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Evaluating Algebraic Expressions

29 questions

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Question 1Question →

If x=−2x = -2, y=5y = 5, and z=−12z = -\frac{1}{2}, what is the value of the algebraic expression x3y+z−2y−2x\frac{x^3 y + z^{-2}}{y - 2x}?

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Answer: -4

Answer

The value of the expression is -4.
Substituting x=−2x = -2, y=5y = 5, and z=−12z = -\frac{1}{2} into the expression yields x3y=(−2)3(5)=−40x^3 y = (-2)^3(5) = -40 and z−2=(−12)−2=4z^{-2} = \left(-\frac{1}{2}\right)^{-2} = 4, making the numerator −40+4=−36-40 + 4 = -36. Evaluating the denominator gives y−2x=5−2(−2)=9y - 2x = 5 - 2(-2) = 9. Dividing −36-36 by 99 results in −4-4.

Step-by-Step Solution

1
Evaluate the terms in the numerator individually.
x3y=(−2)3(5)=−40x^3 y = (-2)^3(5) = -40 and z−2=(−12)−2=4z^{-2} = \left(-\frac{1}{2}\right)^{-2} = 4.
Negative bases raised to odd powers retain a negative sign, while negative exponents represent the reciprocal raised to a positive power.
2
Calculate the total numerator value.
−40+4=−36-40 + 4 = -36.
Summing the two evaluated terms gives the complete numerator.
3
Evaluate the denominator expression.
y−2x=5−2(−2)=5+4=9y - 2x = 5 - 2(-2) = 5 + 4 = 9.
Subtracting a negative quantity is equivalent to adding its positive counterpart.
4
Divide the numerator by the denominator.
−369=−4.\frac{-36}{9} = -4.
Dividing a negative integer by a positive integer yields a negative quotient.

Key Concept

Evaluating Algebraic Expressions with Negative Integers and Negative Exponents
Question 2Question →

If m=−3m = -3, n=18n = \frac{1}{8}, and p=−2p = -2, what is the value of the algebraic expression m−2−n−1/3⋅p−3m^{-2} - n^{-1/3} \cdot p^{-3}? Express your answer as a simplified fraction.

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The value of the expression is .
Show answer & explanation

Answer

The correct answer is 13/36.
Evaluating each term individually gives m−2=19m^{-2} = \frac{1}{9}, n−1/3=2n^{-1/3} = 2, and p−3=−18p^{-3} = -\frac{1}{8}. Substituting these values into the expression yields 19−2⋅(−18)\frac{1}{9} - 2 \cdot \left(-\frac{1}{8}\right). Following the order of operations, we first perform the multiplication: 2⋅(−18)=−142 \cdot \left(-\frac{1}{8}\right) = -\frac{1}{4}. We then subtract this result from the first term: 19−(−14)=19+14=1336\frac{1}{9} - \left(-\frac{1}{4}\right) = \frac{1}{9} + \frac{1}{4} = \frac{13}{36}.

Step-by-Step Solution

1
Evaluate m−2m^{-2} when m=−3m = -3.
m−2=(−3)−2=1(−3)2=19m^{-2} = (-3)^{-2} = \frac{1}{(-3)^2} = \frac{1}{9}
A negative exponent represents the reciprocal of the base raised to the positive power, and a negative base raised to an even power yields a positive result.
2
Evaluate n−1/3n^{-1/3} when n=18n = \frac{1}{8}.
n−1/3=(18)−1/3=(8−1)−1/3=81/3=2n^{-1/3} = \left(\frac{1}{8}\right)^{-1/3} = \left(8^{-1}\right)^{-1/3} = 8^{1/3} = 2
Apply the negative exponent rule to find the reciprocal of the fraction, then find the cube root of 8.
3
Evaluate p−3p^{-3} when p=−2p = -2.
p−3=(−2)−3=1(−2)3=−18p^{-3} = (-2)^{-3} = \frac{1}{(-2)^3} = -\frac{1}{8}
A negative exponent represents the reciprocal, and a negative base raised to an odd power yields a negative result.
4
Substitute the evaluated terms back into the original algebraic expression.
19−2⋅(−18)\frac{1}{9} - 2 \cdot \left(-\frac{1}{8}\right)
Replace each variable expression with its calculated numerical value.
5
Perform the multiplication before subtraction following the order of operations.
2⋅(−18)=−28=−142 \cdot \left(-\frac{1}{8}\right) = -\frac{2}{8} = -\frac{1}{4}
The order of operations (PEMDAS/GEMS) dictates that multiplication must be performed before subtraction.
6
Subtract the product from the first term.
19−(−14)=19+14=436+936=1336\frac{1}{9} - \left(-\frac{1}{4}\right) = \frac{1}{9} + \frac{1}{4} = \frac{4}{36} + \frac{9}{36} = \frac{13}{36}
Subtracting a negative value is equivalent to addition. Find a common denominator to add the fractions.

Key Concept

Evaluating algebraic expressions involving negative bases, negative exponents, fractional exponents, and the order of operations.
Question 3Question →

If a=−27a = -27, b=−14b = -\frac{1}{4}, and c=−2c = -2, what is the value of the algebraic expression a−4/3−b−2⋅c−5a^{-4/3} - b^{-2} \cdot c^{-5}?

