Examkin

Evaluating Algebraic Expressions

29 soru

Tüm türler29Çoktan seçmeli15Sayısal7Boşluk doldurma7
Soru 1Soru →

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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Cevap: -4

Cevap

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.

Adım Adım Çözüm

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.

Anahtar Kavram

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

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.

Aşağıdaki boşlukları doldurun

The value of the expression is .
Cevabı ve açıklamayı göster

Cevap

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}.

Adım Adım Çözüm

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.

Anahtar Kavram

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

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}?

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Cevap: 83162\frac{83}{162}

Cevap

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}.

Adım Adım Çözüm

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.

Anahtar Kavram

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

Evaluate the algebraic expression for the given variable values.

Aşağıdaki boşlukları doldurun

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

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.

Adım Adım Çözüm

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.

Anahtar Kavram

Evaluating Algebraic Expressions with Negative Values
Tahmini Süre:45s
Soru 5Soru →

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

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Cevap: 72

Cevap

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.

Adım Adım Çözüm

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.

Anahtar Kavram

Evaluating Algebraic Expressions with Negative Values
Tahmini Süre:45s
Soru 6Soru →

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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Cevap: 21

Cevap

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.

Adım Adım Çözüm

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.

Anahtar Kavram

Evaluating Algebraic Expressions
Soru 7Soru →

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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Cevap: 4

Cevap

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.

Adım Adım Çözüm

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.

Anahtar Kavram

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.
Soru 8Soru →

Evaluate the algebraic expression for the given variable values.

Aşağıdaki boşlukları doldurun

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

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.

Adım Adım Çözüm

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.

Anahtar Kavram

Evaluating Algebraic Expressions
Tahmini Süre:45s
Soru 9Soru →

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

Cevabı ve açıklamayı göster

Cevap: −5-5

Cevap

−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.

Adım Adım Çözüm

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.

Anahtar Kavram

Evaluating Algebraic Expressions with Signed Numbers
Soru 10Soru →

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}?

Cevabı ve açıklamayı göster

Cevap: −5-5

Cevap

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.

Adım Adım Çözüm

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.

Anahtar Kavram

Evaluating algebraic expressions involving negative exponents, integer substitutions, and order of operations.
Tahmini Süre:1m 30s
Soru 11Soru →

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}}?

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Cevap: -11

Cevap

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.

Adım Adım Çözüm

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.

Anahtar Kavram

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

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}}

Aşağıdaki boşlukları doldurun

The value of the expression is .
Cevabı ve açıklamayı göster

Cevap

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.

Adım Adım Çözüm

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.

Anahtar Kavram

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

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}?

Cevabı ve açıklamayı göster

Cevap: −72-\frac{7}{2}

Cevap

−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}.

Adım Adım Çözüm

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.

Anahtar Kavram

Evaluating Algebraic Expressions with Negative Bases and Negative Exponents
Tahmini Süre:1m 15s
Soru 14Soru →
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}
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Cevap: 763\frac{76}{3}

Cevap

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}.

Adım Adım Çözüm

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.

Anahtar Kavram

Evaluating algebraic expressions with negative bases, fractional exponents, and order of operations
Tahmini Süre:2m 0s
Soru 15Soru →
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}}
Cevabı ve açıklamayı göster

Cevap: 11.5

Cevap

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.

Adım Adım Çözüm

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}).

Anahtar Kavram

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

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?

Cevabı ve açıklamayı göster

Cevap: -10

Cevap

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.

Adım Adım Çözüm

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.

Anahtar Kavram

Evaluating algebraic expressions involving negative numbers, negative exponents, and rational exponents
Tahmini Süre:1m 15s
Soru 17Soru →
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?

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Cevap: 74\frac{7}{4}

Cevap

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.

Adım Adım Çözüm

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.

Anahtar Kavram

Evaluating algebraic expressions with negative numbers, fractions, and negative exponents.
Tahmini Süre:1m 30s
Soru 18Soru →

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.

Aşağıdaki boşlukları doldurun

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

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.

Adım Adım Çözüm

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.

Anahtar Kavram

Evaluating algebraic expressions involving negative numbers, fractions, and order of operations.
Tahmini Süre:1m 30s
Soru 19Soru →

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}?

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Cevap: 267\frac{26}{7}

Cevap

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}.

Adım Adım Çözüm

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.

Anahtar Kavram

Order of operations and handling negative signs when evaluating algebraic expressions involving powers and parentheses.
Tahmini Süre:1m 0s
Soru 20Soru →

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}?

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Cevap: -7

Cevap

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.

Adım Adım Çözüm

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.

Anahtar Kavram

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