All practice questions

Match each unsimplified algebraic expression with its correctly simplified equivalent.

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Answer

The correct pairings map each unsimplified expression to its distributed and combined form.

Each expression on the left is matched with its simplified form on the right by accurately applying the distributive property and combining like terms.

Step-by-Step Solution

1

Simplify the first expression: 2(3y + 4).

6y + 8

Multiply the outer term 2 by both terms inside the parentheses (3y and 4).

2

Simplify the second expression: 5y + 2y - 3.

7y - 3

Add the coefficients of the like terms containing y (5 and 2).

3

Simplify the third expression: -3(y - 2).

-3y + 6

Distribute the -3, noting that a negative multiplied by a negative results in a positive (+6).

4

Simplify the fourth expression: 4y + 2z - y + 3z.

3y + 5z

Group and add the coefficients of the y terms together, and the z terms together.

Key Concept

Simplifying Algebraic Expressions

Question 882Question →

Read the following technical maintenance excerpt:

"When repairing damaged carbon-composite aircraft panels, technicians inspect the cured resin matrix to ensure the patch is cohesive—that is, firmly bonded into a single unified mass. The surface must also remain unblemished, meaning entirely free of cracks or surface defects, before primer application. Additionally, any structural filler used during edge sealing must be malleable, or easily shaped and worked by hand, to guarantee an airtight aerodynamic seal."

Based on the synonymous context clues in the passage, match each vocabulary term with its contextual definition.

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Answer

'cohesive' pairs with 'Firmly bonded into a single unified mass'; 'unblemished' pairs with 'Entirely free of cracks or surface defects'; 'malleable' pairs with 'Easily shaped and worked by hand'.

Each vocabulary term is accompanied by a specific synonymous indicator within the sentence: 'cohesive' is offset by 'that is' followed by 'firmly bonded into a single unified mass'; 'unblemished' is explicitly clarified by 'meaning entirely free of cracks or surface defects'; and 'malleable' is linked via 'or' to 'easily shaped and worked by hand.' These explicit textual clues provide unambiguous matches for each term.

Step-by-Step Solution

1

Locate the word 'cohesive' in the text and inspect the punctuation and transition words immediately following it.

The dash and the phrase 'that is' introduce the direct definition 'firmly bonded into a single unified mass.'

Writers frequently use dashes and 'that is' to restate an unfamiliar technical term with an accessible synonym.

2

Locate the word 'unblemished' and identify the explanatory clause attached to it.

The participle 'meaning' introduces the definition 'entirely free of cracks or surface defects.'

The word 'meaning' functions as an explicit signal that a synonymous explanation is following.

3

Locate the word 'malleable' and evaluate the parallel phrasing offset by commas.

The coordinating conjunction 'or' connects 'malleable' directly to 'easily shaped and worked by hand.'

The conjunction 'or' is often used to present a familiar synonym for a specialized word.

Key Concept

Identifying Synonymous Context Clues (Appositives, Restatement Markers, and Parallel Synonyms)

Alternative Method

Scan the passage directly for structural clue markers such as dashes, 'that is', 'meaning', and introductory 'or' to extract the definitions before evaluating the rest of the text.

Estimated Time:1m 0s

Question 883Question →

Match each unsimplified algebraic expression on the left with its completely simplified equivalent on the right.

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Answer

The unsimplified expressions must be correctly expanded using the distributive property, observing proper handling of negative multipliers and nested grouping symbols, and then reduced by combining like terms.

Each expression correctly matches its fully reduced polynomial form. The reductions require systematic application of the distributive property, careful tracking of positive and negative coefficients, and the grouping and addition of like variables.

Step-by-Step Solution

1

Simplify the first expression:

3(2a - b) - 4(a + 2b)

6a - 3b - 4a - 8b = 2a - 11b

Multiply each term inside the parentheses by the outside factor, then group like terms (

6a - 4a

and

-3b - 8b

).

2

Simplify the second expression:

-2(a - 3b) + 5(2a - b)

-2a + 6b + 10a - 5b = 8a + b

Properly distribute the negative sign in the first term (

-2 \times -3b = 6b

), then combine like terms.