Show answer & explanation

Answer: 83162\frac{83}{162}

Answer

The correct value of the algebraic expression is 83162\frac{83}{162}.
Evaluating each term of the expression with the given values yields: a−4/3=181a^{-4/3} = \frac{1}{81}, b−2=16b^{-2} = 16, and c−5=−132c^{-5} = -\frac{1}{32}. Applying the order of operations, we multiply 1616 by −132-\frac{1}{32} first to obtain −12-\frac{1}{2}. We then subtract this product from 181\frac{1}{81}, which simplifies to 181+12=83162\frac{1}{81} + \frac{1}{2} = \frac{83}{162}.

Step-by-Step Solution

1
Substitute the given values of aa, bb, and cc into the algebraic expression.
The expression is written as (−27)−4/3−(−14)−2⋅(−2)−5(-27)^{-4/3} - (-\frac{1}{4})^{-2} \cdot (-2)^{-5}.
This establishes the numerical expression to be evaluated.
2
Evaluate the first term, (−27)−4/3(-27)^{-4/3}.
(−27)−4/3=1(−27)4/3=1((−27)1/3)4=1(−3)4=181(-27)^{-4/3} = \frac{1}{(-27)^{4/3}} = \frac{1}{((-27)^{1/3})^4} = \frac{1}{(-3)^4} = \frac{1}{81}.
A negative exponent indicates a reciprocal, and a fractional exponent of 4/34/3 indicates taking the cube root and then raising to the fourth power.
3
Evaluate the second term, (−14)−2(-\frac{1}{4})^{-2}.
(−14)−2=(−4)2=16(-\frac{1}{4})^{-2} = (-4)^2 = 16.
Raising a fraction to a negative integer power is equivalent to raising its reciprocal to the corresponding positive integer power.
4
Evaluate the third term, (−2)−5(-2)^{-5}.
(−2)−5=1(−2)5=−132(-2)^{-5} = \frac{1}{(-2)^5} = -\frac{1}{32}.
Evaluating a negative base raised to an odd negative power results in a negative unit fraction.
5
Substitute the evaluated terms back into the original expression and apply the order of operations.
E=181−16⋅(−132)=181−(−1632)=181+12=2+81162=83162E = \frac{1}{81} - 16 \cdot \left(-\frac{1}{32}\right) = \frac{1}{81} - \left(-\frac{16}{32}\right) = \frac{1}{81} + \frac{1}{2} = \frac{2 + 81}{162} = \frac{83}{162}.
Multiplication must be performed before subtraction according to standard mathematical order of operations.

Key Concept

Evaluating expressions containing multiple variables with fractional exponents, negative bases, and standard order of operations.
Estimated Time:2m 0s
Question 4Question →

Evaluate the algebraic expression for the given variable values.

Fill in the blanks below

If x=−3x = -3 and y=4y = 4, the value of the expression 2x2−3y2x^2 - 3y is .
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Answer

6
Substituting x=−3x = -3 into x2x^2 gives (−3)2=9(-3)^2 = 9. Multiplying by 22 yields 1818. Substituting y=4y = 4 into 3y3y gives 1212. Subtracting 1212 from 1818 gives the correct value of 66.

Step-by-Step Solution

1
Substitute the given values x=−3x = -3 and y=4y = 4 into the expression 2x2−3y2x^2 - 3y.
2(−3)2−3(4)2(-3)^2 - 3(4)
Replace each variable with its designated numeric value.
2
Evaluate the exponent (−3)2(-3)^2.
2(9)−3(4)2(9) - 3(4)
Following the order of operations (PEMDAS), exponents are evaluated before multiplication.
3
Perform the multiplication operations.
18−1218 - 12
Multiply 2×9=182 \times 9 = 18 and 3×4=123 \times 4 = 12.
4
Subtract the terms to find the final value.
66
Perform final subtraction: 18−12=618 - 12 = 6.

Key Concept

Evaluating Algebraic Expressions with Negative Values
Estimated Time:45s
Question 5Question →

If a=−4a = -4 and b=3b = 3, what is the value of the expression 3a2−2ab3a^2 - 2ab?

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Answer: 72

Answer

72
Substituting a=−4a = -4 and b=3b = 3 yields 3(−4)2−2(−4)(3)=3(16)−(−24)=48+24=723(-4)^2 - 2(-4)(3) = 3(16) - (-24) = 48 + 24 = 72.

Step-by-Step Solution

1
Substitute the given values into the expression
Replace aa with −4-4 and bb with 33 in 3a2−2ab3a^2 - 2ab to get 3(−4)2−2(−4)(3)3(-4)^2 - 2(-4)(3).
Direct substitution of known variable values.
2
Evaluate the exponent
(−4)2=16(-4)^2 = 16, so the term 3(−4)23(-4)^2 becomes 3(16)=483(16) = 48.
Exponents must be evaluated before multiplication according to PEMDAS.
3
Evaluate the second multiplication term
2ab=2(−4)(3)=−242ab = 2(-4)(3) = -24.
Multiply the numerical factors together.
4
Subtract the terms to find the final value
48−(−24)=48+24=7248 - (-24) = 48 + 24 = 72.
Subtracting a negative number is equivalent to adding its positive value.

