Examkin

Evaluating Algebraic Expressions

29 questions

All types29Multiple choice14Numeric7Fill in the blank8
Question 21Question →

In a mathematical model, the output variable MM is defined by the equation M=2a2−3ab+b3M = 2a^2 - 3ab + b^3. What is the computed value of MM if the input parameters are set to a=−3a = -3 and b=−2b = -2?

Show answer & explanation

Answer: -8

Answer

-8
The correct answer is found by substituting the values and strictly following the order of operations. First, the exponents are evaluated: (−3)2=9(-3)^2 = 9 and (−2)3=−8(-2)^3 = -8. Next, the multiplication is performed: 2(9)=182(9) = 18 and −3(−3)(−2)=−18-3(-3)(-2) = -18. Finally, combining these values gives 18−18−8=−818 - 18 - 8 = -8.

Step-by-Step Solution

1
Substitute the values a=−3a = -3 and b=−2b = -2 into the original equation M=2a2−3ab+b3M = 2a^2 - 3ab + b^3.
M=2(−3)2−3(−3)(−2)+(−2)3M = 2(-3)^2 - 3(-3)(-2) + (-2)^3
Direct substitution is the first step in evaluating any algebraic expression.
2
Evaluate the terms with exponents according to the order of operations (PEMDAS/BODMAS).
(−3)2=9(-3)^2 = 9 and (−2)3=−8(-2)^3 = -8
Exponents must be calculated before multiplication. A negative number squared is positive, while a negative number cubed remains negative.
3
Perform the multiplication for each term in the expression.
2(9)=182(9) = 18 and −3(−3)(−2)=−18-3(-3)(-2) = -18
Multiplication is performed next. For the middle term, multiplying three negative numbers results in a negative product.
4
Combine the resulting values using addition and subtraction.
18−18+(−8)=0−8=−818 - 18 + (-8) = 0 - 8 = -8
This final arithmetic operation yields the evaluated result of the expression.

Key Concept

Evaluating algebraic expressions involving exponents, negative bases, and strict adherence to the order of operations.
Estimated Time:1m 0s
Question 22Question →

What is the numerical value of the algebraic expression −5x2+12yx−y2-5x^2 + \frac{12y}{x} - y^2 when evaluated for x=−2x = -2 and y=−3y = -3?

Show answer & explanation

Answer: -11

Answer

The correct numerical value of the expression is -11.
Substituting x=−2x = -2 and y=−3y = -3 gives −5(−2)2+12(−3)−2−(−3)2-5(-2)^2 + \frac{12(-3)}{-2} - (-3)^2. Evaluating the exponents first results in −5(4)+−36−2−9-5(4) + \frac{-36}{-2} - 9. Multiplying and dividing yields −20+18−9-20 + 18 - 9. Finally, adding and subtracting gives the correct value of −11-11.

Step-by-Step Solution

1
Substitute the given numeric values into the algebraic expression using parentheses.
−5(−2)2+12(−3)−2−(−3)2-5(-2)^2 + \frac{12(-3)}{-2} - (-3)^2
Using parentheses during substitution prevents sign errors, especially when variables have negative values.
2
Evaluate the terms with exponents.
−5(4)+12(−3)−2−(9)-5(4) + \frac{12(-3)}{-2} - (9)
According to the order of operations (PEMDAS), exponents must be evaluated before multiplication and division.
3
Perform the multiplication and division operations.
−20+−36−2−9⇒−20+18−9-20 + \frac{-36}{-2} - 9 \Rightarrow -20 + 18 - 9
Multiplication and division are processed from left to right before addition and subtraction.
4
Complete the evaluation by performing addition and subtraction from left to right.
−20+18−9=−2−9=−11-20 + 18 - 9 = -2 - 9 = -11
Addition and subtraction are the final steps in the order of operations.