3

Simplify the third expression:

a(3 - 2b) - 2(a - ab)

3a - 2ab - 2a + 2ab = a

Distribute the variables and constants, recognizing that

-2ab

and

+2ab

are additive inverses and cancel to zero.

4

Simplify the fourth expression:

4a - [2b - 3(a - b)]

4a - [2b - 3a + 3b] = 4a - [5b - 3a] = 4a - 5b + 3a = 7a - 5b

Work from the innermost parentheses outward. Distribute

-3

, combine like terms inside the bracket, and distribute the outer negative sign.

Key Concept

Simplifying Algebraic Expressions

Estimated Time:2m 0s

Question 884Question →

Read the following technical evaluations regarding specialized industrial equipment. Match each excerpt to the author's primary tone or attitude based on the word choice and phrasing used.

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Answer

Match the excerpt praising the geothermal design as a 'brilliant leap forward' with 'Appreciative and enthusiastic'; match the excerpt criticizing the valve failure and 'reckless disregard' with 'Critical and disapproving'; match the excerpt providing numerical flow rates and pressures with 'Objective and matter-of-fact'.

Tone is determined by examining an author's diction and perspective. The first statement utilizes laudatory language ('brilliant,' 'remarkable') showing appreciation; the second uses condemnatory phrasing ('routinely jammed,' 'reckless disregard') establishing a critical attitude; the third relies strictly on unbiased numerical parameters, creating an objective tone.

Step-by-Step Solution

1

Analyze the descriptive adjectives and emotive language in the first excerpt.

Identified positive descriptors ('brilliant leap forward', 'remarkable quietness') indicating praise.

Word choice reveals the author's supportive and admiring stance toward the design.

2

Examine the diction in the second excerpt for signs of evaluation or judgment.

Identified negative evaluative terms ('routinely jammed', 'reckless disregard') indicating strong disapproval.

The author explicitly condemns the safety oversight and challenges marketing claims.

3

Evaluate the third excerpt for the presence or absence of subjective commentary.

Found only measurable quantities ('twelve liters per minute', 'forty to forty-five pounds per square inch') with zero emotional coloring.

The absence of value judgments indicates a purely factual, neutral presentation.

Key Concept

Inferring Author Tone and Attitude

Estimated Time:1m 15s

Question 885Question →

Read the following technical report on naval submarine trials:

During deep-submergence pressure trials, the experimental hull alloy remained remarkably impervious to corrosive seawater, whereas standard steel plating was vulnerable to oxidation. Furthermore, while the interior seals remained pliable despite freezing depths, older rubber gaskets grew brittle and cracked. Finally, instead of producing a sluggish response, the newly fitted hydraulic steering system was agile under heavy hydrodynamic loads.

Based on the contrast clues in the passage, match each highlighted term to its correct contextual meaning.

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Answer

impervious matches 'Resistant to penetration and unaffected by corrosive forces'; pliable matches 'Flexible and supple under pressure without cracking'; agile matches 'Nimble and capable of quick, efficient movement'.

Each match is established through explicit contrast indicators in the passage: 'impervious' contrasts with 'vulnerable' across 'whereas'; 'pliable' contrasts with 'brittle' across 'while'; and 'agile' contrasts with 'sluggish' across 'instead of'.

Step-by-Step Solution

1

Locate contrast signal words for each target term in the passage.

Identified 'whereas' for impervious, 'while'/'despite' for pliable, and 'instead of' for agile.

Contrast signal words establish opposing relationships between the target vocabulary and known benchmark descriptions.

2

Determine the opposing phrase for each target word to deduce its true definition.

'Impervious' opposes 'vulnerable to oxidation'; 'pliable' opposes 'brittle and cracked'; 'agile' opposes 'sluggish response'.

The contextual definition of each target word is the opposite of the condition described in the contrasting clause.

3

Pair each word with its corresponding contextual definition.

All three terms are successfully matched to their precise contextual definitions.

Matching each term to its positive equivalent verifies complete reading comprehension.

Key Concept

Identifying Antonym and Contrast Context Clues

Estimated Time:1m 0s

Question 886Question →

A military supply officer is preparing cargo for an allied joint exercise. Match the following US Customary measurements to their approximate Metric equivalents based on standard conversion rates ( $1 \text{ inch} = 2.54 \text{ cm}$ , $1 \text{ kg} \approx 2.2 \text{ lbs}$ , $1 \text{ mile} \approx 1.6 \text{ km}$ , $1 \text{ gallon} \approx 3.78 \text{ L}$ ).