Key Concept

Evaluating Algebraic Expressions with Negative Values
Estimated Time:45s
Question 6Question →

If a=−4a = -4 and b=7b = 7, what is the value of the algebraic expression (a+b)2−3a(a + b)^2 - 3a?

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Answer: 21

Answer

21
Substituting a=−4a = -4 and b=7b = 7 into (a+b)2−3a(a + b)^2 - 3a gives ((−4)+7)2−3(−4)=(3)2+12=9+12=21((-4) + 7)^2 - 3(-4) = (3)^2 + 12 = 9 + 12 = 21.

Step-by-Step Solution

1
Substitute the given values into the expression
((−4)+7)2−3(−4)((-4) + 7)^2 - 3(-4)
Replace aa with −4-4 and bb with 77.
2
Simplify inside the parentheses
(3)2−3(−4)(3)^2 - 3(-4)
Following order of operations (PEMDAS), simplify −4+7=3-4 + 7 = 3 first.
3
Evaluate the exponent and multiplication
9+129 + 12
Square 33 to get 99, and multiply −3-3 by −4-4 to get +12+12.
4
Add the terms together
21
Combine 99 and 1212.

Key Concept

Evaluating Algebraic Expressions
Question 7Question →

If x=−2x = -2 and y=5y = 5, what is the value of the algebraic expression x3+4yx+y\frac{x^3 + 4y}{x + y}?

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Answer: 4

Answer

The value of the expression is 4.
Substituting x=−2x = -2 and y=5y = 5 yields (−2)3+4(5)=−8+20=12(-2)^3 + 4(5) = -8 + 20 = 12 in the numerator, and −2+5=3-2 + 5 = 3 in the denominator. Dividing 1212 by 33 gives 44.

Step-by-Step Solution

1
Substitute the given numerical values into the algebraic expression.
(−2)3+4(5)(−2)+5\frac{(-2)^3 + 4(5)}{(-2) + 5}
Replace each occurrence of xx with −2-2 and yy with 55.
2
Evaluate the numerator using order of operations.
−8+20=12-8 + 20 = 12
Calculate (−2)3=−8(-2)^3 = -8 and 4(5)=204(5) = 20, then add the terms together.
3
Evaluate the denominator.
−2+5=3-2 + 5 = 3
Add −2-2 and 55.
4
Divide the numerator by the denominator.
123=4\frac{12}{3} = 4
Simplify the fraction to get the final integer answer.

Key Concept

Evaluating algebraic expressions requires substituting specific numerical values into the expression and carefully applying the order of operations, especially when handling negative numbers raised to powers.
Question 8Question →

Evaluate the algebraic expression for the given variable values.

Fill in the blanks below

When p=−3p = -3 and q=4q = 4, the value of the expression 2p2−5q+12p^2 - 5q + 1 is .
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Answer

The value of the expression is -1.
Substituting p=−3p = -3 and q=4q = 4 into 2p2−5q+12p^2 - 5q + 1 yields 2(−3)2−5(4)+1=2(9)−20+1=18−20+1=−12(-3)^2 - 5(4) + 1 = 2(9) - 20 + 1 = 18 - 20 + 1 = -1.

Step-by-Step Solution

1
Substitute the given values p=−3p = -3 and q=4q = 4 into the expression 2p2−5q+12p^2 - 5q + 1.
2(−3)2−5(4)+12(-3)^2 - 5(4) + 1
Replace each variable with its assigned value.
2
Evaluate the exponent (−3)2(-3)^2.
2(9)−5(4)+12(9) - 5(4) + 1
Exponents must be evaluated before multiplication according to the order of operations.
3
Perform the multiplication operations.
18 - 20 + 1
2×9=182 \times 9 = 18 and 5×4=205 \times 4 = 20.
4
Add and subtract from left to right.
-1
18−20=−218 - 20 = -2, and −2+1=−1-2 + 1 = -1.

Key Concept

Evaluating Algebraic Expressions
Estimated Time:45s
Question 9Question →

If x=−2x = -2 and y=3y = 3, what is the value of the algebraic expression 3x2−y2+4x3x^2 - y^2 + 4x?

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Answer: −5-5

Answer

−5-5
Substituting x=−2x = -2 and y=3y = 3 gives 3(−2)2−(3)2+4(−2)3(-2)^2 - (3)^2 + 4(-2). Squaring −2-2 yields 44, so the first term becomes 3(4)=123(4) = 12. Squaring 33 yields 99, so the second term becomes −9-9. Multiplying 44 by −2-2 yields −8-8. Summing these values gives 12−9−8=−512 - 9 - 8 = -5.

