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Evaluating Algebraic Expressions

29 questions

All types29Multiple choice14Numeric7Fill in the blank8
Question 1Question →

Determine the final numerical value of the expression k2+4jk^2 + 4j when substituting k=−3k = -3 and j=−2j = -2.

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Answer: 1

Answer

The correct value of the expression is 1.
The expression correctly evaluates to 1 by substituting the variables with their respective negative values and strictly following the order of operations (calculating exponents, then multiplication, then addition).

Step-by-Step Solution

1
Substitute the given numerical values into the variables in the expression.
(−3)2+4(−2)(-3)^2 + 4(-2)
To begin evaluating the algebraic expression, each variable must be replaced with its assigned value.
2
Evaluate the exponent portion of the expression.
9+4(−2)9 + 4(-2)
According to the order of operations, exponents are calculated before multiplication or addition. Squaring a negative number results in a positive value.
3
Perform the multiplication.
9−89 - 8
Multiplication precedes addition in the order of operations. Multiplying a positive integer by a negative integer gives a negative result.
4
Calculate the final sum.
11
Combining the remaining terms provides the final evaluated result of the expression.

Key Concept

Evaluating Algebraic Expressions
Question 2Question →

Complete the evaluation of the algebraic expression by filling in the blanks with the correct numerical values.

Fill in the blanks below

Evaluate the algebraic expression 5x2y−3z3x5x^2y - \frac{3z^3}{x} for the given variables x=−2x = -2, y=3y = 3, and z=−2z = -2.

Step 1: The numerical value of the first term, 5x2y5x^2y, is .
Step 2: The numerical value of the second term (including the negative sign), −3z3x-\frac{3z^3}{x}, is .
Step 3: The final evaluated value of the entire expression is .
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Answer

The first term evaluates to 60, the second term evaluates to -12, and the final combined expression value is 48.
The expression requires careful substitution and order of operations. First, squaring x=−2x = -2 yields 44, making the first term 5(4)(3)=605(4)(3) = 60. Next, cubing z=−2z = -2 gives −8-8, resulting in a numerator of −24-24. Dividing by x=−2x = -2 gives 1212. Applying the subtraction sign makes the second term −12-12. Adding these components yields 60−12=4860 - 12 = 48.

Step-by-Step Solution

1
Substitute the given values into the first term 5x2y5x^2y.
5(−2)2(3)5(-2)^2(3)
Variables must be replaced with their assigned numerical values.
2
Evaluate the exponent and multiply to find the value of the first term.
5(4)(3)=605(4)(3) = 60
According to the order of operations, exponents are evaluated before multiplication.
3
Substitute the given values into the second term −3z3x-\frac{3z^3}{x}.
−3(−2)3−2-\frac{3(-2)^3}{-2}
Preparing the rational term for evaluation.
4
Evaluate the exponent in the numerator.
−3(−8)−2-\frac{3(-8)}{-2}
The cube of a negative number remains negative, so (−2)3=−8(-2)^3 = -8.
5
Multiply the numerator and simplify the fraction.
−−24−2=−(12)=−12-\frac{-24}{-2} = - (12) = -12
Multiplying 3 by -8 gives -24. Dividing -24 by -2 gives positive 12. Applying the negative sign in front results in -12.
6
Combine the evaluated terms to find the final result.
60−12=4860 - 12 = 48
Summing the intermediate values yields the final expression value.

Key Concept

Evaluating algebraic expressions involves correctly applying variable substitution, managing negative signs, and strictly following the order of operations.
Estimated Time:1m 0s
Question 3Question →

What is the evaluated result of the algebraic expression a2−4ab+b3a^2 - 4ab + b^3 when a=−5a = -5 and b=−2b = -2?

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Answer: -23

Answer

The correct evaluated result is -23.
The correct answer is found by substituting a=−5a = -5 and b=−2b = -2 into the expression and strictly following the order of operations. Squaring −5-5 results in 2525. The product of −4(−5)(−2)-4(-5)(-2) is −40-40. Cubing −2-2 results in −8-8. Combining these gives 25−40−825 - 40 - 8, which equals −23-23.

