by evaluating the exponents. The length is $2^4 = 2 \times 2 \times 2 \times 2 = 16$ feet, and the width is $2^2 = 2 \times 2 = 4$ feet. The total area of the garden is $16 \times 4 = 64$ square feet. Each square plot has a side length of $2$ feet, so each plot covers $2 \times 2 = 4$ square feet. To find how many plots fit in the garden, divide the total area by the area per plot: $64 ÷ 4 = 16$ plots. This confirms answer B is correct. Looking at the wrong answers: Choice A ( $20$ plots) might come from incorrectly adding the dimensions instead of multiplying them, or making an error when dividing. Choice C ( $24$ plots) could result from miscalculating one of the exponents or the final division. Choice D ( $64$ plots) is a common trap - this is the garden's total area in square feet, but you forgot to divide by the area of each individual plot. When solving area division problems, always work systematically: calculate the total area first, then find the area of each smaller section, and finally divide. Double-check your exponent calculations since $2^4 = 16$ , not $8$ , is a frequent mistake. Remember that area problems require you to think in square units throughout.
A science club records a value using the expression $10^2 \div 5^2$ . What is the value?
Answer: 4 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.
Evaluate: $(2+5)^2-3^3$ .
Answer: 22 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.
Evaluate $2^4 + 3^2$ .
Answer: 25 This question tests evaluating numerical expressions with whole-number exponents like 2⁴ and 3² using the order of operations, often remembered as PEMDAS, where exponents are evaluated before 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 2⁴ + 3², evaluate the exponents: 2⁴ = 16 and 3² = 9, then add 16 + 9 = 25. A common error is treating exponent as multiplication like 2⁴ = 2 × 4 = 8, then 8 + 9 = 17, or adding before exponents which isn't possible here, or arithmetic error like 16 + 9 = 24. To solve these, use this strategy: (1) scan for parentheses and compute inside them first, (2) identify and evaluate all exponents like 2⁴ or 3², (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 16 + 9 = 25. 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, ignoring parentheses, or adding before exponentiating.
Which expression is equivalent to $4^3 \times 4^2$ ?
Answer: $4^5$ Using the rule $a^m \times a^n = a^{m+n}$ , we get $4^3 \times 4^2 = 4^{3+2} = 4^5$ . Choice A incorrectly multiplies the exponents: $4^{3 \times 2} = 4^6$ . Choice C incorrectly changes the base by squaring it: $(4^2)^3 \times 4^2 = 16^3 \times 4^2$ , but this doesn't simplify to $16^5$ . Choice D changes the base incorrectly to $2^3 = 8$ .
Evaluate the expression: $4^2+2^4\div 2^2$ .
Answer: 20 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.

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