Step-by-Step Solution

1
Substitute the given values x=−2x = -2 and y=3y = 3 into the expression 3x2−y2+4x3x^2 - y^2 + 4x.
3(−2)2−(3)2+4(−2)3(-2)^2 - (3)^2 + 4(-2)
Replace each variable with its respective numeric value.
2
Evaluate the exponential terms according to the order of operations (PEMDAS).
3(4)−9+4(−2)3(4) - 9 + 4(-2)
Squaring a negative number yields a positive result: (−2)2=4(-2)^2 = 4, and (3)2=9(3)^2 = 9.
3
Perform multiplication operations.
12−9−812 - 9 - 8
3⋅4=123 \cdot 4 = 12 and 4⋅(−2)=−84 \cdot (-2) = -8.
4
Add and subtract from left to right.
−5-5
12−9=312 - 9 = 3, and 3−8=−53 - 8 = -5.

Key Concept

Evaluating Algebraic Expressions with Signed Numbers
Question 10Question →

If x=−2x = -2, y=12y = \frac{1}{2}, and z=−3z = -3, what is the value of the algebraic expression x3y−2−4z(x+2y)2−z\frac{x^3 y^{-2} - 4z}{(x + 2y)^2 - z}?

Show answer & explanation

Answer: −5-5

Answer

The value of the expression is −5-5.
Substituting x=−2x = -2, y=12y = \frac{1}{2}, and z=−3z = -3 yields a numerator of (−2)3(12)−2−4(−3)=(−8)(4)+12=−20(-2)^3 \left(\frac{1}{2}\right)^{-2} - 4(-3) = (-8)(4) + 12 = -20, and a denominator of (−2+2(12))2−(−3)=(−1)2+3=4\left(-2 + 2\left(\frac{1}{2}\right)\right)^2 - (-3) = (-1)^2 + 3 = 4. Dividing −20-20 by 44 gives the final answer −5-5.

Step-by-Step Solution

1
Evaluate the terms in the numerator: x3y−2−4zx^3 y^{-2} - 4z
Since x=−2x = -2, x3=(−2)3=−8x^3 = (-2)^3 = -8. Since y=12y = \frac{1}{2}, y−2=(12)−2=22=4y^{-2} = \left(\frac{1}{2}\right)^{-2} = 2^2 = 4. So x3y−2=(−8)(4)=−32x^3 y^{-2} = (-8)(4) = -32. Also, −4z=−4(−3)=12-4z = -4(-3) = 12. The numerator simplifies to −32+12=−20-32 + 12 = -20.
Negative exponents indicate reciprocals, and cubing a negative base yields a negative result.
2
Evaluate the terms in the denominator: (x+2y)2−z(x + 2y)^2 - z
Inside the parentheses, x+2y=−2+2(12)=−2+1=−1x + 2y = -2 + 2\left(\frac{1}{2}\right) = -2 + 1 = -1. Squaring this gives (−1)2=1(-1)^2 = 1. Subtracting zz gives 1−(−3)=1+3=41 - (-3) = 1 + 3 = 4.
Operations inside parentheses must be calculated before applying the exponent, and subtracting a negative integer is equivalent to adding its positive.
3
Divide the numerator by the denominator
−204=−5.\frac{-20}{4} = -5.
Dividing a negative integer by a positive integer produces a negative quotient.

Key Concept

Evaluating algebraic expressions involving negative exponents, integer substitutions, and order of operations.
Estimated Time:1m 30s
Question 11Question →

If x=−3x = -3, y=−13y = -\frac{1}{3}, and z=16z = 16, what is the value of the algebraic expression x3y−2+3z1/2(xy+2)3−32z1/2\frac{x^3 y^{-2} + 3 z^{1/2}}{(xy + 2)^3 - \frac{3}{2} z^{1/2}}?

Show answer & explanation

Answer: -11

Answer

The evaluated value of the expression is -11.
Substituting the given values into the numerator yields (−3)3(−13)−2+3(16)1/2=(−27)(9)+3(4)=−243+12=−231(-3)^3 \left(-\frac{1}{3}\right)^{-2} + 3(16)^{1/2} = (-27)(9) + 3(4) = -243 + 12 = -231. Substituting into the denominator yields ((−3)(−13)+2)3−32(16)1/2=(1+2)3−32(4)=27−6=21\left((-3)\left(-\frac{1}{3}\right) + 2\right)^3 - \frac{3}{2}(16)^{1/2} = (1 + 2)^3 - \frac{3}{2}(4) = 27 - 6 = 21. Dividing −231-231 by 2121 produces −11-11.

Step-by-Step Solution

1
Evaluate the terms in the numerator
Numerator = -231
Calculate x3=(−3)3=−27x^3 = (-3)^3 = -27, y−2=(−13)−2=9y^{-2} = \left(-\frac{1}{3}\right)^{-2} = 9, giving x3y−2=−243x^3 y^{-2} = -243. Then calculate 3z1/2=3(4)=123 z^{1/2} = 3(4) = 12. Adding these gives −243+12=−231-243 + 12 = -231.
2
Evaluate the terms in the denominator
Denominator = 21
Calculate xy=(−3)(−13)=1xy = (-3)\left(-\frac{1}{3}\right) = 1, so (xy+2)3=(1+2)3=27(xy + 2)^3 = (1 + 2)^3 = 27. Then calculate 32z1/2=32(4)=6\frac{3}{2} z^{1/2} = \frac{3}{2}(4) = 6. Subtracting gives 27−6=2127 - 6 = 21.
3
Divide the evaluated numerator by the evaluated denominator
-11
−23121=−11\frac{-231}{21} = -11.