Step-by-Step Solution

1
Substitute the given values a=−5a = -5 and b=−2b = -2 into the expression.
(−5)2−4(−5)(−2)+(−2)3(-5)^2 - 4(-5)(-2) + (-2)^3
Direct substitution is the first step to evaluating an algebraic expression.
2
Evaluate the exponential terms.
25−4(−5)(−2)+(−8)25 - 4(-5)(-2) + (-8)
According to the order of operations (PEMDAS), exponents must be calculated before multiplication. Squaring a negative number gives a positive result ((−5)2=25(-5)^2 = 25), and cubing a negative number gives a negative result ((−2)3=−8(-2)^3 = -8).
3
Calculate the multiplication in the middle term.
25−40−825 - 40 - 8
Multiply the three factors: −4×−5×−2-4 \times -5 \times -2. Two negatives make a positive (2020), and multiplying by the third negative makes it negative (−40-40).
4
Combine the simplified terms using addition and subtraction from left to right.
−15−8=−23-15 - 8 = -23
Final arithmetic computation yields the fully evaluated numerical value.

Key Concept

Evaluating Algebraic Expressions
Estimated Time:1m 0s
Question 4Question →

Calculate the exact value of the algebraic expression 3x2y−12yx3x^2y - \frac{12y}{x} when the variables are assigned the values x=−2x = -2 and y=−5y = -5.

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Answer: -90

Answer

The correct evaluated result is -90.
By accurately substituting the values, correctly squaring the negative number (−2)2=4(-2)^2 = 4, evaluating the fractional term as −60−2=30\frac{-60}{-2} = 30, and completing the final subtraction −60−30-60 - 30, the expression correctly evaluates to -90.

Step-by-Step Solution

1
Substitute the given variable values into the algebraic expression.
3(−2)2(−5)−12(−5)−23(-2)^2(-5) - \frac{12(-5)}{-2}
This sets up the expression with numerical values for evaluation.
2
Evaluate the exponent applied to the negative base.
(−2)2=4(-2)^2 = 4, making the expression 3(4)(−5)−12(−5)−23(4)(-5) - \frac{12(-5)}{-2}
According to the order of operations (PEMDAS), exponents must be evaluated before multiplication and division.
3
Perform the multiplication in the first term.
3(4)(−5)=12(−5)=−603(4)(-5) = 12(-5) = -60
Evaluating the first term completely from left to right.
4
Perform the multiplication and division in the second term (the fraction).
12(−5)−2=−60−2=30\frac{12(-5)}{-2} = \frac{-60}{-2} = 30
Evaluating the fractional term by first multiplying the numerator, then dividing by the negative denominator.
5
Subtract the result of the fraction from the first term.
−60−30=−90-60 - 30 = -90
The final operation is the subtraction of the two evaluated terms.

Key Concept

Evaluating algebraic expressions using substitution and the order of operations with signed numbers.
Question 5Question →

If x=−3x = -3, y=4y = 4, and z=−2z = -2, what is the value of the expression 2x2yz+5x\frac{2x^2y}{z} + 5x?

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Answer: -51

Answer

The evaluated result of the algebraic expression is -51.
The correct value is determined by carefully substituting the negative variables, properly applying the exponent to get a positive 99, calculating the fraction 72−2\frac{72}{-2} to get −36-36, and adding the −15-15 from the second term to reach −51-51.

Step-by-Step Solution

1
Substitute the given numerical values for each variable in the expression.
2(−3)2(4)−2+5(−3)\frac{2(-3)^2(4)}{-2} + 5(-3)
Variable substitution replaces letters with their specific numerical values to prepare for arithmetic calculation.
2
Evaluate the exponent according to the order of operations (PEMDAS).
(−3)2=9(-3)^2 = 9
Exponents must be calculated before multiplication or division.
3
Perform the multiplication in the numerator of the fraction and in the second term.
Numerator: 2(9)(4)=722(9)(4) = 72. Second term: 5(−3)=−155(-3) = -15. The expression is now 72−2+(−15)\frac{72}{-2} + (-15).
Multiplication is the next step in the order of operations.
4
Perform the division, then combine the remaining terms.
72−2=−36\frac{72}{-2} = -36. Finally, −36+(−15)=−51-36 + (-15) = -51.
Division occurs before addition, completing the evaluation.

