Algebra Quiz: Deconstructing Complicated Expressions

What this quiz covers

This quiz focuses on Deconstructing Complicated Expressions, giving you a quick way to practice the rules, question types, and explanations that matter most for Algebra.

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.

Question 1

1. A product of $4$ and the power $\left(1-\frac{x}{3}\right)^5$ (correct answer)
2. A power with base $4\left(1-\frac{x}{3}\right)$ and exponent $5$
3. A sum of $4$ and $\left(1-\frac{x}{3}\right)^5$
4. A product of $\left(1-\frac{x}{3}\right)$ and $5$, then multiplied by $4$

Explanation: This question tests your ability to look at a complicated expression and understand its overall structure by seeing certain parts as single 'chunks' rather than getting lost in all the details. When an expression looks overwhelming, we can make sense of it by identifying the main parts and temporarily treating complex subexpressions as single units—like viewing P(1 + r)^n as 'P times [some factor]' where we don't worry about what's inside that factor yet. This 'chunking' helps us see the big picture structure: is it a product? A sum? Something raised to a power? Treating (1 - x/3) as one chunk in 4(1 - x/3)^5 shows it's 4 multiplied by that chunk raised to the 5th power. Choice A correctly views it as a product of 4 and the power (1 - x/3)^5, recognizing that the exponent applies only to the chunk, not to the 4. An option like choice B might apply the exponent to the whole product, but that's a gentle reminder to check where the parentheses are—the 4 is outside! When facing a complicated expression, try this: (1) Identify the outermost operation (product here), (2) Identify what that operation works on (your chunks), (3) If needed, break those chunks down one more level. Don't try to see everything at once—build understanding layer by layer!

Question 2

1. $3a(b+4)^2$
2. $3(b+4)^2$ (correct answer)
3. $a(b+4)$
4. $(b+4)^{2a}$

Explanation: This question tests your ability to look at a complicated expression and understand its overall structure by seeing certain parts as single 'chunks' rather than getting lost in all the details. When an expression looks overwhelming, we can make sense of it by identifying the main parts and temporarily treating complex subexpressions as single units—like viewing $P(1 + r)^n$ as 'P times [some factor]' where we don't worry about what's inside that factor yet. This 'chunking' helps us see the big picture structure: is it a product? A sum? Something raised to a power? In $3a(b+4)^2$, we can chunk it as $a$ multiplied by $3(b+4)^2$, where the factor $3(b+4)^2$ doesn't depend on $a$ at all. Choice C correctly identifies $3(b+4)^2$ as the factor not depending on $a$, recognizing that it's independent and captures the rest of the expression's structure. An option like choice A includes $a$ in the factor, but that's okay—just remind yourself to isolate what's truly independent of the underlined variable. In applied formulas, chunking helps you understand what each factor means: here, it separates the variable $a$ from the constant multiplier and the powered term, revealing how changes in $a$ scale the whole expression.

Question 3

1. The expression is squared after adding $2$ to $x-3$
2. Only the $2$ in the denominator is squared
3. The entire quantity $\frac{x-3}{2}$ is squared (correct answer)
4. Only $x-3$ is squared

Explanation: This question tests your ability to look at a complicated expression and understand its overall structure by seeing certain parts as single 'chunks' rather than getting lost in all the details. When an expression looks overwhelming, we can make sense of it by identifying the main parts and temporarily treating complex subexpressions as single units—like viewing $P(1 + r)^n$ as 'P times [some factor]' where we don't worry about what's inside that factor yet. This 'chunking' helps us see the big picture structure: is it a product? A sum? Something raised to a power? For $g(x) = \left( \frac{x-3}{2} \right)^2$, viewing the fraction $(x-3)/2$ as a single chunk shows that the entire chunk is what's being squared. Choice C correctly identifies that the entire quantity $(x-3)/2$ is squared, recognizing the parentheses enclose the whole fraction for the exponent. Something like choice A might think only the numerator is squared, but remember, the exponent applies to everything inside the parentheses—it's all one unit! In applied formulas, chunking helps you understand what each factor means: here, it clarifies that the squaring operates on the scaled difference $(x-3)/2$, revealing relationships like how it models quadratic behavior.