Key Concept

Evaluating Algebraic Expressions with Negative and Fractional Exponents
Question 12Question →

If p=−3p = -3, q=−12q = -\frac{1}{2}, and r=8r = 8, what is the value of the algebraic expression below?

p2q−3−r4/3(pq−12)2−r1/3\frac{p^2 q^{-3} - r^{4/3}}{\left(pq - \frac{1}{2}\right)^2 - r^{1/3}}

Fill in the blanks below

The value of the expression is .
Show answer & explanation

Answer

88
Evaluating each component using exponent rules and standard order of operations yields −72−16=−88-72 - 16 = -88 in the numerator and 1−2=−11 - 2 = -1 in the denominator. Dividing −88-88 by −1-1 results in the final value of 8888.

Step-by-Step Solution

1
Evaluate the terms in the numerator containing negative and rational exponents.
p2q−3=(−3)2(−12)−3=9⋅(−8)=−72p^2 q^{-3} = (-3)^2 \left(-\frac{1}{2}\right)^{-3} = 9 \cdot (-8) = -72 and r4/3=84/3=(81/3)4=24=16r^{4/3} = 8^{4/3} = (8^{1/3})^4 = 2^4 = 16.
Apply exponent rules for negative bases with integer and fractional powers: a−n=1ana^{-n} = \frac{1}{a^n} and am/n=(an)ma^{m/n} = (\sqrt[n]{a})^m.
2
Subtract the evaluated terms to determine the total numerator value.
\text{Numerator} = -72 - 16 = -88.
Combine the evaluated terms according to the numerator expression p2q−3−r4/3p^2 q^{-3} - r^{4/3}.
3
Evaluate the grouped and exponential terms in the denominator.
pq−12=(−3)(−12)−12=32−12=1pq - \frac{1}{2} = (-3)\left(-\frac{1}{2}\right) - \frac{1}{2} = \frac{3}{2} - \frac{1}{2} = 1. Then (pq−12)2=12=1\left(pq - \frac{1}{2}\right)^2 = 1^2 = 1, and r1/3=81/3=2r^{1/3} = 8^{1/3} = 2.
Follow the order of operations by simplifying inside the parentheses first, then applying the exponent.
4
Calculate the denominator and divide the numerator by the denominator.
\text{Denominator} = 1 - 2 = -1 .Dividingthenumeratorbythedenominatorgives. Dividing the numerator by the denominator gives \frac{-88}{-1} = 88$.
Perform the final subtraction in the denominator and divide to simplify the fraction fully.

Key Concept

Evaluating Algebraic Expressions with Negative and Fractional Exponents
Question 13Question →

If r=−3r = -3, s=2s = 2, and t=−12t = -\frac{1}{2}, what is the value of the algebraic expression r2s−t−2r+st\frac{r^2 s - t^{-2}}{r + s t}?

Show answer & explanation

Answer: −72-\frac{7}{2}

Answer

−72-\frac{7}{2}
Substituting r=−3r = -3, s=2s = 2, and t=−12t = -\frac{1}{2} into the numerator yields (−3)2(2)−(−12)−2=(9)(2)−4=18−4=14(-3)^2(2) - \left(-\frac{1}{2}\right)^{-2} = (9)(2) - 4 = 18 - 4 = 14. Substituting into the denominator yields −3+2(−12)=−3−1=−4-3 + 2\left(-\frac{1}{2}\right) = -3 - 1 = -4. Dividing numerator by denominator gives 14−4=−72\frac{14}{-4} = -\frac{7}{2}.

Step-by-Step Solution

1
Substitute the given variable values into the numerator expression r2s−t−2r^2 s - t^{-2}.
Numerator = (−3)2(2)−(−12)−2(-3)^2 (2) - \left(-\frac{1}{2}\right)^{-2}
Ensure negative values are enclosed in parentheses during substitution.
2
Evaluate the terms in the numerator following order of operations and exponent rules.
(−3)2=9(-3)^2 = 9, so 9⋅2=189 \cdot 2 = 18. Also (−12)−2=(−2)2=4\left(-\frac{1}{2}\right)^{-2} = (-2)^2 = 4. Thus, Numerator = 18−4=1418 - 4 = 14.
A negative base squared yields a positive value, and a negative exponent takes the reciprocal of the base.
3
Substitute values into the denominator expression r+str + s t and simplify.
Denominator = −3+(2)(−12)=−3+(−1)=−4-3 + (2)\left(-\frac{1}{2}\right) = -3 + (-1) = -4
Multiply ss and tt first before adding to rr according to PEMDAS.
4
Divide the evaluated numerator by the evaluated denominator.
14−4=−72\frac{14}{-4} = -\frac{7}{2}
Simplify the fraction by dividing both numerator and denominator by 2.

