Algebra Practice Questions — Armed Services Vocational Aptitude Battery (ASVAB) — Page 8

Match each algebraic expression with its simplified equivalent form.

Click a left item, then click its matching right item

Show answer & explanation

Answer

The matches are:

x^3 \cdot x^2

simplifies to

x^5

,

\frac{x^8}{x^2}

simplifies to

x^6

,

(x^4)^2

simplifies to

x^8

, and

x^{-3}

simplifies to

\frac{1}{x^3}

.

Each expression correctly pairs with its simplified form by applying the four fundamental rules of exponents: the product rule, the quotient rule, the power of a power rule, and the negative exponent rule.

Step-by-Step Solution

1

Apply the product rule for exponents to

x^3 \cdot x^2

.

x^{3+2} = x^5

Multiplying identical bases requires adding the exponents.

2

Apply the quotient rule for exponents to

\frac{x^8}{x^2}

.

x^{8-2} = x^6

Dividing identical bases requires subtracting the denominator's exponent.

3

Apply the power rule for exponents to

(x^4)^2

.

x^{4 \cdot 2} = x^8

Raising an exponent to another power requires multiplying the exponents.

4

Apply the negative exponent rule to

x^{-3}

.

\frac{1}{x^3}

A negative exponent signifies the reciprocal of the corresponding positive exponent.

Key Concept

Operations with Variable Exponents

Estimated Time:1m 0s

Question 142Question →

Match each unsimplified algebraic expression with its correctly simplified equivalent.

Click a left item, then click its matching right item

Show answer & explanation

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 143Question →

Which of the following equations correctly represents the statement: 'Twelve subtracted from five times the sum of a number $y$ and six is equal to three less than twice the number'?

5(y + 6) - 12 = 2y - 3

12 - 5(y + 6) = 3 - 2y

5y + 6 - 12 = 2y - 3

5(y + 6) - 12 = 3 - 2y

Show answer & explanation

Answer: $5(y + 6) - 12 = 2y - 3$

Answer

The equation

5(y + 6) - 12 = 2y - 3

correctly translates all verbal phrases in the appropriate order with necessary grouping symbols.

The translation process requires breaking the sentence into smaller chunks. The left side starts with 'five times the sum of a number

y

and six', written as

5(y + 6)

. Then, 'twelve subtracted from' means we subtract

12

from that product, yielding

5(y + 6) - 12

. The right side is 'three less than twice the number'. Twice the number is

2y

, and three less than that is

2y - 3

. Setting them equal gives

5(y + 6) - 12 = 2y - 3

.

Step-by-Step Solution

1

Translate 'the sum of a number

y

and six'.

(y + 6)

The word 'sum' indicates addition, and because this entire expression will be multiplied in the next step, it must be enclosed in parentheses.

2

Translate 'five times the sum'.

5(y + 6)

The multiplier

5

must apply to the entire grouped quantity of the sum.

3

Translate 'Twelve subtracted from' the previous result.

5(y + 6) - 12

The phrase 'subtracted from' means

12

is taken away from the established quantity, so it is placed after the minus sign.

4

Translate 'three less than twice the number'.

2y - 3

First, 'twice the number' is

2y

. 'Three less than' means

3

is subtracted from

2y

. This forms the right side of the equation.

Key Concept

Translating complex algebraic phrases requires careful attention to grouping words like 'sum' or 'difference' and order-reversing phrases like 'less than' or 'subtracted from'.

Question 144Question →

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

Click a left item, then click its matching right item

Show answer & explanation

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 145Question →

Simplify the following algebraic expression:

8x - 2[x - 3(2x - y)]

Which of the following represents the completely simplified form?

18x + 2y

18x + 6y

18x - 6y

-30x^2 + 18xy

Show answer & explanation

Answer: $18x - 6y$

Answer

The completely simplified form is

18x - 6y

.

The expression is fully simplified by applying the distributive property iteratively, starting from the innermost parentheses, carefully tracking all negative signs, and finally combining like terms to arrive at

18x - 6y

.

Step-by-Step Solution

1

Distribute the -3 into the innermost parentheses.