Key Concept

Evaluating Algebraic Expressions
Estimated Time:1m 0s
Question 6Question →

Evaluate the algebraic expression 3a2b−4bca−c\frac{3a^2b - 4bc}{a - c} given that a=−2a = -2, b=3b = 3, and c=−5c = -5. What is the final numerical value?

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Answer: 32

Answer

The numerical value of the expression is 32.
After properly substituting the given values, evaluating the exponents, and following the order of operations, the numerator calculates to 96 and the denominator simplifies to 3. The final division of 96 by 3 yields 32.

Step-by-Step Solution

1
Substitute the assigned values of aa, bb, and cc into the given algebraic expression.
3(−2)2(3)−4(3)(−5)−2−(−5)\frac{3(-2)^2(3) - 4(3)(-5)}{-2 - (-5)}
Direct variable substitution is required to compute the numerical value.
2
Evaluate the exponent applied to the negative base.
(−2)2=4(-2)^2 = 4
According to the order of operations (PEMDAS), exponents must be resolved prior to multiplication.
3
Carry out the multiplication operations in the numerator.
3(4)(3)=363(4)(3) = 36 and 4(3)(−5)=−604(3)(-5) = -60
Multiplication is processed from left to right within the grouping of the numerator.
4
Complete the subtraction operation in the numerator.
36−(−60)=36+60=9636 - (-60) = 36 + 60 = 96
Subtracting a negative quantity is algebraically identical to adding its positive counterpart.
5
Evaluate the subtraction operation in the denominator.
−2−(−5)=−2+5=3-2 - (-5) = -2 + 5 = 3
The entire denominator acts as a single grouping and must be fully simplified before division.
6
Divide the evaluated numerator by the evaluated denominator.
963=32\frac{96}{3} = 32
This final arithmetic step produces the precise evaluation of the fractional expression.

Key Concept

Evaluating complex rational algebraic expressions using substitution and the order of operations with signed integers.
Question 7Question →

Evaluate the algebraic expression below given that x=−4x = -4, y=−2y = -2, and z=−3z = -3:

x3−y2(x−z)∣x∣−z2 \frac{x^3 - y^2(x - z)}{|x| - z^2}

Calculate the intermediate values and the final result to complete the statements.

Fill in the blanks below

The value of the numerator is .
The value of the denominator is .
The final evaluated result of the expression is .
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Answer

The numerator evaluates to -60, the denominator evaluates to -5, and the final evaluated result of the expression is 12.
By strictly adhering to the order of operations and properly handling negative substitutions, the numerator simplifies to -60 and the denominator simplifies to -5. Dividing these yields a final positive integer result of 12.

Step-by-Step Solution

1
Substitute the given values for xx, yy, and zz into the expression.
(−4)3−(−2)2(−4−(−3))∣−4∣−(−3)2 \frac{(-4)^3 - (-2)^2(-4 - (-3))}{|-4| - (-3)^2}
Direct substitution is the required first step before performing any operations.
2
Evaluate the exponents in the numerator and denominator.
Numerator exponents: (−4)3=−64(-4)^3 = -64 and (−2)2=4(-2)^2 = 4. Denominator exponent: (−3)2=9(-3)^2 = 9.
Following the order of operations, exponents must be evaluated before multiplication and subtraction.
3
Evaluate the expression inside the parentheses in the numerator and the absolute value in the denominator.
Numerator parentheses: (−4−(−3))=(−4+3)=−1(-4 - (-3)) = (-4 + 3) = -1. Denominator absolute value: ∣−4∣=4|-4| = 4.
Operations inside grouping symbols and absolute value operations take precedence.
4
Perform the remaining multiplication in the numerator, then subtract to find the final numerator value.
−64−(4)(−1)=−64−(−4)=−64+4=−60-64 - (4)(-1) = -64 - (-4) = -64 + 4 = -60. (Value for blank 1)
Multiplication is performed before addition or subtraction according to the order of operations.
5
Perform the subtraction in the denominator.
4−9=−54 - 9 = -5. (Value for blank 2)
This fully simplifies the expression in the denominator.
6
Divide the fully simplified numerator by the simplified denominator.
−60−5=12 \frac{-60}{-5} = 12 . (Value for blank 3)
The fraction bar represents division, which is the final operation applied to the evaluated top and bottom terms.

