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Evaluating Algebraic Expressions

29 questions

All types29Multiple choice15Numeric7Fill in the blank7
Question 21Question →

If x=−2x = -2, y=13y = \frac{1}{3}, and z=−3z = -3, what is the value of the algebraic expression 3x2y−z2x+yz\frac{3x^2 y - z^2}{x + yz}?

Show answer & explanation

Answer: 53\frac{5}{3}

Answer

53\frac{5}{3}
Substituting the given values into the expression requires evaluating powers of negative numbers and performing multiplication prior to addition/subtraction. In the numerator, 3(−2)2(13)−(−3)2=3(4)(13)−9=4−9=−53(-2)^2(\frac{1}{3}) - (-3)^2 = 3(4)(\frac{1}{3}) - 9 = 4 - 9 = -5. In the denominator, −2+(13)(−3)=−2−1=−3-2 + (\frac{1}{3})(-3) = -2 - 1 = -3. Simplifying the fraction −5−3\frac{-5}{-3} gives 53\frac{5}{3}.

Step-by-Step Solution

1
Evaluate the numerator 3x2y−z23x^2 y - z^2 by substituting x=−2x = -2, y=13y = \frac{1}{3}, and z=−3z = -3.
3(−2)2(13)−(−3)2=3(4)(13)−9=4−9=−53(-2)^2\left(\frac{1}{3}\right) - (-3)^2 = 3(4)\left(\frac{1}{3}\right) - 9 = 4 - 9 = -5
Squaring negative numbers yields positive values: (−2)2=4(-2)^2 = 4 and (−3)2=9(-3)^2 = 9.
2
Evaluate the denominator x+yzx + yz using the same variable values.
−2+(13)(−3)=−2+(−1)=−3-2 + \left(\frac{1}{3}\right)(-3) = -2 + (-1) = -3
Perform the multiplication y⋅zy \cdot z before adding to xx according to the order of operations.
3
Divide the evaluated numerator by the evaluated denominator.
−5−3=53\frac{-5}{-3} = \frac{5}{3}
Dividing two negative numbers produces a positive quotient.

Key Concept

Evaluating Algebraic Expressions with Negative Values and Rational Numbers
Question 22Question →

If m=−4m = -4 and n=−12n = -\frac{1}{2}, what is the value of the algebraic expression m2−4n3(m+2n)2\frac{m^2 - 4n^3}{(m + 2n)^2}?

Show answer & explanation

Answer: 3350\frac{33}{50}

Answer

3350\frac{33}{50}
Substituting m=−4m = -4 and n=−12n = -\frac{1}{2} into the numerator gives (−4)2−4(−12)3=16−(−12)=332(-4)^2 - 4\left(-\frac{1}{2}\right)^3 = 16 - \left(-\frac{1}{2}\right) = \frac{33}{2}. Substituting into the denominator gives (−4+2(−12))2=(−5)2=25\left(-4 + 2\left(-\frac{1}{2}\right)\right)^2 = (-5)^2 = 25. Dividing the numerator by the denominator yields 33/225=3350\frac{33/2}{25} = \frac{33}{50}.

Step-by-Step Solution

1
Evaluate the terms in the numerator: m2m^2 and 4n34n^3.
m2=(−4)2=16m^2 = (-4)^2 = 16 and 4n3=4(−12)3=4(−18)=−124n^3 = 4\left(-\frac{1}{2}\right)^3 = 4\left(-\frac{1}{8}\right) = -\frac{1}{2}.
Substitute m=−4m = -4 and n=−12n = -\frac{1}{2} while applying powers before multiplication according to PEMDAS.
2
Calculate the complete numerator by subtracting the evaluated terms.
m2−4n3=16−(−12)=16+12=332m^2 - 4n^3 = 16 - \left(-\frac{1}{2}\right) = 16 + \frac{1}{2} = \frac{33}{2}.
Subtracting a negative quantity is equivalent to adding its positive counterpart.
3
Evaluate the expression inside the denominator parentheses, then square it.
m+2n=−4+2(−12)=−4−1=−5m + 2n = -4 + 2\left(-\frac{1}{2}\right) = -4 - 1 = -5, and (−5)2=25(-5)^2 = 25.
Grouped operations inside parentheses must be evaluated prior to applying outer exponents.
4
Divide the numerator by the denominator to get the final simplified fraction.
33225=332×25=3350\frac{\frac{33}{2}}{25} = \frac{33}{2 \times 25} = \frac{33}{50}.
Dividing a fraction by an integer combines the denominators.

Key Concept

Evaluating algebraic expressions involving negative bases, fractions, and order of operations
Estimated Time:1m 30s
Question 23Question →

Evaluate the algebraic expression 3x2−2xy+y23x^2 - 2xy + y^2 for x=−3x = -3 and y=−2y = -2.