Question 4

1. A sum: $2\pi r + (r+h)$
2. A product: $(2\pi r)\cdot(r+h)$ (correct answer)
3. A power: $(2\pi r)^{(r+h)}$
4. A difference: $2\pi(r-h)$

Explanation: This question tests your ability to look at a complicated expression and understand its overall structure by seeing certain parts as single 'chunks' rather than getting lost in all the details. When an expression looks overwhelming, we can make sense of it by identifying the main parts and temporarily treating complex subexpressions as single units—like viewing $P(1 + r)^n$ as 'P times [some factor]' where we don't worry about what's inside that factor yet. This 'chunking' helps us see the big picture structure: is it a product? A sum? Something raised to a power? For $2\pi r(r+h)$, viewing $(r+h)$ as one chunk reveals it's $2\pi r$ multiplied by that chunk, showing a product structure. Choice B correctly views the expression as the product $(2\pi r) \cdot (r+h)$, recognizing the key insight that chunking $(r+h)$ highlights the multiplicative nature without expanding everything. Something like choice A might see it as a sum, but that's a common mix-up—look for the lack of a plus sign outside the chunk! When facing a complicated expression, try this: (1) Identify the outermost operation (here, multiplication), (2) Identify what that operation works on (your chunks), (3) If needed, break those chunks down one more level. Don't try to see everything at once—build understanding layer by layer!

Question 5

1. $3a(b+4)^2$
2. $3(b+4)^2$ (correct answer)
3. $a(b+4)$
4. $(b+4)^{2a}$

Explanation: This question tests your ability to look at a complicated expression and understand its overall structure by seeing certain parts as single 'chunks' rather than getting lost in all the details. When an expression looks overwhelming, we can make sense of it by identifying the main parts and temporarily treating complex subexpressions as single units—like viewing $P(1 + r)^n$ as 'P times [some factor]' where we don't worry about what's inside that factor yet. This 'chunking' helps us see the big picture structure: is it a product? A sum? Something raised to a power? In $3a(b+4)^2$, we can chunk it as a multiplied by $3(b+4)^2$, where the factor $3(b+4)^2$ doesn't depend on a at all. Choice C correctly identifies $3(b+4)^2$ as the factor not depending on a, recognizing that it's independent and captures the rest of the expression's structure. An option like choice A includes a in the factor, but that's okay—just remind yourself to isolate what's truly independent of the underlined variable. In applied formulas, chunking helps you understand what each factor means: here, it separates the variable a from the constant multiplier and the powered term, revealing how changes in a scale the whole expression.

Question 6

1. Outer operation: addition; inner expressions: $\sqrt{(x-4)^2}$ and $\sqrt{9}$
2. Outer operation: square root; inner expression: $(x-4)^2+9$ (correct answer)
3. Outer operation: subtraction; inner expression: $(x-4)^2-9$
4. Outer operation: squaring; inner expression: $\sqrt{x-4}+9$

Explanation: This question tests your ability to look at a complicated expression and understand its overall structure by seeing certain parts as single 'chunks' rather than getting lost in all the details. When an expression looks overwhelming, we can make sense of it by identifying the main parts and temporarily treating complex subexpressions as single units—like viewing P(1 + r)^n as 'P times [some factor]' where we don't worry about what's inside that factor yet. This 'chunking' helps us see the big picture structure: is it a product? A sum? Something raised to a power? In $\sqrt{(x-4)^2+9}$, the outermost operation is the square root—it applies to everything inside. Inside the square root, we have $(x-4)^2+9$, which is a sum. If we treat $(x-4)$ as a single unit, then $(x-4)^2$ means 'square that unit,' and we're adding 9 to that squared result. Choice A correctly identifies 'Outer operation: square root; inner expression: $(x-4)^2+9$,' recognizing the nested structure where we first compute what's inside the radical, then take its square root. Choice C incorrectly suggests we can split the square root across addition as $\sqrt{(x-4)^2} + \sqrt{9}$, but that's not how square roots work—$\sqrt{a+b} \neq \sqrt{a} + \sqrt{b}$! When facing a complicated expression, try this: (1) Identify the outermost operation (is the whole thing a product? a sum? a power?), (2) Identify what that operation works on (these are your main 'chunks'), (3) If needed, break those chunks down one more level. Don't try to see everything at once—build understanding layer by layer!