Key Concept

Evaluating Complex Algebraic Expressions
Question 8Question →

Evaluate the algebraic expression:

−3a2+4b−∣a−c∣-3a^2 + 4b - |a - c|

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

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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=−2a = -2 gives 44, which multiplied by −3-3 results in −12-12. The term 4b4b becomes 4(−3)=−124(-3) = -12. The absolute value of a−ca - c evaluates to ∣−2−5∣=∣−7∣=7|-2 - 5| = |-7| = 7. Subtracting this 77 from the sum of the first two terms yields −12−12−7=−31-12 - 12 - 7 = -31.

Step-by-Step Solution

1
Substitute the given values for aa, bb, and cc into the expression.
−3(−2)2+4(−3)−∣−2−5∣-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(-2)^2 = 4. The absolute value expression evaluates to ∣−7∣=7|-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)=−12-3(4) = -12 and 4(−3)=−124(-3) = -12. The expression simplifies to −12−12−7-12 - 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=−31-12 - 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 9Question →

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

x2−y3z∣x+z∣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., y3z∣x+z∣\frac{y^3 z}{|x + z|}) is , which makes the final value of the entire expression equal to .
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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|-3 + -5| = 8). Next, exponents are calculated ((−3)2=9(-3)^2 = 9 and 23=82^3 = 8), followed by multiplication and division to simplify the fraction to −5-5. Finally, subtracting this negative value gives 9−(−5)=149 - (-5) = 14.

Step-by-Step Solution

1
Substitute the given values into the expression.
(−3)2−23(−5)∣−3+(−5)∣(-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.
23(−5)=8(−5)=−402^3(-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|-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.
−408=−5\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=149 - (-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 10Question →

Find the value of the following expression for x=−3x = -3, y=−2y = -2, and z=12z = \frac{1}{2}:

x2y−y3x−y+4z \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.
(−3)2(−2)−(−2)3(−3)−(−2)+4(12)\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(-3)^2 = 9 and (−2)3=−8(-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)=−189(-2) = -18. The z-term becomes 4(12)=24(\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)=−10-18 - (-8) = -10. Denominator: −3−(−2)=−1-3 - (-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.
−10−1=10\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=1210 + 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.
Question 11Question →

If x=−2x = -2 and y=−3y = -3, what is the value of the expression 2x2−5y2x^2 - 5y?

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Answer: 23

Answer

23
The expression is evaluated by first substituting the values and applying the exponent to the base, giving (−2)2=4(-2)^2 = 4. Then, perform the multiplication to get 2(4)=82(4) = 8 and 5(−3)=−155(-3) = -15. Finally, subtracting negative 15 from 8 is the same as adding 15, resulting in 8+15=238 + 15 = 23.

Step-by-Step Solution

1
Substitute the given values for xx and yy into the expression.
2(−2)2−5(−3)2(-2)^2 - 5(-3)
To evaluate the expression, each variable must be replaced with its corresponding numerical value.
2
Evaluate the exponent.
2(4)−5(−3)2(4) - 5(-3)
According to the order of operations (PEMDAS), exponents must be resolved before multiplication. The square of any non-zero real number is positive, so (−2)2=4(-2)^2 = 4.
3
Perform the multiplication operations.
8−(−15)8 - (-15)
Multiply 22 by 44 to get 88, and 55 by −3-3 to get −15-15.
4
Perform the subtraction.
8+15=238 + 15 = 23
Subtracting a negative number is mathematically identical to adding its positive counterpart.

Key Concept

Evaluating expressions using substitution and the order of operations.
Question 12Question →

Calculate the intermediate and final values for the given algebraic expression.

