Evaluating Algebraic Expressions Alıştırma Soruları — ACT — Sayfa 2

Soru 21Soru →

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

\frac{5}{3}

-\frac{13}{3}

\frac{13}{3}

-1

5

Cevabı ve açıklamayı göster

Cevap: $\frac{5}{3}$

Cevap

\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(\frac{1}{3}) - (-3)^2 = 3(4)(\frac{1}{3}) - 9 = 4 - 9 = -5

. In the denominator,

-2 + (\frac{1}{3})(-3) = -2 - 1 = -3

. Simplifying the fraction

\frac{-5}{-3}

gives

\frac{5}{3}

.

Adım Adım Çözüm

1

Evaluate the numerator

3x^2 y - z^2

by substituting

x = -2

,

y = \frac{1}{3}

, and

z = -3

.

3(-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

and

(-3)^2 = 9

.

2

Evaluate the denominator

x + yz

using the same variable values.

-2 + \left(\frac{1}{3}\right)(-3) = -2 + (-1) = -3

Perform the multiplication

y \cdot z

before adding to

x

according to the order of operations.

3

Divide the evaluated numerator by the evaluated denominator.

\frac{-5}{-3} = \frac{5}{3}

Dividing two negative numbers produces a positive quotient.

Anahtar Kavram

Evaluating Algebraic Expressions with Negative Values and Rational Numbers

Soru 22Soru →

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

\frac{31}{50}

\frac{33}{50}

\frac{33}{34}

-\frac{31}{50}

\frac{24}{25}

Cevabı ve açıklamayı göster

Cevap: $\frac{33}{50}$

Cevap

\frac{33}{50}

Substituting

m = -4

and

n = -\frac{1}{2}

into the numerator gives

(-4)^2 - 4\left(-\frac{1}{2}\right)^3 = 16 - \left(-\frac{1}{2}\right) = \frac{33}{2}

. Substituting into the denominator gives

\left(-4 + 2\left(-\frac{1}{2}\right)\right)^2 = (-5)^2 = 25

. Dividing the numerator by the denominator yields

\frac{33/2}{25} = \frac{33}{50}

.

Adım Adım Çözüm

1

Evaluate the terms in the numerator:

m^2

and

4n^3

.

m^2 = (-4)^2 = 16

and

4n^3 = 4\left(-\frac{1}{2}\right)^3 = 4\left(-\frac{1}{8}\right) = -\frac{1}{2}

.

Substitute

m = -4

and

n = -\frac{1}{2}

while applying powers before multiplication according to PEMDAS.

2

Calculate the complete numerator by subtracting the evaluated terms.

m^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\left(-\frac{1}{2}\right) = -4 - 1 = -5

, and

(-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.

\frac{\frac{33}{2}}{25} = \frac{33}{2 \times 25} = \frac{33}{50}

.

Dividing a fraction by an integer combines the denominators.

Anahtar Kavram

Evaluating algebraic expressions involving negative bases, fractions, and order of operations

Tahmini Süre:1m 30s

Soru 23Soru →

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

Aşağıdaki boşlukları doldurun

The value of the expression

3x^2 - 2xy + y^2

when

x = -3

and

y = -2

is .

Cevabı ve açıklamayı göster

Cevap

19

Substituting

x = -3

and

y = -2

into

3x^2 - 2xy + y^2

yields

3(-3)^2 - 2(-3)(-2) + (-2)^2 = 3(9) - 12 + 4 = 27 - 12 + 4 = 19

.

Adım Adım Çözüm

1

Substitute

x = -3

and

y = -2

into the expression

3(-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) + 4

Follow the order of operations (PEMDAS) by evaluating powers first:

(-3)^2 = 9

and

(-2)^2 = 4

.

3

Perform the multiplications

27 - 12 + 4

Multiply the numerical factors:

3 \times 9 = 27

and

-2 \times (-3) \times (-2) = -12

.

4

Perform addition and subtraction from left to right

19

Subtract

12

from

27

to get

15

, then add

4

to arrive at

19

.

Anahtar Kavram

Evaluating Algebraic Expressions with Negative Values

Soru 24Soru →

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

Cevabı ve açıklamayı göster

Cevap: -1.6

Cevap

The numerical value of the expression is

-1.6

.

Substituting

x = -3

,

y = -\frac{1}{2}

, and

z = 4

into

\frac{x^2 - 4y^2}{x + yz}

gives a numerator of

(-3)^2 - 4\left(-\frac{1}{2}\right)^2 = 9 - 4\left(\frac{1}{4}\right) = 8

and a denominator of

-3 + \left(-\frac{1}{2}\right)(4) = -3 - 2 = -5

. Evaluating

\frac{8}{-5}

yields

-1.6

.

Adım Adım Çözüm

1

Evaluate the numerator

x^2 - 4y^2

8

Squaring

-3

gives

9

, and squaring

-\frac{1}{2}

gives

\frac{1}{4}

. Thus,

9 - 4\left(\frac{1}{4}\right) = 9 - 1 = 8

.

2

Evaluate the denominator

x + yz

-5

Multiplying

-\frac{1}{2}

by

4

gives

-2

. Adding

-3 + (-2)

yields

-5

.

3

Compute the final fraction quotient

-1.6

Dividing the numerator

8

by the denominator

-5

yields

-1.6

.