Question 7

1. Inner part: $x^2$; outer operation: add $1$ then cube
2. Inner part: $x^2+1$; outer operation: cube the result (correct answer)
3. Inner part: $x+1$; outer operation: square then cube
4. Inner part: $x^2+1$; outer operation: multiply by $3$

Explanation: This question tests your ability to look at a complicated expression and understand its overall structure by seeing certain parts as single 'chunks' rather than getting lost in all the details. When an expression looks overwhelming, we can make sense of it by identifying the main parts and temporarily treating complex subexpressions as single units—like viewing P(1 + r)^n as 'P times [some factor]' where we don't worry about what's inside that factor yet. This 'chunking' helps us see the big picture structure: is it a product? A sum? Something raised to a power? For f(x) = (x^2 + 1)^3, we identify the inner chunk as (x^2 + 1) and the outer operation as cubing that chunk. Choice B correctly views the inner part as x^2 + 1 and the outer operation as cubing the result, recognizing the composition where addition happens before the power. An option like choice A might separate the add 1 from the cubing incorrectly, but gently check the parentheses—they group x^2 + 1 together! A helpful trick: circle or box the parts you want to treat as units. For example, box (x^2 + 1) and think '[box] cubed.' This visual chunking helps your brain organize the structure.

Question 8

1. It is the product $r\cdot(\pi r+2\pi h)$, so $r$ is a common factor in both terms (correct answer)
2. It is the product $\pi\cdot(r^2+2rh)$, so the common factor is $\pi$
3. It is a difference: $\pi r^2-2\pi rh$
4. It is a power: $(\pi r h)^2$

Explanation: This question tests your ability to look at a complicated expression and understand its overall structure by seeing certain parts as single 'chunks' rather than getting lost in all the details. When an expression looks overwhelming, we can make sense of it by identifying the main parts and temporarily treating complex subexpressions as single units—like viewing P(1 + r)^n as 'P times [some factor]' where we don't worry about what's inside that factor yet. This 'chunking' helps us see the big picture structure: is it a product? A sum? Something raised to a power? In A = π r^2 + 2 π r h, we can chunk it by factoring out r, seeing it as r times (π r + 2 π h), highlighting the common r in both terms. Choice B correctly views it as the product r · (π r + 2 π h), recognizing that r is a common factor involving r without fully simplifying. Something like choice A focuses on π instead, but the question emphasizes a factor involving r—so that's a supportive nudge to match the prompt! When facing a complicated expression, try this: (1) Identify the outermost operation (a sum here), (2) Look for common factors in the terms (like r), (3) If needed, break those chunks down one more level. Don't try to see everything at once—build understanding layer by layer!

Question 9

1. A product: $(2\pi r)\times(r+h)$ (correct answer)
2. A sum: $2\pi r + (r+h)$
3. A product: $2\pi\times(rh)$
4. A power: $(2\pi r)^{(r+h)}$