Key Concept

Evaluating Algebraic Expressions with Negative Bases and Negative Exponents
Estimated Time:1m 15s
Question 14Question →
If m=−2m = -2, n=13n = \frac{1}{3}, and p=−4p = -4, what is the value of the algebraic expression below?
m3n−2−−p⋅m2m2−n−1+p\frac{m^3 n^{-2} - \sqrt{-p \cdot m^2}}{m^2 - n^{-1} + p}
Show answer & explanation

Answer: 763\frac{76}{3}

Answer

763\frac{76}{3}
Substituting m=−2m = -2, n=13n = \frac{1}{3}, and p=−4p = -4 into the numerator gives (−2)3(13)−2−−(−4)(−2)2=−8(9)−16=−72−4=−76(-2)^3 \left(\frac{1}{3}\right)^{-2} - \sqrt{-(-4)(-2)^2} = -8(9) - \sqrt{16} = -72 - 4 = -76. Substituting into the denominator gives (−2)2−(13)−1+(−4)=4−3−4=−3(-2)^2 - \left(\frac{1}{3}\right)^{-1} + (-4) = 4 - 3 - 4 = -3. Dividing −76-76 by −3-3 gives 763\frac{76}{3}.

Step-by-Step Solution

1
Evaluate the terms in the numerator
m3=(−2)3=−8m^3 = (-2)^3 = -8, n−2=(13)−2=9n^{-2} = \left(\frac{1}{3}\right)^{-2} = 9, and −p⋅m2=−(−4)⋅(−2)2=4⋅4=4\sqrt{-p \cdot m^2} = \sqrt{-(-4) \cdot (-2)^2} = \sqrt{4 \cdot 4} = 4. Thus, Numerator =(−8)(9)−4=−72−4=−76= (-8)(9) - 4 = -72 - 4 = -76.
Simplify each term in the numerator using proper exponent rules and sign conventions.
2
Evaluate the terms in the denominator
m2=(−2)2=4m^2 = (-2)^2 = 4, n−1=(13)−1=3n^{-1} = \left(\frac{1}{3}\right)^{-1} = 3, and p=−4p = -4. Thus, Denominator =4−3+(−4)=1−4=−3= 4 - 3 + (-4) = 1 - 4 = -3.
Substitute the variable values into the denominator expression and evaluate left to right.
3
Divide the numerator by the denominator
−76−3=763.\frac{-76}{-3} = \frac{76}{3}.
Dividing two negative numbers yields a positive quotient.

Key Concept

Evaluating algebraic expressions with negative bases, fractional exponents, and order of operations
Estimated Time:2m 0s
Question 15Question →
If p=−3p = -3, q=12q = \frac{1}{2}, and r=−27r = -27, what is the value of the algebraic expression below?
p2q−3+r1/3(2pq+5)2+q−1\frac{p^2 q^{-3} + r^{1/3}}{(2pq + 5)^2 + q^{-1}}
Show answer & explanation

Answer: 11.5

Answer

11.5
Evaluating each component of the expression step-by-step using order of operations:
1. Numerator: p2q−3+r1/3=(−3)2(12)−3+(−27)1/3=9⋅8+(−3)=72−3=69p^2 q^{-3} + r^{1/3} = (-3)^2 \left(\frac{1}{2}\right)^{-3} + (-27)^{1/3} = 9 \cdot 8 + (-3) = 72 - 3 = 69.
2. Denominator: (2pq+5)2+q−1=(2(−3)(12)+5)2+(12)−1=(−3+5)2+2=22+2=4+2=6(2pq + 5)^2 + q^{-1} = \left(2(-3)\left(\frac{1}{2}\right) + 5\right)^2 + \left(\frac{1}{2}\right)^{-1} = (-3 + 5)^2 + 2 = 2^2 + 2 = 4 + 2 = 6.
3. Final Quotient: 696=11.5\frac{69}{6} = 11.5.

Step-by-Step Solution

1
Evaluate the terms in the numerator
Numerator = 69
Since p=−3p = -3, p2=9p^2 = 9. With q=12q = \frac{1}{2}, q−3=23=8q^{-3} = 2^3 = 8. Thus, p2q−3=9⋅8=72p^2 q^{-3} = 9 \cdot 8 = 72. Also, r1/3=(−27)1/3=−3r^{1/3} = (-27)^{1/3} = -3. Adding these values gives 72+(−3)=6972 + (-3) = 69.
2
Evaluate the terms in the denominator
Denominator = 6
First, 2pq=2(−3)(12)=−32pq = 2(-3)\left(\frac{1}{2}\right) = -3. Then (2pq+5)2=(−3+5)2=22=4(2pq + 5)^2 = (-3 + 5)^2 = 2^2 = 4. Next, q−1=(12)−1=2q^{-1} = \left(\frac{1}{2}\right)^{-1} = 2. Adding these components gives 4+2=64 + 2 = 6.
3
Divide the numerator by the denominator
11.5
Dividing the numerator (69) by the denominator (6) yields 696=11.5\frac{69}{6} = 11.5 (or 232\frac{23}{2}).

Key Concept

Evaluating algebraic expressions with negative bases, negative exponents, fractional exponents, and order of operations.
Question 16Question →

An engineering formula used to calculate a structural load index is given by L=a3b−2−c3/4a2+12bL = \frac{a^3 b^{-2} - c^{3/4}}{a^2 + 12b}. What is the value of LL when a=−2a = -2, b=13b = \frac{1}{3}, and c=16c = 16?