-3(2x - y) = -6x + 3y

According to the order of operations, innermost grouping symbols must be evaluated first. Distributing a negative number flips the signs of the terms inside.

2

Substitute this back and combine like terms inside the brackets.

x - 6x + 3y = -5x + 3y

Combining the like variable terms simplifies the expression inside the brackets before moving to the next operation.

3

Distribute the -2 across the newly simplified terms inside the bracket.

-2(-5x + 3y) = 10x - 6y

The distributive property applies the outside multiplier to every term within the brackets.

4

Combine this result with the remaining term outside the brackets.

8x + 10x - 6y = 18x - 6y

Adding the like x terms yields the final, completely simplified algebraic expression.

Key Concept

Simplifying Nested Algebraic Expressions

Question 146Question →

Simplify the following algebraic expression:

(3y^4)(-5y^2)

Which of the following is the correctly simplified expression?

15y^6

-15y^6

-15y^8

-2y^6

Show answer & explanation

Answer: $-15y^6$

Answer

-15y^6

The expression is simplified by first multiplying the numerical coefficients (

3 \times -5 = -15

). Then, the product rule for exponents is applied to the variables by adding their exponents (

y^4 \times y^2 = y^{4+2} = y^6

). Combining these two parts yields the correct simplified expression,

-15y^6

.

Step-by-Step Solution

1

Group and multiply the numerical coefficients.

3 \times (-5) = -15

The commutative property allows us to group the constant numbers together for multiplication.

2

Multiply the variable terms using the product rule for exponents.

y^4 \times y^2 = y^{4+2} = y^6

When multiplying exponential terms with the same base, their exponents must be added.

3

Combine the resulting coefficient and variable term.

-15y^6

This represents the final simplified form of the expression.

Key Concept

Product Rule for Exponents and Integer Multiplication

Question 147Question →

Determine the numerical value of the variable $x$ that satisfies the algebraic equation below.

Fill in the blanks below

\frac{3}{8}(16x - 24) + 5 = -2(2x - 1) + 4x + 6

x =

Show answer & explanation

Answer

The value of x that satisfies the equation is 2.

By successfully applying the distributive property to both sides of the equation, the expression simplifies to 6x - 9 + 5 = -4x + 2 + 4x + 6. Combining like terms reduces this to 6x - 4 = 8. Adding 4 to both sides yields 6x = 12, which gives exactly 2.

Step-by-Step Solution

1

Distribute the fraction on the left side and the negative integer on the right side.

6x - 9 + 5 = -4x + 2 + 4x + 6

Applying the distributive property removes the grouping parentheses, which is necessary before combining any terms.

2

Combine the constant terms on the left side and the like terms on the right side.

6x - 4 = 8

Simplifying both sides of the equation by grouping like terms makes it straightforward to isolate the variable. The -4x and +4x cancel each other out.

3

Isolate the term containing the variable by adding 4 to both sides of the equation.

6x = 12

Using the addition property of equality maintains the balance while moving constants to one side.

4

Divide both sides by 6 to solve for x.

x = 2

Dividing by the coefficient of x isolates the variable completely and provides the final solution.

Key Concept

Solving multi-step linear equations involving fractional distribution and combining like terms across both sides of the equals sign.

Question 148Question →

A word problem describes a quantity as being 'ten less than three times a number, $x$ '. What is the correct algebraic expression for this quantity?

3x - 10

10 - 3x

3(x - 10)

3x + 10

Show answer & explanation

Answer: $3x - 10$

Answer

The correct mathematical expression is

3x - 10

.

The phrase 'three times a number,

x

' translates directly to

3x

. The phrase 'ten less than' indicates a subtraction of

10

from that initial quantity. Therefore, the correct combined expression is

3x - 10

.

Step-by-Step Solution

1

Translate the multiplication part of the phrase: 'three times a number,

x

'.

3x

The word 'times' indicates multiplication between

3

and the variable

x

.

2

Translate the subtraction part of the phrase: 'ten less than'.

Subtract

10

from the result of step 1.

The phrase 'less than' is a turnaround phrase, meaning the value that comes before 'less than' must be subtracted from the value that comes after it.

3

Combine the translated parts into a single expression.