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Answer

The correct pairings are:

10 \text{ inches}

matches with

25.4 \text{ centimeters}

,

5 \text{ miles}

matches with

8 \text{ kilometers}

,

110 \text{ pounds}

matches with

50 \text{ kilograms}

, and

10 \text{ gallons}

matches with

37.8 \text{ liters}

.

Each measurement is paired with its metric equivalent by correctly applying the given conversion factors. Multiplying is used to convert inches to centimeters, miles to kilometers, and gallons to liters. Dividing is required to convert pounds to kilograms.

Step-by-Step Solution

1

Identify the conversion factors provided in the prompt.

1 \text{ inch} = 2.54 \text{ cm}

,

1 \text{ kg} = 2.2 \text{ lbs}

,

1 \text{ mile} = 1.6 \text{ km}

,

1 \text{ gallon} = 3.78 \text{ L}

These rates determine whether to multiply or divide.

2

Convert the clearance height from inches to centimeters.

10 \times 2.54 = 25.4 \text{ cm}

Multiplying by the factor converts US standard length to metric.

3

Convert the marching distance from miles to kilometers.

5 \times 1.6 = 8 \text{ km}

Multiplying miles by

1.6

yields the correct distance in kilometers.

4

Convert the crate weight from pounds to kilograms.

110 \div 2.2 = 50 \text{ kg}

Since

1 \text{ kg}

is

2.2 \text{ lbs}

, dividing the total pounds by

2.2

gives the number of kilograms.

5

Convert the fuel capacity from gallons to liters.

10 \times 3.78 = 37.8 \text{ L}

Multiplying gallons by the conversion factor

3.78

yields the volume in liters.

Key Concept

Inter-System Unit Conversions

Question 887Question →

Match each verbal phrase with its correct algebraic expression translation.

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Answer

The phrases correctly match with their algebraic translations based on the operational keywords: 'more than' maps to addition (

y + 4

), 'less than' maps to subtraction (

y - 4

), 'product' maps to multiplication (

4y

), and 'quotient' maps to division (

\frac{y}{4}

).

Matching each phrase requires identifying the key operation word: 'more than' for addition, 'less than' for subtraction (with reversed order), 'product' for multiplication, and 'quotient' for division. Applying these directly to the variable

y

and the number 4 yields the correct algebraic expressions.

Step-by-Step Solution

1

Translate 'Four more than a number

y

'.

y + 4

The words 'more than' specify adding to the given variable.

2

Translate 'Four less than a number

y

'.

y - 4

The words 'less than' specify subtracting from the given variable.

3

Translate 'The product of four and a number

y

'.

4y

The word 'product' specifies multiplying the values.

4

Translate 'The quotient of a number

y

and four'.

\frac{y}{4}

The word 'quotient' specifies dividing the first value by the second value.

Key Concept

Translating basic arithmetic operation keywords into their corresponding algebraic symbols.

Question 888Question →

Read the three brief excerpts from naval engineering field evaluations below. Which author tone or attitude correctly matches each excerpt?

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Answer

The excerpt evaluating the titanium-reinforced hull matches with 'Skeptical and cautionary'; the excerpt describing the autonomous sonar navigation suite matches with 'Appreciative and commendatory'; and the excerpt recording fuel consumption data matches with 'Objective and matter-of-fact'.

The author's tone is determined by analyzing descriptive word choices: the first excerpt uses qualifying words ('failed to account', 'questionable') to urge prudence; the second excerpt utilizes complimentary language ('flawlessly', 'exceptional', 'invaluable') to commend performance; and the third excerpt presents unembellished measurements ('consumed 420 liters', 'aligning directly') to deliver an unbiased report.

Step-by-Step Solution

1

Analyze the diction in the first excerpt regarding the titanium-reinforced hull.

Identified reservations in phrases such as 'failed to account' and 'reliability questionable.'

Negative evaluative phrases and calls for caution indicate a skeptical and cautionary attitude.

2

Analyze the diction in the second excerpt regarding the autonomous sonar suite.