Explanation: This question tests your ability to look at a complicated expression and understand its overall structure by seeing certain parts as single 'chunks' rather than getting lost in all the details. When an expression looks overwhelming, we can make sense of it by identifying the main parts and temporarily treating complex subexpressions as single units—like viewing $P(1 + r)^n$ as 'P times [some factor]' where we don't worry about what's inside that factor yet. This 'chunking' helps us see the big picture structure: is it a product? A sum? Something raised to a power? In the cylinder formula part $2\pi r(r+h)$, treating $(r+h)$ as a single unit shows it's $2\pi r$ multiplied by that unit, making the whole thing a product of two chunks. Choice B correctly views the expression as a product of ($2\pi r$) and ($r+h$), recognizing that these chunks combine multiplicatively to represent the lateral surface area factor. It's easy to mistake it for a sum like in choice A, but by chunking $(r+h)$, we see the multiplication is key—great job spotting that distinction! A helpful trick: circle or box the parts you want to treat as units. For example, in $P(1 + r)^n$, box the $(1 + r)^n$ part and think 'P times [box].' This visual chunking helps your brain organize the structure. Once you understand the structure, then you can dive into the details of each part if needed!

Question 10

1. A product of $4$ and the power $\left(1-\frac{x}{3}\right)^5$ (correct answer)
2. A power with base $4\left(1-\frac{x}{3}\right)$ and exponent $5$
3. A sum of $4$ and $\left(1-\frac{x}{3}\right)^5$
4. A product of $\left(1-\frac{x}{3}\right)$ and $5$, then multiplied by $4$

Explanation: This question tests your ability to look at a complicated expression and understand its overall structure by seeing certain parts as single 'chunks' rather than getting lost in all the details. When an expression looks overwhelming, we can make sense of it by identifying the main parts and temporarily treating complex subexpressions as single units—like viewing P(1 + r)^n as 'P times [some factor]' where we don't worry about what's inside that factor yet. This 'chunking' helps us see the big picture structure: is it a product? A sum? Something raised to a power? Treating (1 - x/3) as one chunk in 4(1 - x/3)^5 shows it's 4 multiplied by that chunk raised to the 5th power. Choice A correctly views it as a product of 4 and the power (1 - x/3)^5, recognizing that the exponent applies only to the chunk, not to the 4. An option like choice B might apply the exponent to the whole product, but that's a gentle reminder to check where the parentheses are—the 4 is outside! When facing a complicated expression, try this: (1) Identify the outermost operation (product here), (2) Identify what that operation works on (your chunks), (3) If needed, break those chunks down one more level. Don't try to see everything at once—build understanding layer by layer!

Question 11

1. Two raised to the power $5(x+1)$
2. Five times two, plus $(x+1)$
3. Five times a power of 2, where the exponent is $(x+1)$ (correct answer)
4. The power $(5\cdot 2)^{x}+1$

Explanation: This question tests your ability to look at a complicated expression and understand its overall structure by seeing certain parts as single 'chunks' rather than getting lost in all the details. When an expression looks overwhelming, we can make sense of it by identifying the main parts and temporarily treating complex subexpressions as single units—like viewing P(1 + r)^n as 'P times [some factor]' where we don't worry about what's inside that factor yet. This 'chunking' helps us see the big picture structure: is it a product? A sum? Something raised to a power? In $5\,2^{(x+1)}$, we need to carefully identify what's an exponent and what's a coefficient. The expression means $5 \times 2^{(x+1)}$, where 5 is multiplied by the power $2^{(x+1)}$. The chunk $(x+1)$ is the exponent on the base 2, not something that affects the 5. Choice C correctly views this as 'Five times a power of 2, where the exponent is $(x+1)$,' recognizing that we multiply 5 by the entire exponential expression $2^{(x+1)}$. Choice B incorrectly suggests the base is 2 and the exponent is $5(x+1)$, which would be written as $2^{5(x+1)}$—a very different expression! A helpful trick: circle or box the parts you want to treat as units. For example, in $5\,2^{(x+1)}$, box the $2^{(x+1)}$ part and think '5 times [box].' This visual chunking helps your brain organize the structure. Once you understand the structure, then you can dive into the details of each part if needed!