Fill in the blanks below

To evaluate the expression 5x−y25x - y^2 for x=3x = 3 and y=−4y = -4, we first substitute the given numbers. The term 5x5x evaluates to . The term y2y^2 evaluates to . Subtracting the second result from the first gives a final answer of .
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Answer

The intermediate values are 15 and 16, resulting in a final answer of -1.
Substituting the given values into the expression 5x−y25x - y^2 yields 5(3)−(−4)25(3) - (-4)^2. Following the order of operations, first evaluate the exponent: (−4)2=16(-4)^2 = 16. Next, perform the multiplication: 5(3)=155(3) = 15. Finally, perform the subtraction: 15−16=−115 - 16 = -1.

Step-by-Step Solution

1
Substitute x = 3 into the first term, 5x.
5(3) = 15
Variable substitution replaces the variable with its assigned numerical value.
2
Substitute y = -4 into the second term, y^2.
(-4)^2 = 16
Squaring a negative number results in a positive value, since negative multiplied by negative is positive.
3
Subtract the evaluated second term from the evaluated first term.
15 - 16 = -1
The original expression requires subtracting the y^2 term from the 5x term.

Key Concept

Evaluating Algebraic Expressions
Question 13Question →

Calculate the value of the algebraic expression x2−4yx^2 - 4y given that x=−5x = -5 and y=−2y = -2. What is the final value?

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Answer: 33

Answer

33
By correctly applying substitution and the order of operations, squaring −5-5 gives 2525, and multiplying 44 by −2-2 gives −8-8. Subtracting −8-8 from 2525 results in 3333.

Step-by-Step Solution

1
Substitute the given variable values into the expression
(−5)2−4(−2)(-5)^2 - 4(-2)
To replace the variables with their corresponding numerical values.
2
Evaluate the exponent
25−4(−2)25 - 4(-2)
Following the order of operations (PEMDAS), exponents must be calculated before multiplication and subtraction. The square of a negative number is positive.
3
Evaluate the multiplication
25−(−8)25 - (-8)
Multiply 4 by -2 to find the term to be subtracted.
4
Perform the final subtraction
25+8=3325 + 8 = 33
Subtracting a negative number is equivalent to adding its positive counterpart.

Key Concept

Evaluating Algebraic Expressions
Question 14Question →

Follow the steps to evaluate the algebraic expression given the variable assignments. What are the intermediate and final values?

Fill in the blanks below

For the expression p3−6qp^3 - 6q, given that p=−2p = -2 and q=3q = 3:

The value of the first term, p3p^3, is .
The value of the second term, 6q6q, is .
The final evaluated result of the entire expression is .
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Answer

The intermediate values are -8 and 18, which subtract to a final result of -26.
By accurately substituting p=−2p = -2 and q=3q = 3 into the given expression, applying the exponent correctly, and performing the final subtraction, we arrive at the correct values for each step and the final answer of -26.

Step-by-Step Solution

1
Substitute the given value for pp into the first term.
(−2)3=−8(-2)^3 = -8
Replacing the variable pp with −2-2 evaluates the exponent. A negative base raised to an odd power remains negative.
2
Substitute the given value for qq into the second term.
6(3)=186(3) = 18
Replacing the variable qq with 33 completes the multiplication required for the second term.
3
Combine the evaluated terms according to the original expression.
−8−18=−26-8 - 18 = -26
Following the original expression p3−6qp^3 - 6q, we subtract the value of the second term from the first term.

Key Concept

Evaluating algebraic expressions involving exponents and negative numbers
Estimated Time:45s
Question 15Question →

If x=−3x = -3 and y=−4y = -4, what does the expression 5x−2y5x - 2y equal?

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Answer: -7

Answer

-7
Substituting the given values into the expression yields 5(−3)−2(−4)5(-3) - 2(-4). Multiplying the terms results in −15−(−8)-15 - (-8), which simplifies to −15+8-15 + 8. Adding these numbers correctly produces −7-7.

Step-by-Step Solution

1
Substitute x=−3x = -3 and y=−4y = -4 into the algebraic expression.
5(−3)−2(−4)5(-3) - 2(-4)
Variable substitution replaces each variable with its given numerical value using parentheses.
2
Multiply the terms according to the order of operations.
−15−(−8)-15 - (-8)
Multiplication is performed before subtraction: 5×−3=−155 \times -3 = -15 and 2×−4=−82 \times -4 = -8.
3
Simplify the double negative and add the integers.
−15+8=−7-15 + 8 = -7
Subtracting a negative number is equivalent to adding its positive counterpart.