Key Concept

Evaluating algebraic expressions requires careful substitution of values using parentheses and strict adherence to the order of operations (PEMDAS).
Question 23Question →

Given the values x=12x = \frac{1}{2}, y=−4y = -4, and z=−2z = -2, determine the value of the expression 4x2y−yz34x^2y - yz^3.

Show answer & explanation

Answer: -36

Answer

-36
The correct answer is derived by substituting the variables accurately, applying the exponents (especially noting that a negative base to an odd power remains negative), and strictly following the order of operations to multiply and then subtract.

Step-by-Step Solution

1
Substitute x=12x = \frac{1}{2}, y=−4y = -4, and z=−2z = -2 into the expression.
4(12)2(−4)−(−4)(−2)34(\frac{1}{2})^2(-4) - (-4)(-2)^3
Variable substitution must be done exactly as given to prepare for evaluation.
2
Evaluate the terms with exponents.
(12)2=14(\frac{1}{2})^2 = \frac{1}{4} and (−2)3=−8(-2)^3 = -8. The expression becomes 4(14)(−4)−(−4)(−8)4(\frac{1}{4})(-4) - (-4)(-8).
According to the order of operations (PEMDAS), exponents must be resolved before multiplication.
3
Perform the multiplication operations from left to right.
4(14)(−4)=1(−4)=−44(\frac{1}{4})(-4) = 1(-4) = -4 and (−4)(−8)=32(-4)(-8) = 32. The expression is now −4−32-4 - 32.
Multiplication is the next step in the order of operations. Note that multiplying two negative numbers results in a positive product.
4
Perform the final subtraction.
−4−32=−36-4 - 32 = -36
Subtracting a positive number from a negative number moves further left on the number line.

Key Concept

Evaluating algebraic expressions using rational numbers, negative bases, and the correct order of operations.
Estimated Time:1m 0s
Question 24Question →

Calculate the result of the following algebraic fraction when x=−2x = -2 and y=2y = 2.

Fill in the blanks below

The evaluated value of the expression 4x3+2y2x−y\frac{4x^3 + 2y^2}{x - y} is .
Show answer & explanation

Answer

6
The correct answer is 6 because substituting the values x=−2x = -2 and y=2y = 2 into the expression yields a numerator of −24-24 and a denominator of −4-4. Dividing a negative by a negative results in a positive 6.

Step-by-Step Solution

1
Substitute the given values for xx and yy into the expression.
4(−2)3+2(2)2−2−2\frac{4(-2)^3 + 2(2)^2}{-2 - 2}
Variable substitution is the first required step in evaluating any algebraic expression.
2
Evaluate the terms with exponents in the numerator.
(−2)3=−8(-2)^3 = -8 and 22=42^2 = 4
According to the order of operations (PEMDAS), exponents must be resolved before multiplication.
3
Perform the multiplication operations in the numerator.
4(−8)=−324(-8) = -32 and 2(4)=82(4) = 8
Multiplication is performed after exponents.
4
Add the resulting terms in the numerator and subtract the terms in the denominator.
Numerator: −32+8=−24-32 + 8 = -24. Denominator: −2−2=−4-2 - 2 = -4.
Simplify the numerator and denominator completely before dividing.
5
Divide the simplified numerator by the simplified denominator.
−24−4=6\frac{-24}{-4} = 6
Dividing a negative number by a negative number results in a positive quotient.

Key Concept

Evaluating algebraic expressions with negative integers and exponents
Estimated Time:1m 0s
Question 25Question →

Given the variable assignments x=−4x = -4, y=2y = 2, and z=−6z = -6, which of the following represents the evaluated result of the mathematical expression below?

x2y−∣x−z∣y+x x^2 y - \frac{|x - z|}{y + x}
Show answer & explanation

Answer: 33

Answer

33
Substituting the given values into the expression yields (−4)2(2)−∣−4−(−6)∣2+(−4)(-4)^2(2) - \frac{|-4 - (-6)|}{2 + (-4)}. Squaring −4-4 gives 1616, and multiplying by 22 results in 3232. Inside the absolute value, −4−(−6)-4 - (-6) becomes −4+6=2-4 + 6 = 2, so the numerator is 22. The denominator is 2+(−4)=−22 + (-4) = -2. The fraction simplifies to 2/−2=−12 / -2 = -1. Subtracting this from the first term gives 32−(−1)=3332 - (-1) = 33.