Identified high praise in terms like 'flawlessly,' 'exceptional precision,' and 'invaluable.'

Strongly favorable descriptive words convey an appreciative and commendatory tone.

3

Analyze the diction in the third excerpt regarding diesel-electric fuel consumption.

Identified plain quantitative statements ('consumed 420 liters', 'aligning directly with baseline parameters') without qualitative bias.

Reporting verified figures without personal opinion or emotional color denotes an objective and matter-of-fact tone.

Key Concept

Inferring Author Tone and Attitude through Diction and Phrasing

Estimated Time:1m 30s

Question 889Question →

During cold-weather evaluation of an expeditionary filtration unit, technicians recorded the following observations:

• While the secondary filter cartridge was porous, the primary silicone membrane proved completely impermeable.
• Thermal fluctuations caused only a transient drop in line pressure, whereas the baseline operating pressure remained enduring and constant.
• In contrast to the sluggish motorized exhaust valve, the pneumatic relief flap exhibited an instantaneous, rapid release.
• Despite the pungent odor emitted by the solvent wash, the neutralizing rinse was entirely mild and odorless.

Based on the contrast context clues in the passage, match each bolded word to its correct meaning.

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Answer

Porous matches 'Permeable and allowing fluids to pass through'; Transient matches 'Temporary and lasting for only a short period'; Sluggish matches 'Slow-moving and delayed in action'; Pungent matches 'Sharp, intense, and biting in smell'.

Each match correctly inverts the stated antonym clue signaled by the transition words: 'porous' is the opposite of impermeable (permeable), 'transient' is the opposite of enduring (temporary), 'sluggish' is the opposite of instantaneous/rapid (slow-moving), and 'pungent' is the opposite of mild/odorless (sharp and intense).

Step-by-Step Solution

1

Identify the contrast signal words in each observation sentence.

Signal words identified: 'While', 'whereas', 'In contrast to', and 'Despite'.

Contrast signal words establish that the target bolded term has the opposite meaning of the descriptive clue provided in the same sentence.

2

Locate the contrasting counterpart words for each target term.

Porous opposes 'impermeable'; Transient opposes 'enduring and constant'; Sluggish opposes 'instantaneous, rapid'; Pungent opposes 'mild and odorless'.

Extracting the explicit antonym in the text defines the boundary of the unknown word's meaning.

3

Match each bolded term to the definition representing the opposite of its textual counterpart.

Porous = permeable; Transient = temporary; Sluggish = slow-moving; Pungent = sharp/intense in smell.

Inverting the contrasting clues produces the precise contextual definitions.

Key Concept

Using explicit contrast signal words and antonymous counterparts to infer vocabulary definitions in technical context

Estimated Time:1m 15s

Question 890Question →

Match each unsimplified algebraic expression on the left with its completely simplified equivalent on the right.

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Answer

The correctly simplified expressions are:

5x - 3(2x - 4)

matches with

-x + 12

;

3(2m - 5n) - 2(m - 3n)

matches with

4m - 9n

;

4x(x - 2) - x^2 + 5x

matches with

3x^2 - 3x

; and

7y - 3[y - 2(y + 1)]

matches with

10y + 6

.

Each unsimplified expression must be correctly expanded using the distributive property, ensuring that negative multipliers are applied to every term inside the parentheses. After expanding, grouping and adding the coefficients of like terms produces the single correct simplified equivalent provided in the match list.

Step-by-Step Solution

1

Distribute any coefficients or negative signs immediately outside of parentheses or brackets.

Terms are expanded (e.g.,

-3(2x - 4)

becomes

-6x + 12

).

Distribution removes grouping symbols, which is required before combining terms.

2

For expressions with nested grouping symbols, simplify from the innermost parentheses outward.

Complex expressions like

7y - 3[y - 2(y + 1)]

simplify step-by-step to

7y + 3y + 6

.

Following the order of operations ensures accurate distribution of multipliers.

3

Group and combine like terms (terms with the exact same variable and exponent).

The final canonical form is reached for each expression.

Combining like terms fully simplifies the polynomial into standard form.

Key Concept

Simplifying Algebraic Expressions

Estimated Time:1m 30s

Question 891Question →

Pair each basic algebraic expression with its correctly simplified result by applying the fundamental laws of exponents.