Question 12

1. Both $15$ and $2.5n$ are constant
2. Only $2.5$ varies with $n$, and $15n$ is constant
3. $2.5n$ varies with $n$, and $15$ is constant (correct answer)
4. $15$ varies with $n$, and $2.5n$ is constant

Explanation: This question tests your ability to look at a complicated expression and understand its overall structure by seeing certain parts as single 'chunks' rather than getting lost in all the details. When an expression looks overwhelming, we can make sense of it by identifying the main parts and temporarily treating complex subexpressions as single units—like viewing P(1 + r)^n as 'P times [some factor]' where we don't worry about what's inside that factor yet. This 'chunking' helps us see the big picture structure: is it a product? A sum? Something raised to a power? In C(n) = 15 + 2.5n, we can chunk it as a sum where 15 is a fixed chunk and 2.5n is the part that changes with n. Choice C correctly identifies that 2.5n varies with n and 15 is constant, recognizing the independence of the fixed cost from the variable one. Something like choice A might think both are constant, but that's okay—just remember to spot which terms include the variable n! In applied formulas, chunking helps you understand what each factor means: here, the constant 15 might be a base fee, while 2.5n scales with quantity, showing how total cost grows linearly.

Question 13

1. A sum: $2\pi r + (r+h)$
2. A product: $(2\pi r)\cdot(r+h)$ (correct answer)
3. A power: $(2\pi r)^{(r+h)}$
4. A difference: $2\pi(r-h)$

Explanation: This question tests your ability to look at a complicated expression and understand its overall structure by seeing certain parts as single 'chunks' rather than getting lost in all the details. When an expression looks overwhelming, we can make sense of it by identifying the main parts and temporarily treating complex subexpressions as single units—like viewing P(1 + r)^n as 'P times [some factor]' where we don't worry about what's inside that factor yet. This 'chunking' helps us see the big picture structure: is it a product? A sum? Something raised to a power? For 2πr(r+h), viewing (r+h) as one chunk reveals it's 2πr multiplied by that chunk, showing a product structure. Choice B correctly views the expression as the product (2πr) · (r+h), recognizing the key insight that chunking (r+h) highlights the multiplicative nature without expanding everything. Something like choice A might see it as a sum, but that's a common mix-up—look for the lack of a plus sign outside the chunk! When facing a complicated expression, try this: (1) Identify the outermost operation (here, multiplication), (2) Identify what that operation works on (your chunks), (3) If needed, break those chunks down one more level. Don't try to see everything at once—build understanding layer by layer!

Question 14

1. The cube of $5(2x-1)$
2. A sum of $5$ and $(2x-1)^3$
3. A product of $5$ and the cube of $(2x-1)$ (correct answer)
4. The cube of $2x$ minus $1$, then multiplied by $5$

Explanation: This question tests your ability to look at a complicated expression and understand its overall structure by seeing certain parts as single 'chunks' rather than getting lost in all the details. When an expression looks overwhelming, we can make sense of it by identifying the main parts and temporarily treating complex subexpressions as single units—like viewing $P(1 + r)^n$ as 'P times [some factor]' where we don't worry about what's inside that factor yet. This 'chunking' helps us see the big picture structure: is it a product? A sum? Something raised to a power? For $5(2x-1)^3$, viewing $(2x-1)$ as a single unit reveals it's 5 multiplied by that unit cubed, so the structure is a product with a constant and a power. Choice C correctly views the expression as a product of 5 and the cube of $(2x-1)$, recognizing that the cubing applies only to the chunk inside. You might think it's just cubing the whole thing like in choice A, but chunking shows the 5 is outside the power—nice catch on that subtlety! When facing a complicated expression, try this: (1) Identify the outermost operation (is the whole thing a product? a sum? a power?), (2) Identify what that operation works on (these are your main 'chunks'), (3) If needed, break those chunks down one more level. Don't try to see everything at once—build understanding layer by layer!

