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 $$3^4$$.

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.

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.

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.

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.

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 $$2^3 = 8$$.

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.

When you encounter problems comparing areas and surface areas of different shapes, you need to calculate each measurement separately, then find their difference.

First, let's find the area of the square. The side length is $$3^2 = 9$$ units, so the area is $$9 \times 9 = 81$$ square units.

Next, calculate the surface area of the cube. A cube has 6 identical square faces, each with area $$3 \times 3 = 9$$ square units. The total surface area is $$6 \times 9 = 54$$ square units.

The difference between the square's area and the cube's surface area is $$81 - 54 = 27$$ square units.

Looking at the wrong answers: Choice A gives $$54$$ square units, which is just the cube's surface area without subtracting it from the square's area. Choice B shows $$35$$ square units, which might result from incorrectly calculating $$54 - 19$$ if you mistakenly found the square's side length as $$19$$ instead of $$9$$. Choice D gives $$81$$ square units, which is only the square's area without performing the subtraction.

The key strategy here is to work systematically: calculate each measurement completely before finding the difference. Also, remember that $$3^2$$ means $$3 \times 3 = 9$$, not $$3 \times 2 = 6$$. When comparing geometric measurements, always double-check that you're using the correct formulas—area for 2D shapes and surface area for 3D shapes.

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.

This question tests evaluating numerical expressions with whole-number exponents like 9² 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 ÷ 2² + 3, compute 16 ÷ 4 = 4, then +3 = 7. For example, to evaluate 2³ ÷ 2² + 1, step 1: 2³ = 8, step 2: 2² = 4, step 3: 8 ÷ 4 = 2, step 4: 2 + 1 = 3; 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 9² ÷ 3² + 4, first compute exponents: 9² = 81 and 3² = 9, then divide 81 ÷ 9 = 9, and add 9 + 4 = 13. A common error is adding first like 9² ÷ (3² + 4) = 81 ÷ 13 ≈ 6.23, or violating order by dividing bases first incorrectly, or treating exponents as multiplication like 9² = 18 leading to 18 ÷ 6 + 4 = 3 + 4 = 7. The strategy is to (1) scan for parentheses and compute inside first, (2) evaluate all exponents like 9² and 3², (3) multiply/divide left to right, (4) add/subtract left to right, and (5) verify reasonable, such as 81 ÷ 9 = 9 plus 4 equaling 13. Common exponents to know include 3² = 9, 9² = 81; mistakes often involve adding before dividing.