Key Concept

Evaluating Algebraic Expressions
Estimated Time:45s
Question 16Question →

Given the expression 7x−2y27x - 2y^2, what is its numerical value when x=5x = 5 and y=3y = 3?

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Answer: 17

Answer

17
The expression evaluates to 17 by successfully substituting the values into the variables and properly applying the order of operations.

Step-by-Step Solution

1
Substitute the values x=5x = 5 and y=3y = 3 into the algebraic expression.
7(5)−2(3)27(5) - 2(3)^2
Variables must be replaced with their numerical values before evaluating.
2
Calculate the exponent.
7(5)−2(9)7(5) - 2(9)
According to the order of operations (PEMDAS), exponents must be evaluated before multiplication.
3
Perform the multiplication operations.
35−1835 - 18
Multiplication is evaluated before subtraction.
4
Subtract the resulting numbers.
17
This final arithmetic step yields the evaluated expression.

Key Concept

Evaluating Algebraic Expressions
Question 17Question →

In a specific algorithm, the output value is determined by the algebraic rule 2m2n−3n22m^2n - 3n^2. Which of the following is the resulting output when the algorithm processes the inputs m=−4m = -4 and n=−3n = -3?

Show answer & explanation

Answer: -123

Answer

The correct evaluated result is -123.
The correct answer is found by substituting the numerical values for the variables and strictly adhering to the order of operations. First, squaring the negative integers results in positive values, transforming the expression to 2(16)(-3) - 3(9). Multiplying the terms gives -96 and 27, respectively. Finally, subtracting 27 from -96 produces a final value of -123.

Step-by-Step Solution

1
Substitute the given values m=−4m = -4 and n=−3n = -3 into the expression 2m2n−3n22m^2n - 3n^2.
2(−4)2(−3)−3(−3)22(-4)^2(-3) - 3(-3)^2
Direct substitution of variables with their numerical values is the required first step to evaluate any algebraic expression.
2
Evaluate the exponents according to the order of operations (PEMDAS).
(−4)2=16(-4)^2 = 16 and (−3)2=9(-3)^2 = 9. The expression becomes 2(16)(−3)−3(9)2(16)(-3) - 3(9).
Exponents must be calculated before multiplication to maintain mathematical accuracy. A negative number raised to an even power results in a positive number.
3
Perform the multiplication operations from left to right.
For the first term: 2(16)=322(16) = 32, and 32(−3)=−9632(-3) = -96. For the second term: 3(9)=273(9) = 27. The expression is now −96−27-96 - 27.
Multiplication precedes subtraction in the order of operations.
4
Perform the final subtraction to find the total value.
−96−27=−123-96 - 27 = -123
Combining the remaining terms yields the final evaluated numerical result of the polynomial.

Key Concept

Evaluating algebraic expressions with negative integers using substitution and the order of operations.
Question 18Question →

Determine the value of the polynomial −2a2+5ab−b3-2a^2 + 5ab - b^3 under the condition that a=−2a = -2 and b=−3b = -3. Which of the following represents the correct evaluation?

Show answer & explanation

Answer: 49

Answer

49
The correct answer is found by substituting the variable values, properly applying the exponents to the negative numbers (where (−2)2=4(-2)^2 = 4 and (−3)3=−27(-3)^3 = -27), and adhering to the order of operations to multiply before adding and subtracting.

Step-by-Step Solution

1
Substitute −2-2 for aa and −3-3 for bb into the given polynomial.
−2(−2)2+5(−2)(−3)−(−3)3-2(-2)^2 + 5(-2)(-3) - (-3)^3
To prepare the expression for numerical evaluation.
2
Evaluate the exponential terms, noting that a negative number raised to an even power is positive, and to an odd power is negative.
−2(4)+5(−2)(−3)−(−27)-2(4) + 5(-2)(-3) - (-27)
The order of operations (PEMDAS) requires resolving exponents before multiplication.
3
Perform the multiplication operations from left to right.
−8+30−(−27)-8 + 30 - (-27)
The order of operations dictates that multiplication must be completed before addition and subtraction.
4
Convert the subtraction of a negative into addition and calculate the final sum.
−8+30+27=22+27=49-8 + 30 + 27 = 22 + 27 = 49
This is the final step to find the total value of the polynomial.