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Answer

The correct pairings are:

m^6 \cdot m^2

with

m^8

;

\frac{n^8}{n^5}

with

n^3

;

(p^4)^3

with

p^{12}

; and

q^{-5}

with

\frac{1}{q^5}

.

Each expression can be simplified by directly applying one specific fundamental property of exponents.

m^6 \cdot m^2

uses the product rule.

\frac{n^8}{n^5}

uses the quotient rule.

(p^4)^3

uses the power rule.

q^{-5}

uses the negative exponent definition.

Step-by-Step Solution

1

Simplify the product expression

m^6 \cdot m^2

.

m^8

When bases are identical and multiplied, their exponents are added together.

2

Simplify the quotient expression

\frac{n^8}{n^5}

.

n^3

When dividing identical bases, the exponent of the denominator is subtracted from the exponent of the numerator.

3

Simplify the nested power expression

(p^4)^3

.

p^{12}

When raising an exponent to another power, the two exponents are multiplied.

4

Rewrite the negative exponent expression

q^{-5}

.

\frac{1}{q^5}

A negative exponent indicates taking the reciprocal of the base raised to the positive exponent.

Key Concept

Applying fundamental exponent rules (product, quotient, power, and negative exponents) to simplify single-variable expressions.

Question 892Question →

Match each unsimplified algebraic expression on the left with its equivalent fully simplified form on the right. Assume all variables represent non-zero real numbers.

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Answer

The product rule matches to -12a^6/b^3, the quotient rule matches to 2b^4/a^4, the power of a product rule matches to 16a^12b^20, and the negative exponent simplification matches to a^4/25.

Matching each pair correctly requires correctly identifying and applying four distinct fundamental laws of exponents: the product rule (adding exponents), the quotient rule (subtracting exponents), the power of a product rule (multiplying exponents), and the rules for evaluating zero and negative exponents.

Step-by-Step Solution

1

Simplify

(3a^4b^2)(-4a^2b^{-5})

using the product rule for exponents.

-12a^6b^{-3}

, which simplifies to

\frac{-12a^6}{b^3}

.

When multiplying expressions with the same base, add their exponents and multiply their numerical coefficients.

2

Simplify

\frac{16a^3b^6}{8a^7b^2}

using the quotient rule for exponents.

2a^{-4}b^4

, which simplifies to

\frac{2b^4}{a^4}

.

When dividing expressions with the same base, subtract the denominator's exponent from the numerator's exponent and divide the numerical coefficients.

3

Simplify

(2a^3b^5)^4

using the power of a product rule.

16a^{12}b^{20}

.

When raising a product to a power, raise the numerical coefficient to that power and multiply each variable's exponent by the outer power.

4

Simplify

(5a^{-2}b^0)^{-2}

using the zero and negative exponent rules.

First,

b^0 = 1

. Then evaluate

5^{-2}a^4

, which simplifies to

\frac{a^4}{25}

.

Any non-zero term raised to the zero power equals 1. Negative exponents indicate the reciprocal of the base.

Key Concept

Operations with Variable Exponents

Question 893Question →

Match each verbal phrase or word problem to its correct algebraic equation translation.

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Answer

The correct matches pair each verbal description with its mathematically equivalent equation by carefully translating order of operations, rates, geometric formulas, and grouping words.

Each verbal phrase uniquely defines a single algebraic relationship. By identifying key terms such as 'less than' for reverse-order subtraction, rate contexts for variable multiplication, geometric formulas for perimeter, and grouping words for parentheses, each scenario is mapped accurately to its corresponding equation.

Step-by-Step Solution

1

Analyze the first phrase: 'Eight less than five times a number

x

is equal to twenty.'

Translates to

5x - 8 = 20

.

'Five times a number' is

5x

. The phrase 'less than' dictates that 8 is subtracted from

5x

.

2

Analyze the second phrase: 'A service charges 8 dollars per hour plus a 5 dollar flat fee. The total cost for

x

hours is 20 dollars.'

Translates to

8x + 5 = 20

.

The variable

x

is multiplied by the hourly rate of 8, and the one-time flat fee of 5 is added to this product.