3x - 10

This correctly places the subtraction after the multiplied term.

Key Concept

Translating Word Phrases into Algebraic Equations

Question 149Question →

Find the value of $n$ that makes the following equation true:

\frac{x^9}{x^3} = x^n

Show answer & explanation

Answer: 6; six

Answer

6

According to the quotient rule for exponents, when you divide two exponential terms that have the same base, you subtract the exponent of the denominator from the exponent of the numerator. In this equation, both terms have the base

x

. Subtracting the denominator's exponent (3) from the numerator's exponent (9) gives

9 - 3 = 6

. Therefore, the left side simplifies to

x^6

. Equating this to

x^n

means that

n

must equal 6.

Step-by-Step Solution

1

Apply the quotient rule for exponents, which states that when dividing terms with identical bases, you subtract the exponent in the denominator from the exponent in the numerator.

x^{9-3} = x^n

To simplify the left side of the equation into a single term with base

x

.

2

Perform the subtraction in the exponent.

x^6 = x^n

Calculating

9 - 3

gives the simplified exponent.

3

Set the resulting exponent equal to

n

.

n = 6

Since the bases are identical (

x

), their exponents must be equal for the equation to hold true.

Key Concept

Quotient Rule for Exponents

Question 150Question →

Consider the following linear inequality:

3(x - 2) - 7x \geq 2(x + 3)

Which of the following represents the correct solution set for $x$ ?

x \leq -2

x \geq -2

x \leq 0

x \geq 6

Show answer & explanation

Answer: $x \leq -2$

Answer

The correct solution is

x \leq -2

.

To find the solution, expand the terms to get

3x - 6 - 7x \geq 2x + 6

. Combine the like terms on the left to form

-4x - 6 \geq 2x + 6

. Subtracting

2x

and adding

6

to both sides isolates the variable, resulting in

-6x \geq 12

. Finally, divide by

-6

. Because we are dividing by a negative number, the inequality sign must flip, giving the correct answer of

x \leq -2

.

Step-by-Step Solution

1

Distribute the constants on both sides of the inequality.

3x - 6 - 7x \geq 2x + 6

Expanding the expressions clears the parentheses and allows us to combine like terms.

2

Combine the like terms on the left side of the inequality.

-4x - 6 \geq 2x + 6

3x - 7x

simplifies to

-4x

.

3

Move all terms containing

x

to one side and constants to the other side.

-6x \geq 12

Subtracting

2x

from both sides gives

-6x - 6 \geq 6

. Then, adding

6

to both sides isolates the variable term.

4

Divide both sides by

-6

and reverse the inequality symbol.

x \leq -2

When multiplying or dividing an inequality by a negative number, the inequality direction must be flipped to remain mathematically true.

Key Concept

Solving Linear Inequalities with Variables on Both Sides

Question 151Question →

Match each verbal phrase with its correct algebraic expression translation.

Click a left item, then click its matching right item

Show answer & explanation

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 152Question →

Evaluate the algebraic expression:

-3a^2 + 4b - |a - c|

Given the values $a = -2$ , $b = -3$ , and $c = 5$ , which of the following is the correct result?

-31

-27

-17

17

Show answer & explanation

Answer: -31

Answer

The correct result is -31.

The expression requires precise substitution and adherence to the order of operations. Squaring the negative value

a = -2

gives

4

, which multiplied by

-3

results in

-12

. The term

4b

becomes

4(-3) = -12

. The absolute value of

a - c

evaluates to

|-2 - 5| = |-7| = 7

. Subtracting this

7

from the sum of the first two terms yields

-12 - 12 - 7 = -31

.

Step-by-Step Solution

1

Substitute the given values for

a

,

b

, and

c

into the expression.

-3(-2)^2 + 4(-3) - |-2 - 5|

Correct substitution requires maintaining negative signs within parentheses to prevent sign errors.

2

Evaluate the exponent and the grouping symbol (absolute value).

The exponent term evaluates to

(-2)^2 = 4

. The absolute value expression evaluates to

|-7| = 7

.

According to the order of operations, exponents and operations inside grouping symbols must be resolved before multiplication.

3

Perform the multiplication.