Key Concept

Evaluating Algebraic Expressions
Estimated Time:45s
Question 19Question →

Evaluate the algebraic expression P=4v3−3uv+2u2P = 4v^3 - 3uv + 2u^2 using the assigned values u=−4u = -4 and v=−2v = -2. Fill in the missing numerical values for the intermediate calculation step and the final evaluated result.

Fill in the blanks below

The calculated value for the isolated term 4v34v^3 is , which makes the final evaluated total for the entire expression PP equal to .
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Answer

The first term evaluates to -32, and the entire expression evaluates to -24.
The expression requires careful substitution of u=−4u = -4 and v=−2v = -2. Following the order of operations (PEMDAS), we first calculate the exponents: (−2)3=−8(-2)^3 = -8 and (−4)2=16(-4)^2 = 16. Next, we perform the multiplication for each term: 4(−8)=−324(-8) = -32, −3(−4)(−2)=−24-3(-4)(-2) = -24, and 2(16)=322(16) = 32. Finally, combining these intermediate values gives −32−24+32=−24-32 - 24 + 32 = -24.

Step-by-Step Solution

1
Substitute the value of v into the first term 4v34v^3 and evaluate according to the order of operations.
4(−2)3=4(−8)=−324(-2)^3 = 4(-8) = -32
Exponents must be evaluated before multiplication. The cube of a negative number remains negative.
2
Substitute the values of u and v into the second term −3uv-3uv and evaluate.
−3(−4)(−2)=−3(8)=−24-3(-4)(-2) = -3(8) = -24
Multiplying two negative numbers yields a positive product, which is then multiplied by the negative coefficient.
3
Substitute the value of u into the third term 2u22u^2 and evaluate.
2(−4)2=2(16)=322(-4)^2 = 2(16) = 32
The square of a negative number is positive. This must be calculated before multiplying by the coefficient.
4
Combine the evaluated terms to find the final total for P.
−32−24+32=−24-32 - 24 + 32 = -24
Adding and subtracting the values from left to right yields the final result.

Key Concept

Evaluating multi-variable algebraic expressions by correctly substituting negative integers and rigorously applying the order of operations.
Question 20Question →

Consider the algebraic expression 2x2−yzx+y\frac{2x^2 - yz}{x + y}. If the variables are assigned the values x=−3x = -3, y=5y = 5, and z=−2z = -2, what are the intermediate evaluated values for the numerator and denominator, and what is the final result?

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After substituting the values, the evaluated result of the numerator is and the denominator is . Dividing these intermediate values yields a final result of .
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Answer

The intermediate numerator evaluates to 28, the denominator to 2, and the final evaluated result is 14.
The correct final value is 14. This is determined by carefully substituting the negative and positive variable values into the numerator and denominator, strictly applying the order of operations to handle exponents and negative signs, and finally dividing the intermediate numerator 28 by the denominator 2.

Step-by-Step Solution

1
Substitute the value of xx into the first term of the numerator, 2x22x^2.
2(−3)2=2(9)=182(-3)^2 = 2(9) = 18
Applying the order of operations, exponentiation must be evaluated before multiplication.
2
Substitute the values of yy and zz into the second term of the numerator, yzyz.
(5)(−2)=−10(5)(-2) = -10
Multiplying a positive integer by a negative integer results in a negative integer.
3
Calculate the complete numerator by subtracting the second term from the first.
18−(−10)=18+10=2818 - (-10) = 18 + 10 = 28
Subtracting a negative number is mathematically equivalent to adding its positive counterpart.
4
Substitute the values of xx and yy to evaluate the denominator, x+yx + y.
−3+5=2-3 + 5 = 2
Combining the negative and positive integers.
5
Divide the fully evaluated numerator by the evaluated denominator.
282=14\frac{28}{2} = 14
The final step in evaluating the rational expression.

Key Concept

Evaluating Algebraic Expressions
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