3

Analyze the third phrase: 'The perimeter of a rectangle is 20. Its length is 5 units more than its width,

x

.'

Translates to

2x + 2(x + 5) = 20

.

The perimeter formula is

2W + 2L

. Substituting

x

for width and

(x + 5)

for length provides the correct equation.

4

Analyze the fourth phrase: 'Five times the difference of a number

x

and eight results in twenty.'

Translates to

5(x - 8) = 20

.

The word 'difference' combined with 'times' indicates that the subtraction

(x - 8)

must occur first, requiring parentheses before multiplying by 5.

Key Concept

Translating complex word phrases into algebraic equations requires strict attention to operation order, grouping symbols, and contextual formulas.

Question 894Question →

Match each unsimplified algebraic expression on the left with its fully simplified equivalent form on the right. Assume $p$ and $q$ represent non-zero numbers.

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Answer

The correct matches are: (4p^5 q^-2)(2p^-1 q^6) matches 8p^4 q^4; the fraction 15p^7 q^2 over 3p^2 q^-3 matches 5p^5 q^5; (3p^3 q^2)^2 matches 9p^6 q^4; and the fraction (2p^2 q^3)^3 over 4p^4 q^7 matches 2p^2 q^2.

Each expression simplifies to exactly one of the given target forms by consistently applying the laws of exponents (product, quotient, and power rules) and handling numerical coefficients accurately.

Step-by-Step Solution

1

Simplify the first expression using the product rule for exponents.

The coefficients multiply to 8. Adding the exponents for p gives 5 + (-1) = 4. Adding the exponents for q gives -2 + 6 = 4. The result is 8p^4 q^4.

When multiplying terms with the same base, you add their exponents.

2

Simplify the second expression using the quotient rule for exponents.

The coefficients divide to 5. Subtracting the p exponents gives 7 - 2 = 5. Subtracting the q exponents gives 2 - (-3) = 5. The result is 5p^5 q^5.

When dividing terms with the same base, you subtract the denominator's exponent from the numerator's exponent.

3

Simplify the third expression using the power of a product rule.

The coefficient squared is 3^2 = 9. Multiplying the p exponent by 2 gives 3 * 2 = 6. Multiplying the q exponent by 2 gives 2 * 2 = 4. The result is 9p^6 q^4.

When raising a power to a power, multiply the exponents. Remember to also apply the power to the numerical coefficient.

4

Simplify the fourth expression by applying the power rule first, then the quotient rule.

The numerator becomes 8p^6 q^9. Dividing this by 4p^4 q^7 gives a coefficient of 2, p exponent of 6 - 4 = 2, and q exponent of 9 - 7 = 2. The result is 2p^2 q^2.

Order of operations requires handling the exponent on the entire numerator before performing division.

Key Concept

Applying the product, quotient, and power rules of exponents to simplify algebraic expressions.

Question 895Question →

Match each verbal phrase describing a mathematical relationship or scenario to its correct algebraic expression.

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Answer

The correct pairings match each contextual or mathematical description with its exact symbolic translation, respecting order of operations, geometric formulas, and grouping words.

Each correct match successfully identifies key translation words ('product', 'quotient', 'less than', 'quantity') and applies proper algebraic syntax, including necessary parentheses, geometric formulas, and the correct order for subtraction.

Step-by-Step Solution

1

Translate the rate and discount scenario.

The phrase '

k

items at twelve dollars each' translates to

12k

. The phrase 'minus a five-dollar discount' makes it

12k - 5

.

Multiplication represents a per-item rate, and a discount implies subtraction.

2

Translate the quotient phrase.

The 'quotient of

k

and five' is

\frac{k}{5}

. 'Twelve less than' means subtracting

12

from this term, yielding

\frac{k}{5} - 12

.

The phrase 'less than' dictates reversing the order of the terms in the algebraic subtraction.

3

Translate the grouped product phrase.

The 'quantity of five less than

k

' is translated as

(k - 5)

. The 'product of twelve and the quantity' is

12(k - 5)

.

The word 'quantity' signals that the addition or subtraction must be grouped in parentheses before applying the multiplication.

4

Translate the geometric scenario.

The 'area of a rectangle with length twelve and width

k

' translates to

12k

. 'Increased by five' adds

5

, yielding

12k + 5

.