-3(4) = -12

and

4(-3) = -12

. The expression simplifies to

-12 - 12 - 7

.

Multiplication must be executed before addition and subtraction.

4

Perform the final subtraction operations from left to right.

-12 - 12 - 7 = -24 - 7 = -31

.

Combining the remaining terms provides the final evaluated result.

Key Concept

Evaluating algebraic expressions involving negative integers, exponents, and absolute values

Estimated Time:1m 0s

Question 153Question →

A technician is configuring a power supply. The required output is described as twice the difference between the baseline voltage, $v$ , and six. What expression represents the required output?

2(v - 6)

2v - 6

2(6 - v)

6 - 2v

Show answer & explanation

Answer: $2(v - 6)$

Answer

The expression

2(v - 6)

correctly represents the required output.

The phrase 'difference between the baseline voltage,

v

, and six' translates to

(v - 6)

. The word 'twice' means multiplying this entire difference by

2

. Therefore, the correct translation must use parentheses to apply the multiplication to the whole difference, resulting in

2(v - 6)

.

Step-by-Step Solution

1

Translate 'the difference between the baseline voltage, v, and six'.

The expression becomes

v - 6

.

The word 'difference' indicates subtraction in the order the terms are given.

2

Translate 'twice' applied to the entire difference.

The expression becomes

2(v - 6)

.

Multiplying an entire binomial expression requires placing it in parentheses to ensure the multiplication applies to both terms.

Key Concept

Translating Word Phrases into Algebraic Equations

Question 154Question →

An algebraic relationship is given by the equation:

\frac{3}{5}(10k - 25) - \frac{1}{4}(8 - 16k) = \frac{1}{2}(14k - 10) - 2(k - 9)

What value of $k$ balances the given equation?

Show answer & explanation

Answer: 6

Answer

The value that satisfies the equation is 6.

The correct value is found by carefully distributing the fractional multipliers and negative signs to eliminate all parentheses, simplifying both sides to get

10k - 17 = 5k + 13

, and finally isolating

k

by performing balanced operations to find

k = 6

.

Step-by-Step Solution

1

Distribute the fractional coefficients into the parentheses on both sides of the equation.

(6k - 15) - (2 - 4k) = (7k - 5) - 2(k - 9)

Distribution eliminates the parentheses and clears the fractional multipliers, allowing the terms to be evaluated and combined.

2

Distribute the negative signs for the remaining terms in the equation.

6k - 15 - 2 + 4k = 7k - 5 - 2k + 18

Properly applying the negative sign ensures all terms are correctly added or subtracted, especially noting that

- \frac{1}{4}(-16k) = +4k

and

-2(-9) = +18

.

3

Combine the constant terms and the variable terms independently on each side.

10k - 17 = 5k + 13

Simplifying each side makes it easier to isolate the variable.

4

Isolate the variable term by moving all terms containing k to one side and constants to the other side.

10k - 5k = 13 + 17 \Rightarrow 5k = 30 \Rightarrow k = 6

Solving for k provides the final numerical value that balances the equation.

Key Concept

Solving complex one-variable linear equations by distributing fractional coefficients, managing negative signs, and combining like terms.

Question 155Question →

A square metal plate has a side length of $3x^4$ centimeters. Which of the following expressions represents the area of the plate in square centimeters?

9x^8

6x^8

9x^6

3x^8

Show answer & explanation

Answer: $9x^8$

Answer

9x^8

The area of a square is calculated by squaring its side length. When squaring a term like

3x^4

, the power of a product rule states that both the coefficient and the variable must be raised to the second power. Squaring the coefficient 3 yields 9, and raising

x^4

to the second power requires multiplying the exponents (

4 \times 2 = 8

), resulting in a final area of

9x^8

.

Step-by-Step Solution

1

Recall the formula for the area of a square.

A = s^2

To find the area, the side length must be squared.

2

Substitute the given side length into the area formula.

A = (3x^4)^2

This sets up the mathematical expression that needs to be simplified.

3

Apply the power of a product rule and power of a power rule.

3^2 \cdot (x^4)^2 = 9x^8

The outer exponent (2) must be applied to both the numerical coefficient (

3^2=9

) and the variable exponent (

4 \times 2 = 8

).