Fill in the blanks below

The value of the expression 3x2−2xy+y23x^2 - 2xy + y^2 when x=−3x = -3 and y=−2y = -2 is .
Show answer & explanation

Answer

19
Substituting x=−3x = -3 and y=−2y = -2 into 3x2−2xy+y23x^2 - 2xy + y^2 yields 3(−3)2−2(−3)(−2)+(−2)2=3(9)−12+4=27−12+4=193(-3)^2 - 2(-3)(-2) + (-2)^2 = 3(9) - 12 + 4 = 27 - 12 + 4 = 19.

Step-by-Step Solution

1
Substitute x=−3x = -3 and y=−2y = -2 into the expression
3(−3)2−2(−3)(−2)+(−2)23(-3)^2 - 2(-3)(-2) + (-2)^2
Replace each variable with its assigned value, using parentheses to properly preserve negative signs.
2
Evaluate the exponent terms
3(9)−2(−3)(−2)+43(9) - 2(-3)(-2) + 4
Follow the order of operations (PEMDAS) by evaluating powers first: (−3)2=9(-3)^2 = 9 and (−2)2=4(-2)^2 = 4.
3
Perform the multiplications
27−12+427 - 12 + 4
Multiply the numerical factors: 3×9=273 \times 9 = 27 and −2×(−3)×(−2)=−12-2 \times (-3) \times (-2) = -12.
4
Perform addition and subtraction from left to right
1919
Subtract 1212 from 2727 to get 1515, then add 44 to arrive at 1919.

Key Concept

Evaluating Algebraic Expressions with Negative Values
Question 24Question →

If x=−3x = -3, y=−12y = -\frac{1}{2}, and z=4z = 4, what is the numerical value of the algebraic expression x2−4y2x+yz\frac{x^2 - 4y^2}{x + yz}?

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Answer: -1.6

Answer

The numerical value of the expression is −1.6-1.6.
Substituting x=−3x = -3, y=−12y = -\frac{1}{2}, and z=4z = 4 into x2−4y2x+yz\frac{x^2 - 4y^2}{x + yz} gives a numerator of (−3)2−4(−12)2=9−4(14)=8(-3)^2 - 4\left(-\frac{1}{2}\right)^2 = 9 - 4\left(\frac{1}{4}\right) = 8 and a denominator of −3+(−12)(4)=−3−2=−5-3 + \left(-\frac{1}{2}\right)(4) = -3 - 2 = -5. Evaluating 8−5\frac{8}{-5} yields −1.6-1.6.

Step-by-Step Solution

1
Evaluate the numerator x2−4y2x^2 - 4y^2
8
Squaring −3-3 gives 99, and squaring −12-\frac{1}{2} gives 14\frac{1}{4}. Thus, 9−4(14)=9−1=89 - 4\left(\frac{1}{4}\right) = 9 - 1 = 8.
2
Evaluate the denominator x+yzx + yz
-5
Multiplying −12-\frac{1}{2} by 44 gives −2-2. Adding −3+(−2)-3 + (-2) yields −5-5.
3
Compute the final fraction quotient
-1.6
Dividing the numerator 88 by the denominator −5-5 yields −1.6-1.6.

Key Concept

Evaluating algebraic expressions using order of operations with negative bases and fractional values.
Estimated Time:1m 30s
Question 25Question →

If a=−2a = -2, b=−12b = -\frac{1}{2}, and c=3c = 3, what is the value of the algebraic expression a3b−2c2a−b−2\frac{a^3 b - 2c^2}{a - b^{-2}}?

Show answer & explanation

Answer: 73\frac{7}{3}

Answer

The correct answer is 73\frac{7}{3}.
Substituting a=−2a = -2, b=−12b = -\frac{1}{2}, and c=3c = 3 gives a numerator of (−2)3(−12)−2(3)2=(−8)(−12)−18=4−18=−14(-2)^3 \left(-\frac{1}{2}\right) - 2(3)^2 = (-8)\left(-\frac{1}{2}\right) - 18 = 4 - 18 = -14, and a denominator of −2−(−12)−2=−2−4=−6-2 - \left(-\frac{1}{2}\right)^{-2} = -2 - 4 = -6. Dividing −14-14 by −6-6 simplifies to 73\frac{7}{3}.