Area of a rectangle is calculated by multiplying length and width, and 'increased by' signifies addition.

Key Concept

Translating Word Phrases into Algebraic Equations

Question 896Question →

Match each unsimplified algebraic expression on the left with its equivalent, fully simplified form on the right. Assume all variables represent non-zero real numbers.

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Answer

The expressions match as follows: (3a^4b^2)(4a^2b) is equivalent to 12a^6b^3. The quotient (24a^7b^3)/(6a^3b^6) simplifies to 4a^4b^-3. The power (2a^3b^-2)^4 expands to 16a^12b^-8. Finally, (5a^0b^3)^-2 simplifies to 1/(25b^6).

Each expression simplifies directly using the fundamental rules of exponents: product rule (add exponents), quotient rule (subtract exponents), power rule (multiply exponents), and zero/negative exponent rules.

Step-by-Step Solution

1

Simplify the first expression by multiplying the coefficients and applying the product rule for exponents.

3 * 4 = 12, a^(4+2) = a^6, b^(2+1) = b^3. Combined: 12a^6b^3

When multiplying terms with the same base, add their exponents.

2

Simplify the second expression by dividing the coefficients and applying the quotient rule for exponents.

24 / 6 = 4, a^(7-3) = a^4, b^(3-6) = b^-3. Combined: 4a^4b^-3

When dividing terms with the same base, subtract the denominator's exponent from the numerator's exponent.

3

Simplify the third expression by applying the power of a product rule.

2^4 = 16, a^(3*4) = a^12, b^(-2*4) = b^-8. Combined: 16a^12b^-8

When raising a power to a power, multiply the exponents. Apply the exponent to the numerical coefficient as well.

4

Simplify the fourth expression by applying the zero exponent rule, then the power rule.

a^0 = 1. 5^-2 = 1/25. b^(3*-2) = b^-6. Combined: 1/(25b^6)

Any non-zero term raised to the power of 0 is 1. Negative exponents indicate taking the reciprocal.

Key Concept

Applying the product, quotient, power, zero, and negative rules of exponents to simplify algebraic expressions with multiple variables and coefficients.

Question 897Question →

Match each algebraic expression on the left with its fully simplified equivalent form on the right. Assume all variables represent non-zero real numbers.

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Show answer & explanation

Answer

The quotient matches (3a^6)/b^5, the squared term matches (9b^8)/a^4, the product matches 8a^2b^2, and the fraction raised to a negative power matches (8b^6)/a^6.

Each expression is thoroughly simplified by applying the fundamental laws of exponents: adding exponents for multiplication, subtracting exponents for division, multiplying exponents for a power to a power, and reciprocating terms to handle negative exponents.

Step-by-Step Solution

1

Simplify the first expression: (15a^4b^-3) / (5a^-2b^2).

3a^6b^-5, which becomes (3a^6)/b^5.

When dividing terms with the same base, divide the coefficients and subtract the exponents. Move negative exponents to the denominator.

2

Simplify the second expression: (3a^-2b^4)^2.

9a^-4b^8, which becomes (9b^8)/a^4.

When raising a power to a power, raise the coefficient to the outer power and multiply the variable exponents.

3

Simplify the third expression: (2a^3b^-1)(4a^-1b^3).

8a^2b^2.

When multiplying terms with the same base, multiply the coefficients and add the respective exponents.

4

Simplify the fourth expression: ((a^2b^-2)/2)^-3.

a^-6b^6 / 2^-3, which becomes (8b^6)/a^6.

Apply the outer negative exponent to all factors inside the parentheses, including the denominator, then move negative powers across the fraction bar to make them positive.

Key Concept

Exponent rules: Product of powers, quotient of powers, power of a product, and negative exponents.

Question 898Question →

Match each biological kingdom to its primary cellular and nutritional characteristics.

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Answer

Plantae matches multicellular autotrophs with cellulose walls; Fungi matches absorptive heterotrophs with chitin walls; Animalia matches multicellular heterotrophs lacking cell walls; Protista matches mostly unicellular eukaryotes.

Each kingdom in Domain Eukarya is distinguished by specific cellular structures and modes of nutrition: plants are autotrophic with cellulose walls, fungi absorb nutrients through chitin walls, animals ingest nutrients and lack cell walls, and protists represent simple, predominantly single-celled eukaryotes.