Key Concept

Power of a Product and Power of a Power rules

Question 156Question →

Simplify the following algebraic expression completely by distributing and combining like terms:

3a(2a - 4b) - 2b(a - 5b)

Show answer & explanation

Answer: 6a^2 - 14ab + 10b^2; 6a^2-14ab+10b^2; 10b^2 - 14ab + 6a^2; 6a^2+10b^2-14ab; 6a^2 - 14ba + 10b^2

Answer

6a^2 - 14ab + 10b^2

The expression is correctly simplified by first applying the distributive property to both parts:

3a(2a - 4b)

becomes

6a^2 - 12ab

, and

-2b(a - 5b)

becomes

-2ab + 10b^2

. Next, the like terms (

-12ab

and

-2ab

) are combined to form

-14ab

. The resulting expression in standard form is

6a^2 - 14ab + 10b^2

.

Step-by-Step Solution

1

Distribute

3a

across the terms in the first parenthesis

(2a - 4b)

.

6a^2 - 12ab

Applying the distributive property to eliminate the first set of parentheses.

2

Distribute

-2b

across the terms in the second parenthesis

(a - 5b)

.

-2ab + 10b^2

Multiplying by a negative requires changing the signs. Specifically,

(-2b) \times (-5b) = +10b^2

.

3

Write out the full expanded expression and identify like terms.

6a^2 - 12ab - 2ab + 10b^2

Preparing to combine terms with identical variable parts.

4

Combine the like terms (

-12ab

and

-2ab

).

6a^2 - 14ab + 10b^2

-12 - 2 = -14

, resulting in the final simplified standard form.

Key Concept

Distributive Property and Combining Like Terms with Multiple Variables

Question 157Question →

Which of the following represents the value of $p$ that satisfies the given equation?

-2(4p - 5) + \frac{1}{3}(15p - 12) = 2p - (7p - 14)

-10

4

10

14

Show answer & explanation

Answer: 4

Answer

4

The correct answer is 4. To solve the equation, first distribute the outer terms across the parentheses on both sides:

-2(4p - 5)

becomes

-8p + 10

,

\frac{1}{3}(15p - 12)

becomes

5p - 4

, and

-(7p - 14)

becomes

-7p + 14

. Combining like terms on the left side (

-8p + 5p

and

10 - 4

) gives

-3p + 6

. On the right side, combining

2p

and

-7p

gives

-5p + 14

. Setting these equal gives

-3p + 6 = -5p + 14

. Adding

5p

to both sides yields

2p + 6 = 14

, and subtracting 6 from both sides gives

2p = 8

. Dividing by 2 results in

p = 4

.

Step-by-Step Solution

1

Distribute the coefficients to remove parentheses on the left side.

-8p + 10 + 5p - 4 = 2p - (7p - 14)

Applying the distributive property is necessary to eliminate grouping symbols.

2

Distribute the negative sign to remove parentheses on the right side.

-8p + 10 + 5p - 4 = 2p - 7p + 14

Distributing the subtraction over the group allows all terms to be evaluated.

3

Combine like terms on each side of the equation independently.

-3p + 6 = -5p + 14

Simplifying both sides makes it easier to balance the equation and isolate the variable.

4

Add

5p

to both sides and subtract 6 from both sides.

2p = 8

Performing these inverse operations groups all variable terms on one side and all constant terms on the other.

5

Divide both sides by 2.

p = 4

Dividing by the coefficient isolates the variable

p

, revealing its final value.

Key Concept

Solving complex multi-step linear equations using the distributive property and combining like terms.

Question 158Question →

A shipment contains a number of crates equal to twice the quantity of seven less than the target value, $v$ . What expression defines the number of crates?

2(v - 7)

2v - 7

2(7 - v)

7 - 2v

Show answer & explanation

Answer: $2(v - 7)$

Answer

The correct expression is

2(v - 7)

.

The phrase 'seven less than the target value,

v

' establishes a baseline amount of

v - 7

. The problem then states we need 'twice the quantity of' that baseline amount. To multiply the entire baseline amount by two, we must enclose

v - 7

in parentheses, resulting in

2(v - 7)

.