Show answer & explanation

Answer: -10

Answer

The value of the expression is -10.
Substituting the values into the formula gives L=(−2)3⋅(1/3)−2−163/4(−2)2+12(1/3)=−8⋅9−84+4=−72−88=−808=−10L = \frac{(-2)^3 \cdot (1/3)^{-2} - 16^{3/4}}{(-2)^2 + 12(1/3)} = \frac{-8 \cdot 9 - 8}{4 + 4} = \frac{-72 - 8}{8} = \frac{-80}{8} = -10.

Step-by-Step Solution

1
Evaluate the terms in the numerator containing powers and negative exponents
a3=−8a^3 = -8, b−2=9b^{-2} = 9, and c3/4=8c^{3/4} = 8
Negative bases raised to odd powers remain negative: (−2)3=−8(-2)^3 = -8. A negative exponent represents the reciprocal raised to a positive exponent: (1/3)−2=32=9(1/3)^{-2} = 3^2 = 9. Fractional exponent c3/4=(164)3=23=8c^{3/4} = (\sqrt[4]{16})^3 = 2^3 = 8.
2
Compute the full numerator
−72−8=−80-72 - 8 = -80
Multiply a3a^3 and b−2b^{-2} to get (−8)(9)=−72(-8)(9) = -72, then subtract c3/4=8c^{3/4} = 8.
3
Evaluate the denominator
(−2)2+12(13)=4+4=8(-2)^2 + 12\left(\frac{1}{3}\right) = 4 + 4 = 8
Squaring a negative number yields a positive value: (−2)2=4(-2)^2 = 4. Multiplying 12⋅13=412 \cdot \frac{1}{3} = 4.
4
Divide the numerator by the denominator
−808=−10\frac{-80}{8} = -10
Dividing a negative integer by a positive integer yields a negative result.

Key Concept

Evaluating algebraic expressions involving negative numbers, negative exponents, and rational exponents
Estimated Time:1m 15s
Question 17Question →
An algebraic function F(a,b)F(a, b) is defined as:
F(a,b)=a2−3bab−1+2F(a, b) = \frac{a^2 - 3b}{a b^{-1} + 2}

The values for variables aa and bb are given in the table below:

VariableValue
aa−4-4
bb−12-\frac{1}{2}

What is the value of F(a,b)F(a, b) for these given values?

Show answer & explanation

Answer: 74\frac{7}{4}

Answer

The value of F(a,b)F(a, b) is 74\frac{7}{4}.
Substituting a=−4a = -4 and b=−1/2b = -1/2 into the expression gives a numerator of (−4)2−3(−1/2)=16+3/2=35/2(-4)^2 - 3(-1/2) = 16 + 3/2 = 35/2 and a denominator of (−4)(−2)+2=8+2=10(-4)(-2) + 2 = 8 + 2 = 10. Dividing 35/235/2 by 1010 simplifies to 35/2035/20, which equals 7/47/4.

Step-by-Step Solution

1
Evaluate the reciprocal term b−1b^{-1}
b−1=(−12)−1=−2b^{-1} = \left(-\frac{1}{2}\right)^{-1} = -2
A negative exponent indicates taking the reciprocal of the base.
2
Evaluate the numerator a2−3ba^2 - 3b
(-4)^2 - 3\left(-\frac{1}{2}\right) = 16 + \frac{3}{2} = \frac{35}{2}
Squaring a negative number yields a positive value ((−4)2=16(-4)^2 = 16), and subtracting a negative value is equivalent to addition.
3
Evaluate the denominator ab−1+2a b^{-1} + 2
(-4)(-2) + 2 = 8 + 2 = 10
Multiplying two negative numbers yields a positive product.
4
Divide the numerator by the denominator and simplify
35210=3520=74\frac{\frac{35}{2}}{10} = \frac{35}{20} = \frac{7}{4}
Dividing 352\frac{35}{2} by 1010 gives 3520\frac{35}{20}, which simplifies to 74\frac{7}{4} when dividing both numerator and denominator by 55.

Key Concept

Evaluating algebraic expressions with negative numbers, fractions, and negative exponents.
Estimated Time:1m 30s
Question 18Question →

If u=−3u = -3 and v=−12v = -\frac{1}{2}, what is the value of the algebraic expression below? Enter your numerical answer in the blank.

Fill in the blanks below

The value of the expression 4uv2−u22v−u\frac{4uv^2 - u^2}{2v - u} is .
Show answer & explanation

Answer

The value of the expression when evaluated at u=−3u = -3 and v=−12v = -\frac{1}{2} is −6-6.
Substituting u=−3u = -3 and v=−12v = -\frac{1}{2} into the numerator gives 4(−3)(−12)2−(−3)2=−3−9=−124(-3)(-\frac{1}{2})^2 - (-3)^2 = -3 - 9 = -12. Substituting into the denominator gives 2(−12)−(−3)=−1+3=22(-\frac{1}{2}) - (-3) = -1 + 3 = 2. Dividing −12-12 by 22 results in −6-6.