Question 15

1. It is the product $r\cdot(\pi r+2\pi h)$, so $r$ is a common factor in both terms (correct answer)
2. It is the product $\pi\cdot(r^2+2rh)$, so the common factor is $\pi$
3. It is a power: $(\pi r h)^2$
4. It is a difference: $\pi r^2-2\pi rh$

Explanation: This question tests your ability to look at a complicated expression and understand its overall structure by seeing certain parts as single 'chunks' rather than getting lost in all the details. When an expression looks overwhelming, we can make sense of it by identifying the main parts and temporarily treating complex subexpressions as single units—like viewing P(1 + r)^n as 'P times [some factor]' where we don't worry about what's inside that factor yet. This 'chunking' helps us see the big picture structure: is it a product? A sum? Something raised to a power? In A = π r^2 + 2 π r h, we can chunk it by factoring out r, seeing it as r times (π r + 2 π h), highlighting the common r in both terms. Choice B correctly views it as the product r · (π r + 2 π h), recognizing that r is a common factor involving r without fully simplifying. Something like choice A focuses on π instead, but the question emphasizes a factor involving r—so that's a supportive nudge to match the prompt! When facing a complicated expression, try this: (1) Identify the outermost operation (a sum here), (2) Look for common factors in the terms (like r), (3) If needed, break those chunks down one more level. Don't try to see everything at once—build understanding layer by layer!

Question 16

1. Take the square root of $5x$, then add $1$, then add $2$
2. Take the square root of $(5x+1)$, then add $2$ (correct answer)
3. Add $2$ to $x$, multiply by $5$, then take the square root, then add $1$
4. Take the square root of $(5x+1+2)$

Explanation: This question tests your ability to look at a complicated expression and understand its overall structure by seeing certain parts as single 'chunks' rather than getting lost in all the details. When an expression looks overwhelming, we can make sense of it by identifying the main parts and temporarily treating complex subexpressions as single units—like viewing P(1 + r)^n as 'P times [some factor]' where we don't worry about what's inside that factor yet. This 'chunking' helps us see the big picture structure: is it a product? A sum? Something raised to a power? In √(5x+1) + 2, we chunk the inside of the square root as (5x+1), then take the square root of that chunk and add 2. Choice B correctly views the expression as taking the square root of (5x+1) then adding 2, recognizing that the addition of 2 happens outside the root. An option like choice A might add 1 after the root, but gently note that the +1 is inside the chunk under the square root—parentheses matter! A helpful trick: circle or box the parts you want to treat as units. For example, box (5x+1) and think 'square root of [box] plus 2.' This visual chunking helps your brain organize the structure.

Question 17

1. Take the square root of $5x$, then add $1$, then add $2$
2. Take the square root of $(5x+1)$, then add $2$ (correct answer)
3. Add $2$ to $x$, multiply by $5$, then take the square root, then add $1$
4. Take the square root of $(5x+1+2)$

Explanation: This question tests your ability to look at a complicated expression and understand its overall structure by seeing certain parts as single 'chunks' rather than getting lost in all the details. When an expression looks overwhelming, we can make sense of it by identifying the main parts and temporarily treating complex subexpressions as single units—like viewing $P(1 + r)^n$ as 'P times [some factor]' where we don't worry about what's inside that factor yet. This 'chunking' helps us see the big picture structure: is it a product? A sum? Something raised to a power? In $\sqrt{5x+1} + 2$, we chunk the inside of the square root as $(5x+1)$, then take the square root of that chunk and add 2. Choice B correctly views the expression as taking the square root of $(5x+1)$ then adding 2, recognizing that the addition of 2 happens outside the root. An option like choice A might add 1 after the root, but gently note that the +1 is inside the chunk under the square root—parentheses matter! A helpful trick: circle or box the parts you want to treat as units. For example, box $(5x+1)$ and think 'square root of [box] plus 2.' This visual chunking helps your brain organize the structure.