Step-by-Step Solution

1
Substitute the given values for x, y, and z into the expression.
(−4)2(2)−∣−4−(−6)∣2+(−4)(-4)^2 (2) - \frac{|-4 - (-6)|}{2 + (-4)}
Direct substitution is the first requirement to evaluate an algebraic expression.
2
Evaluate the first term with exponents and multiplication.
(−4)2=16(-4)^2 = 16, then 16×2=3216 \times 2 = 32.
Following the order of operations (PEMDAS), exponents are calculated before multiplication.
3
Simplify the expression inside the absolute value in the numerator.
∣−4−(−6)∣=∣−4+6∣=∣2∣=2|-4 - (-6)| = |-4 + 6| = |2| = 2.
Subtracting a negative is equivalent to adding a positive. Absolute value makes the result positive.
4
Simplify the denominator and evaluate the fraction.
Denominator is 2+(−4)=−22 + (-4) = -2. The fraction evaluates to 2−2=−1\frac{2}{-2} = -1.
The operations within the implied grouping of the fraction must be fully resolved before dividing.
5
Subtract the evaluated fraction from the first term.
32−(−1)=32+1=3332 - (-1) = 32 + 1 = 33.
Final subtraction, ensuring that subtracting a negative value is treated as addition.

Key Concept

Evaluating Complex Algebraic Expressions with Integers
Question 26Question →

Compute the value of the algebraic expression 3p2−2pq3p^2 - 2pq utilizing the parameters p=−3p = -3 and q=5q = 5.

Show answer & explanation

Answer: 57

Answer

57
The correct answer accurately substitutes the negative and positive integers into the formula and follows PEMDAS. Squaring −3-3 yields 99, which is multiplied by 33 to give 2727. The second term involves multiplying 22, −3-3, and 55 to yield −30-30. Subtracting −30-30 from 2727 is equivalent to 27+3027 + 30, which totals 5757.

Step-by-Step Solution

1
Substitute the assigned values for p and q into the expression.
3(−3)2−2(−3)(5)3(-3)^2 - 2(-3)(5)
Variable substitution requires replacing every instance of a letter with its numerical value.
2
Evaluate the exponent according to the order of operations.
3(9)−2(−3)(5)3(9) - 2(-3)(5)
Exponents must be calculated before multiplication. A negative number squared becomes positive.
3
Perform the multiplications for both terms.
27−(−30)27 - (-30)
Multiply 3×93 \times 9 to get 27, and multiply 2×(−3)×52 \times (-3) \times 5 to get −30-30.
4
Subtract the second term from the first term.
57
Subtracting a negative number is the same as adding its positive equivalent (27+30=5727 + 30 = 57).

Key Concept

Evaluating Algebraic Expressions
Estimated Time:45s
Question 27Question →

When p=−2p = -2 and q=−5q = -5, which of the following represents the value of 4p−3q4p - 3q?

Show answer & explanation

Answer: 7

Answer

7
The correct value is found by substituting p=−2p = -2 and q=−5q = -5 into the expression 4p−3q4p - 3q. Applying the order of operations, first multiply to get −8-8 and −15-15. Finally, evaluate −8−(−15)-8 - (-15), which correctly simplifies to −8+15=7-8 + 15 = 7.

Step-by-Step Solution

1
Substitute −2-2 for pp and −5-5 for qq into the algebraic expression.
4(−2)−3(−5)4(-2) - 3(-5)
Variables must be replaced with their given numerical values to evaluate the expression.
2
Multiply the terms following the order of operations.
−8−(−15)-8 - (-15)
Multiplication must be performed before subtraction.
3
Simplify the resulting subtraction.
−8+15=7-8 + 15 = 7
Subtracting a negative number is mathematically identical to adding a positive number.