Step-by-Step Solution

1
Substitute a=−2a = -2, b=−12b = -\frac{1}{2}, and c=3c = 3 into the numerator a3b−2c2a^3 b - 2c^2.
Numerator =(−2)3⋅(−12)−2(3)2=(−8)⋅(−12)−2(9)=4−18=−14= (-2)^3 \cdot \left(-\frac{1}{2}\right) - 2(3)^2 = (-8) \cdot \left(-\frac{1}{2}\right) - 2(9) = 4 - 18 = -14.
Apply exponents first, then multiplication, and finally subtraction.
2
Substitute a=−2a = -2 and b=−12b = -\frac{1}{2} into the denominator a−b−2a - b^{-2}.
Denominator =−2−(−12)−2=−2−(−2)2=−2−4=−6= -2 - \left(-\frac{1}{2}\right)^{-2} = -2 - (-2)^2 = -2 - 4 = -6.
A negative exponent indicates the reciprocal of the base, so (−12)−2=(−2)2=4\left(-\frac{1}{2}\right)^{-2} = (-2)^2 = 4.
3
Divide the numerator by the denominator and simplify the fraction.
−14−6=146=73.\frac{-14}{-6} = \frac{14}{6} = \frac{7}{3}.
Dividing two negative numbers yields a positive quotient, which simplifies by dividing the numerator and denominator by 2.

Key Concept

Evaluating algebraic expressions with negative bases and negative integer exponents
Estimated Time:1m 30s
Question 26Question →

A chemist uses the algebraic formula E=2x2−yzx+y2E = \frac{2x^2 - yz}{x + y^2} to determine the stability index of a synthesized compound. If x=−4x = -4, y=−3y = -3, and z=23z = \frac{2}{3}, what is the value of EE?

Show answer & explanation

Answer: 345\frac{34}{5}

Answer

345\frac{34}{5}
Substituting x=−4x = -4, y=−3y = -3, and z=23z = \frac{2}{3} into the numerator yields 2(−4)2−(−3)(23)=2(16)−(−2)=32+2=342(-4)^2 - (-3)\left(\frac{2}{3}\right) = 2(16) - (-2) = 32 + 2 = 34. Substituting into the denominator yields −4+(−3)2=−4+9=5-4 + (-3)^2 = -4 + 9 = 5. Thus, the value of the expression is 345\frac{34}{5}.

Step-by-Step Solution

1
Substitute the given values into the numerator 2x2−yz2x^2 - yz.
2(−4)2−(−3)(23)=2(16)−(−2)=32+2=342(-4)^2 - (-3)\left(\frac{2}{3}\right) = 2(16) - (-2) = 32 + 2 = 34
Squaring a negative number yields a positive result (−4)2=16(-4)^2 = 16, and multiplying −3-3 by 23\frac{2}{3} gives −2-2, which is then subtracted.
2
Substitute the given values into the denominator x+y2x + y^2.
−4+(−3)2=−4+9=5-4 + (-3)^2 = -4 + 9 = 5
Evaluating the exponent first gives (−3)2=9(-3)^2 = 9, then adding −4-4 gives 55.
3
Divide the numerator by the denominator.
345\frac{34}{5}
Combine the evaluated numerator and denominator to get the final value of the expression.

Key Concept

Evaluating algebraic expressions involving multiple variables with negative bases and fractional terms using standard order of operations.
Estimated Time:1m 15s
Question 27Question →

If a=−2a = -2, b=3b = 3, and c=−12c = -\frac{1}{2}, what is the value of the algebraic expression a3b+4ca2−2b\frac{a^3 b + 4c}{a^2 - 2b}?

Fill in the blanks below

If a=−2a = -2, b=3b = 3, and c=−12c = -\frac{1}{2}, the value of the algebraic expression a3b+4ca2−2b\frac{a^3 b + 4c}{a^2 - 2b} is .
Show answer & explanation

Answer

13
Substituting a=−2a = -2, b=3b = 3, and c=−12c = -\frac{1}{2} into the given expression yields a numerator of (−2)3(3)+4(−12)=−24−2=−26(-2)^3(3) + 4(-\frac{1}{2}) = -24 - 2 = -26 and a denominator of (−2)2−2(3)=4−6=−2(-2)^2 - 2(3) = 4 - 6 = -2. Dividing the numerator by the denominator gives −26−2=13\frac{-26}{-2} = 13.

Step-by-Step Solution

1
Evaluate the terms in the numerator individually.
a3b=(−2)3⋅3=−8⋅3=−24a^3 b = (-2)^3 \cdot 3 = -8 \cdot 3 = -24 and 4c=4(−12)=−24c = 4\left(-\frac{1}{2}\right) = -2.
Exponents take precedence before multiplication, and multiplying a positive by a negative yields a negative number.
2
Combine the terms to calculate the numerator.
Numerator =−24+(−2)=−26= -24 + (-2) = -26.
Adding two negative numbers sums their magnitudes with a negative sign.
3
Evaluate the terms in the denominator.
a2=(−2)2=4a^2 = (-2)^2 = 4 and 2b=2(3)=62b = 2(3) = 6.
Squaring a negative base results in a positive value.
4
Calculate the denominator.
Denominator =4−6=−2= 4 - 6 = -2.
Subtracting a larger number from a smaller number produces a negative result.
5
Divide the numerator by the denominator to find the final value.
−26−2=13\frac{-26}{-2} = 13.
Dividing a negative number by a negative number yields a positive quotient.