Question 18

1. $2.5n$ varies with $n$, and $15$ is constant (correct answer)
2. Only $2.5$ varies with $n$, and $15n$ is constant
3. $15$ varies with $n$, and $2.5n$ is constant
4. Both $15$ and $2.5n$ are constant

Explanation: This question tests your ability to look at a complicated expression and understand its overall structure by seeing certain parts as single 'chunks' rather than getting lost in all the details. When an expression looks overwhelming, we can make sense of it by identifying the main parts and temporarily treating complex subexpressions as single units—like viewing $P(1 + r)^n$ as 'P times [some factor]' where we don't worry about what's inside that factor yet. This 'chunking' helps us see the big picture structure: is it a product? A sum? Something raised to a power? In $C(n) = 15 + 2.5n$, we can chunk it as a sum where 15 is a fixed chunk and $2.5n$ is the part that changes with $n$. Choice C correctly identifies that $2.5n$ varies with $n$ and 15 is constant, recognizing the independence of the fixed cost from the variable one. Something like choice A might think both are constant, but that's okay—just remember to spot which terms include the variable $n$! In applied formulas, chunking helps you understand what each factor means: here, the constant 15 might be a base fee, while $2.5n$ scales with quantity, showing how total cost grows linearly.

Question 19

1. $A = (1 + r)^t[P + \frac{500}{(1 + r)^2}]$, showing a common factor of $(1 + r)^t$ with a bracketed expression
2. $A = P(1 + r)^t + \frac{500}{(1 + r)^2}(1 + r)^t$, showing both terms as multiples of $(1 + r)^t$ with different coefficients
3. $A = (P + 500)(1 + r)^t - 500(1 + r)^2$, showing the sum distributed across exponential terms
4. $A = (1 + r)^{t-2}[P(1 + r)^2 + 500]$, showing a common exponential factor and a bracketed sum depending on $P$ (correct answer)

Explanation: When you encounter algebraic expressions with multiple exponential terms, look for opportunities to factor out common elements to reveal the underlying structure. This process helps simplify complex expressions and makes relationships between terms clearer. To find the best decomposition, you need to identify the greatest common factor among the exponential terms. In $A = P(1 + r)^t + 500(1 + r)^{t-2}$, both terms contain powers of $(1 + r)$. Since $t-2$ is smaller than $t$, you can factor out $(1 + r)^{t-2}$ from both terms. From the first term: $P(1 + r)^t = P(1 + r)^{t-2} \cdot (1 + r)^2$ From the second term: $500(1 + r)^{t-2}$ remains unchanged. Factoring gives: $A = (1 + r)^{t-2}[P(1 + r)^2 + 500]$ This matches answer choice D, which correctly shows $(1 + r)^{t-2}$ as the common exponential factor with a bracketed expression containing the remaining terms. Answer A incorrectly factors out $(1 + r)^t$, which would require dividing the second term by $(1 + r)^2$, but this creates an incorrect negative exponent situation. Answer B rewrites the expression without actually factoring, just rearranging terms. Answer C attempts to distribute incorrectly and changes the mathematical meaning entirely. Study tip: When factoring exponential expressions, always factor out the term with the smallest exponent first. This ensures you're pulling out the true greatest common factor and reveals the clearest structural relationships.

Question 20

1. A quadratic expression in $u$ with leading coefficient 3, linear coefficient -5, and constant term 7 (correct answer)
2. A quadratic expression in $u$ with leading coefficient 3, linear coefficient 5, and constant term 7
3. A linear expression in $u$ with slope -5 and y-intercept equal to $3u^2 + 7$
4. A cubic expression in $u$ because the original expression contains both $u^2$ and $u$ terms

Explanation: When we substitute $u = x + 2$, the expression becomes $3u^2 - 5u + 7$. This is a quadratic in $u$ with leading coefficient 3, linear coefficient -5, and constant term 7. Choice B incorrectly states the linear coefficient as positive 5. Choice C incorrectly describes it as linear. Choice D incorrectly identifies it as cubic when it's clearly quadratic.