Step-by-Step Solution

1
Evaluate the terms containing exponents in the numerator.
v2=(−12)2=14v^2 = \left(-\frac{1}{2}\right)^2 = \frac{1}{4} and u2=(−3)2=9u^2 = (-3)^2 = 9.
Exponents must be calculated before multiplication according to the order of operations.
2
Evaluate the numerator 4uv2−u24uv^2 - u^2.
4(−3)(14)−9=−3−9=−124(-3)\left(\frac{1}{4}\right) - 9 = -3 - 9 = -12.
Multiply 4×(−3)×144 \times (-3) \times \frac{1}{4} to get −3-3, then subtract 99.
3
Evaluate the denominator 2v−u2v - u.
2(−12)−(−3)=−1+3=22\left(-\frac{1}{2}\right) - (-3) = -1 + 3 = 2.
Multiplying 22 by −12-\frac{1}{2} gives −1-1, and subtracting negative 33 is equivalent to adding 33.
4
Divide the numerator by the denominator.
−122=−6\frac{-12}{2} = -6.
Dividing −12-12 by 22 yields the final result −6-6.

Key Concept

Evaluating algebraic expressions involving negative numbers, fractions, and order of operations.
Estimated Time:1m 30s
Question 19Question →

If a=−3a = -3 and b=−2b = -2, what is the value of the expression 2a2−b3(a−b)2+ab\frac{2a^2 - b^3}{(a - b)^2 + ab}?

Show answer & explanation

Answer: 267\frac{26}{7}

Answer

267\frac{26}{7}
Substituting a=−3a = -3 and b=−2b = -2 directly into the numerator yields 2(−3)2−(−2)3=2(9)−(−8)=262(-3)^2 - (-2)^3 = 2(9) - (-8) = 26. Substituting into the denominator gives ((−3)−(−2))2+(−3)(−2)=(−1)2+6=7((-3) - (-2))^2 + (-3)(-2) = (-1)^2 + 6 = 7. Combining these results gives 267\frac{26}{7}.

Step-by-Step Solution

1
Substitute a=−3a = -3 and b=−2b = -2 into the numerator 2a2−b32a^2 - b^3.
2(−3)2−(−2)3=2(9)−(−8)=18+8=262(-3)^2 - (-2)^3 = 2(9) - (-8) = 18 + 8 = 26
Squaring −3-3 gives 99, and cubing −2-2 gives −8-8. Subtracting −8-8 is equivalent to adding 88.
2
Substitute a=−3a = -3 and b=−2b = -2 into the denominator (a−b)2+ab(a - b)^2 + ab.
((−3)−(−2))2+(−3)(−2)=(−3+2)2+6=(−1)2+6=1+6=7((-3) - (-2))^2 + (-3)(-2) = (-3 + 2)^2 + 6 = (-1)^2 + 6 = 1 + 6 = 7
Subtracting a negative number becomes addition, and squaring −1-1 yields 11.
3
Form the fraction by dividing the numerator by the denominator.
267\frac{26}{7}
The evaluated numerator is 2626 and the evaluated denominator is 77.

Key Concept

Order of operations and handling negative signs when evaluating algebraic expressions involving powers and parentheses.
Estimated Time:1m 0s
Question 20Question →

If x=−3x = -3, y=−12y = -\frac{1}{2}, and z=8z = 8, what is the value of the algebraic expression x2y−1+z2/32xy−1\frac{x^2 y^{-1} + z^{2/3}}{2xy - 1}?

Show answer & explanation

Answer: -7

Answer

The value of the expression is -7.
Substituting the given values into the expression requires careful application of exponent rules and order of operations. First, (−3)2=9(-3)^2 = 9 and (−12)−1=−2(-\frac{1}{2})^{-1} = -2, so x2y−1=9×(−2)=−18x^2 y^{-1} = 9 \times (-2) = -18. Second, 82/3=(83)2=22=48^{2/3} = (\sqrt[3]{8})^2 = 2^2 = 4. This makes the numerator −18+4=−14-18 + 4 = -14. The denominator evaluates to 2(−3)(−12)−1=3−1=22(-3)(-\frac{1}{2}) - 1 = 3 - 1 = 2. Dividing −14-14 by 22 gives the correct answer of −7-7.

Step-by-Step Solution

1
Substitute x=−3x = -3 and y=−12y = -\frac{1}{2} into x2y−1x^2 y^{-1}
(−3)2⋅(−2)=−18(-3)^2 \cdot (-2) = -18
Squaring −3-3 yields 99, and taking the reciprocal of −12-\frac{1}{2} yields −2-2.
2
Substitute z=8z = 8 into z2/3z^{2/3}
82/3=48^{2/3} = 4
Taking the cube root of 88 gives 22, and squaring 22 gives 44.
3
Evaluate the numerator
−18+4=−14-18 + 4 = -14
Adding the evaluated terms together.
4
Substitute values into the denominator 2xy−12xy - 1
2(−3)(−12)−1=3−1=22(-3)\left(-\frac{1}{2}\right) - 1 = 3 - 1 = 2
Multiplying 22, −3-3, and −12-\frac{1}{2} produces 33, then subtracting 11 yields 22.
5
Divide the numerator by the denominator
−142=−7\frac{-14}{2} = -7
Simplifying the rational expression yields the final numeric answer.

Key Concept

Evaluating algebraic expressions with negative bases, rational exponents, and negative exponents
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