Key Concept

Evaluating Algebraic Expressions
Estimated Time:1m 30s
Question 28Question →

If x=−3x = -3 and y=2y = 2, what is the value of the algebraic expression x2y−(x−y)22x+y2\frac{x^2 y - (x - y)^2}{2x + y^2}?

Show answer & explanation

Answer: 72\frac{7}{2}

Answer

72\frac{7}{2}
Substituting x=−3x = -3 and y=2y = 2 gives a numerator of (−3)2(2)−(−3−2)2=18−25=−7(-3)^2(2) - (-3 - 2)^2 = 18 - 25 = -7 and a denominator of 2(−3)+22=−6+4=−22(-3) + 2^2 = -6 + 4 = -2. Dividing −7-7 by −2-2 gives 72\frac{7}{2}.

Step-by-Step Solution

1
Evaluate the terms in the numerator
x2y=(−3)2(2)=9⋅2=18x^2 y = (-3)^2(2) = 9 \cdot 2 = 18 and (x−y)2=(−3−2)2=(−5)2=25(x - y)^2 = (-3 - 2)^2 = (-5)^2 = 25
Apply order of operations by performing parenthetical subtraction before squaring negative numbers.
2
Subtract the terms to find the total numerator value
18−25=−718 - 25 = -7
Subtract the squared binomial result from the first product.
3
Evaluate the denominator
2x+y2=2(−3)+22=−6+4=−22x + y^2 = 2(-3) + 2^2 = -6 + 4 = -2
Multiply and square the variable values before adding.
4
Divide the numerator by the denominator and simplify
−7−2=72\frac{-7}{-2} = \frac{7}{2}
Dividing a negative number by a negative number yields a positive result.

Key Concept

Evaluating Algebraic Expressions with Negative Bases and Parentheses
Estimated Time:1m 0s
Question 29Question →

If x=−2x = -2, y=13y = \frac{1}{3}, and z=−4z = -4, what is the value of the algebraic expression x3y−1−(x−z)2x+yz\frac{x^3 y^{-1} - (x - z)^2}{x + y z}?

Show answer & explanation

Answer: 425\frac{42}{5}

Answer

425\frac{42}{5}
Substituting x=−2x = -2, y=13y = \frac{1}{3}, and z=−4z = -4 into the expression yields (−2)3=−8(-2)^3 = -8, (13)−1=3(\frac{1}{3})^{-1} = 3, and (−2−(−4))2=22=4(-2 - (-4))^2 = 2^2 = 4. Thus, the numerator equals −8×3−4=−28-8 \times 3 - 4 = -28. The denominator equals −2+(13)(−4)=−2−43=−103-2 + (\frac{1}{3})(-4) = -2 - \frac{4}{3} = -\frac{10}{3}. Dividing the numerator by the denominator gives −28−103=8410=425\frac{-28}{-\frac{10}{3}} = \frac{84}{10} = \frac{42}{5}.

Step-by-Step Solution

1
Substitute the given values into the numerator of the expression: x3y−1−(x−z)2x^3 y^{-1} - (x - z)^2.
Numerator = (−2)3(13)−1−(−2−(−4))2=(−8)(3)−(2)2=−24−4=−28(-2)^3 \left(\frac{1}{3}\right)^{-1} - (-2 - (-4))^2 = (-8)(3) - (2)^2 = -24 - 4 = -28.
First apply exponent rules and basic arithmetic inside the parentheses following standard order of operations.
2
Substitute the given values into the denominator of the expression: x+yzx + y z.
Denominator = −2+(13)(−4)=−2−43=−63−43=−103-2 + \left(\frac{1}{3}\right)(-4) = -2 - \frac{4}{3} = -\frac{6}{3} - \frac{4}{3} = -\frac{10}{3}.
Multiply yy and zz first, then find a common denominator to add the fraction to the integer.
3
Divide the simplified numerator by the simplified denominator.
\frac{-28}{-\frac{10}{3}} = -28 \times \left(-\frac{3}{10}\right) = \frac{84}{10} = \frac{42}{5}.
Dividing by a fraction is equivalent to multiplying by its reciprocal.

Key Concept

Evaluating Algebraic Expressions with Negative Numbers and Negative Exponents
Estimated Time:1m 15s
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