Step-by-Step Solution

1

Identify the defining features of Kingdom Plantae

Matched with multicellular autotrophs containing cellulose cell walls

Plants perform photosynthesis using chlorophyll and have structural cell walls made of cellulose.

2

Identify the defining features of Kingdom Fungi

Matched with absorptive heterotrophs containing chitin cell walls

Fungi cannot perform photosynthesis and rely on external absorption of nutrients through chitin-reinforced walls.

3

Identify the defining features of Kingdom Animalia

Matched with multicellular ingestive heterotrophs without cell walls

Animals ingest food internally and possess flexible cells without rigid walls.

4

Identify the defining features of Kingdom Protista

Matched with mostly unicellular eukaryotes

Protists are primarily microscopic, single-celled eukaryotes that lack the tissue specialization of plants, animals, or fungi.

Key Concept

Major diagnostic characteristics separating eukaryotic kingdoms based on cell structure, nutritional mode, and cell wall composition.

Estimated Time:45s

Question 899Question →

Match each foundational genetic term on the left with its corresponding biological definition on the right.

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Answer

Genotype matches the specific genetic makeup or combination of alleles; Phenotype matches the observable physical or biochemical characteristics; Allele matches an alternative or variant form of a single gene; Chromosome matches a condensed structure of DNA and histone proteins.

Each term is accurately matched to its precise biological definition: Genotype is the organism's allele combination, Phenotype is the observable physical trait, Allele is a specific variant form of a gene, and Chromosome is the organized thread-like structure of DNA and protein.

Step-by-Step Solution

1

Differentiate between genetic code and physical expression.

Genotype corresponds to the underlying genetic makeup and allele combination, whereas phenotype represents the observable physical traits.

Genotype is the internal genetic instructions, while phenotype is the outward result.

2

Distinguish between individual gene variations and macroscopic cellular structures.

An allele is an individual variant of a gene, whereas a chromosome is the large organized package composed of DNA and proteins containing many genes.

An allele is a sub-unit of genetic sequence variation, while a chromosome is the structural vehicle for genomic DNA.

3

Pair each genetic term with its exact corresponding definition.

All four pairs are accurately aligned based on precise biological terminology.

Each definition uniquely identifies the specific level of genetic organization and function.

Key Concept

Foundational Genetics Terminology and Structural Hierarchy

Estimated Time:1m 0s

Question 900Question →

Match each eukaryotic cellular structure with its primary physiological role within the cell.

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Answer

Golgi apparatus matches packaging and sorting; Plasma membrane matches selective permeability and transport control; Smooth endoplasmic reticulum matches lipid synthesis and detoxification; Nucleus matches genetic storage and transcriptional regulation.

Each cellular component has a distinct biochemical and structural specialization: the Golgi apparatus serves as the cellular packaging center, the plasma membrane maintains selective barrier integrity, the smooth endoplasmic reticulum executes lipid synthesis and detoxification, and the nucleus stores genetic instructions.

Step-by-Step Solution

1

Identify the primary role of the Golgi apparatus in the endomembrane system.

The Golgi apparatus processes, tags, and packages proteins and lipids for dispatch.

Vesicular transport from the endoplasmic reticulum is received at the cis face of the Golgi apparatus and modified before export at the trans face.

2

Determine the barrier and transport function of the plasma membrane.

The plasma membrane maintains homeostasis through selective transport.

The amphipathic phospholipid bilayer prevents passive diffusion of large polar molecules while allowing specific transport.

3

Distinguish smooth endoplasmic reticulum functions from rough endoplasmic reticulum functions.

The smooth endoplasmic reticulum is specialized for lipid metabolic pathways and detoxification.

Unlike the rough endoplasmic reticulum which synthesizes proteins via attached ribosomes, the smooth portion handles non-protein biochemical syntheses.

4

Identify the control and storage center of the eukaryotic cell.

The nucleus preserves genomic DNA and governs cellular operations.

The nuclear envelope encloses hereditary material and coordinates RNA synthesis.

Key Concept

Eukaryotic organelle compartmentalization and functional specialization

Estimated Time:1m 0s

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