Key Concept

Evaluating Algebraic Expressions
Estimated Time:45s
Question 28Question →
Find the numerical value of the rational expression 3a2−4bc+2\frac{3a^2 - 4b}{c + 2} using the parameter values a=−2a = -2, b=−5b = -5, and c=−6c = -6.
Show answer & explanation

Answer: -8

Answer

The final numerical value is -8.
The correct value is found by carefully substituting the negative integer values into the rational expression, strictly applying the order of operations (exponents, then multiplication, then addition/subtraction), and accurately tracking the signs at each step to get 32 in the numerator and -4 in the denominator, which yields -8.

Step-by-Step Solution

1
Substitute the given values into the expression.
3(−2)2−4(−5)−6+2\frac{3(-2)^2 - 4(-5)}{-6 + 2}
Replace variables aa, bb, and cc with their assigned numerical equivalents to begin evaluation.
2
Evaluate the exponent in the numerator.
3(4)−4(−5)−6+2\frac{3(4) - 4(-5)}{-6 + 2}
According to the order of operations, exponents must be evaluated before multiplication. The square of a negative number is positive: (−2)2=4(-2)^2 = 4.
3
Perform the multiplications in the numerator.
12−(−20)−6+2\frac{12 - (-20)}{-6 + 2}
Calculate the products: 3×4=123 \times 4 = 12 and 4×(−5)=−204 \times (-5) = -20.
4
Simplify the subtraction in the numerator.
12+20−6+2=32−6+2\frac{12 + 20}{-6 + 2} = \frac{32}{-6 + 2}
Subtracting a negative number is equivalent to adding a positive number: 12−(−20)=12+20=3212 - (-20) = 12 + 20 = 32.
5
Evaluate the addition in the denominator.
32−4\frac{32}{-4}
Combine the constant terms in the denominator: −6+2=−4-6 + 2 = -4.
6
Divide the numerator by the denominator.
-8
A positive number divided by a negative number results in a negative quotient: 32÷(−4)=−832 \div (-4) = -8.

Key Concept

Evaluating Algebraic Expressions
Question 29Question →

A technician calculates a load factor using the expression x2−2yx+2\frac{x^2 - 2y}{x + 2}. If the inputs are x=−4x = -4 and y=3y = 3, what is the resulting load factor?

Show answer & explanation

Answer: -5

Answer

The resulting load factor is -5.
Substituting x=−4x = -4 and y=3y = 3 into the numerator gives (−4)2−2(3)=16−6=10(-4)^2 - 2(3) = 16 - 6 = 10. Substituting x=−4x = -4 into the denominator gives −4+2=−2-4 + 2 = -2. Finally, dividing the numerator by the denominator yields 10/−2=−510 / -2 = -5.

Step-by-Step Solution

1
Substitute the values x=−4x = -4 and y=3y = 3 into the numerator of the expression.
(−4)2−2(3)(-4)^2 - 2(3)
Variable substitution requires replacing the letters with their corresponding numerical values, using parentheses for negative numbers.
2
Evaluate the numerator following the standard order of operations.
16−6=1016 - 6 = 10
Squaring the negative integer −4-4 results in positive 1616. Multiplication (2×3=62 \times 3 = 6) must be performed before subtraction.
3
Substitute the value of xx into the denominator and evaluate.
−4+2=−2-4 + 2 = -2
Adding a positive integer to a negative integer reduces its absolute value while maintaining the negative sign.
4
Divide the evaluated numerator by the evaluated denominator.
10/−2=−510 / -2 = -5
A positive number divided by a negative number results in a negative quotient.

Key Concept

Evaluating rational algebraic expressions with negative integers
PreviousPage 2 / 2
Evaluating Algebraic Expressions Practice Questions — Armed Services Vocational Aptitude Battery (ASVAB) — Page 2 | Examkin