Step-by-Step Solution

1

Translate the phrase 'seven less than the target value,

v

'.

The expression becomes

v - 7

.

The phrase 'less than' is a turnaround phrase, meaning the seven must be subtracted from the variable

v

.

2

Translate the phrase 'twice the quantity of' and apply it to the first step.

The final expression is

2(v - 7)

.

The word 'quantity' indicates that the entire previous expression (

v - 7

) must be grouped in parentheses before multiplying by two.

Key Concept

Translating Word Phrases into Algebraic Equations

Question 159Question →

Evaluate the following algebraic expression given $x = -3$ , $y = 2$ , and $z = -5$ :

x^2 - \frac{y^3 z}{|x + z|}

Fill in the blanks with the correct numerical values.

Fill in the blanks below

The evaluated value of the fraction term alone (i.e.,

\frac{y^3 z}{|x + z|}

) is , which makes the final value of the entire expression equal to .

Show answer & explanation

Answer

The fraction term evaluates to -5, and the final expression evaluates to 14.

By following the order of operations (PEMDAS), we first substitute the values and handle operations inside grouping symbols like absolute value (

|-3 + -5| = 8

). Next, exponents are calculated (

(-3)^2 = 9

and

2^3 = 8

), followed by multiplication and division to simplify the fraction to

-5

. Finally, subtracting this negative value gives

9 - (-5) = 14

.

Step-by-Step Solution

1

Substitute the given values into the expression.

(-3)^2 - \frac{2^3(-5)}{|-3 + (-5)|}

Direct variable substitution according to the given values is the first step in evaluating any algebraic expression.

2

Evaluate the numerator of the fraction.

2^3(-5) = 8(-5) = -40

According to the order of operations, exponents must be evaluated before multiplication.

3

Evaluate the denominator of the fraction.

|-3 + (-5)| = |-8| = 8

Grouping symbols, such as absolute value bars, must be evaluated from the inside out.

4

Divide the numerator by the denominator to find the fraction term's value.

\frac{-40}{8} = -5

Division within the fraction is performed before interacting with the rest of the expression.

5

Evaluate the squared term and subtract the fraction's value.

9 - (-5) = 9 + 5 = 14

Exponents are evaluated first, and subtracting a negative number is mathematically equivalent to adding a positive number.

Key Concept

Evaluating multi-step algebraic expressions involving negative bases, absolute values, and fractions.

Estimated Time:1m 0s

Question 160Question →

Find the value of the following expression for $x = -3$ , $y = -2$ , and $z = \frac{1}{2}$ :

\frac{x^2 y - y^3}{x - y} + 4z

Show answer & explanation

Answer: 12

Answer

12

The correct result of 12 is obtained by substituting the given values, correctly applying exponents to negative bases, properly combining terms with double negatives, and following the standard order of operations.

Step-by-Step Solution

1

Substitute the values of x, y, and z into the algebraic expression.

\frac{(-3)^2(-2) - (-2)^3}{(-3) - (-2)} + 4(\frac{1}{2})

Direct substitution establishes the numerical expression to be evaluated.

2

Evaluate the terms containing exponents.

(-3)^2 = 9

and

(-2)^3 = -8

Following the order of operations, exponents are calculated before multiplication or addition.

3

Multiply the factors in the numerator and the separate z-term.

The numerator term becomes

9(-2) = -18

. The z-term becomes

4(\frac{1}{2}) = 2

.

Multiplication is the next step in simplifying the expression according to the order of operations.

4

Simplify the fraction's numerator and denominator by performing the subtractions.

Numerator:

-18 - (-8) = -10

. Denominator:

-3 - (-2) = -1

.

The fraction bar acts as a grouping symbol; the numerator and denominator must be simplified completely before division.

5

Divide the simplified numerator by the simplified denominator.

\frac{-10}{-1} = 10

Completing the division of the rational part of the expression.

6

Add the result of the division to the remaining term.

10 + 2 = 12

Addition is the final operation required to find the total value.

Key Concept

Evaluating complex algebraic expressions using substitution and the order of operations with negative integers and fractions.

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