Anahtar Kavram

Evaluating algebraic expressions using order of operations with negative bases and fractional values.

Tahmini Süre:1m 30s

Soru 25Soru →

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

\frac{7}{3}

\frac{11}{3}

-7

\frac{16}{3}

-\frac{7}{3}

$

Cevabı ve açıklamayı göster

Cevap: $\frac{7}{3}$

Cevap

The correct answer is

\frac{7}{3}

.

Substituting

a = -2

,

b = -\frac{1}{2}

, and

c = 3

gives a numerator of

(-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 - \left(-\frac{1}{2}\right)^{-2} = -2 - 4 = -6

. Dividing

-14

by

-6

simplifies to

\frac{7}{3}

.

Adım Adım Çözüm

1

Substitute

a = -2

,

b = -\frac{1}{2}

, and

c = 3

into the numerator

a^3 b - 2c^2

.

Numerator

= (-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 = -2

and

b = -\frac{1}{2}

into the denominator

a - b^{-2}

.

Denominator

= -2 - \left(-\frac{1}{2}\right)^{-2} = -2 - (-2)^2 = -2 - 4 = -6

.

A negative exponent indicates the reciprocal of the base, so

\left(-\frac{1}{2}\right)^{-2} = (-2)^2 = 4

.

3

Divide the numerator by the denominator and simplify the fraction.

\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.

Anahtar Kavram

Evaluating algebraic expressions with negative bases and negative integer exponents

Tahmini Süre:1m 30s

Soru 26Soru →

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

-6

6

\frac{34}{49}

\frac{34}{5}

-\frac{34}{13}

Cevabı ve açıklamayı göster

Cevap: $\frac{34}{5}$

Cevap

\frac{34}{5}

Substituting

x = -4

,

y = -3

, and

z = \frac{2}{3}

into the numerator yields

2(-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

. Thus, the value of the expression is

\frac{34}{5}

.

Adım Adım Çözüm

1

Substitute the given values into the numerator

2x^2 - yz

.

2(-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

, and multiplying

-3

by

\frac{2}{3}

gives

-2

, which is then subtracted.

2

Substitute the given values into the denominator

x + y^2

.

-4 + (-3)^2 = -4 + 9 = 5

Evaluating the exponent first gives

(-3)^2 = 9

, then adding

-4

gives

5

.

3

Divide the numerator by the denominator.

\frac{34}{5}

Combine the evaluated numerator and denominator to get the final value of the expression.

Anahtar Kavram

Evaluating algebraic expressions involving multiple variables with negative bases and fractional terms using standard order of operations.

Tahmini Süre:1m 15s

Soru 27Soru →

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

Aşağıdaki boşlukları doldurun

If

a = -2

,

b = 3

, and

c = -\frac{1}{2}

, the value of the algebraic expression

\frac{a^3 b + 4c}{a^2 - 2b}

is .

Cevabı ve açıklamayı göster

Cevap

13

Substituting

a = -2

,

b = 3

, and

c = -\frac{1}{2}

into the given expression yields a numerator of

(-2)^3(3) + 4(-\frac{1}{2}) = -24 - 2 = -26

and a denominator of

(-2)^2 - 2(3) = 4 - 6 = -2

. Dividing the numerator by the denominator gives

\frac{-26}{-2} = 13

.

Adım Adım Çözüm

1

Evaluate the terms in the numerator individually.

a^3 b = (-2)^3 \cdot 3 = -8 \cdot 3 = -24

and

4c = 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

.

Adding two negative numbers sums their magnitudes with a negative sign.

3

Evaluate the terms in the denominator.

a^2 = (-2)^2 = 4

and

2b = 2(3) = 6

.

Squaring a negative base results in a positive value.

4

Calculate the denominator.

Denominator

= 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.

\frac{-26}{-2} = 13

.

Dividing a negative number by a negative number yields a positive quotient.

Anahtar Kavram

Evaluating Algebraic Expressions

Tahmini Süre:1m 30s

Soru 28Soru →

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

-\frac{17}{2}

-\frac{13}{2}

\frac{7}{10}

\frac{7}{2}

\frac{43}{2}

Cevabı ve açıklamayı göster

Cevap: $\frac{7}{2}$

Cevap

\frac{7}{2}

Substituting

x = -3

and

y = 2

gives a numerator of

(-3)^2(2) - (-3 - 2)^2 = 18 - 25 = -7

and a denominator of

2(-3) + 2^2 = -6 + 4 = -2

. Dividing

-7

by

-2

gives

\frac{7}{2}

.

Adım Adım Çözüm

1

Evaluate the terms in the numerator

x^2 y = (-3)^2(2) = 9 \cdot 2 = 18

and

(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 = -7

Subtract the squared binomial result from the first product.

3

Evaluate the denominator

2x + 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

\frac{-7}{-2} = \frac{7}{2}

Dividing a negative number by a negative number yields a positive result.

Anahtar Kavram

Evaluating Algebraic Expressions with Negative Bases and Parentheses

Tahmini Süre:1m 0s

Soru 29Soru →

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

\frac{42}{5}

\frac{18}{5}

\frac{33}{5}

\frac{2}{5}

14

Cevabı ve açıklamayı göster

Cevap: $\frac{42}{5}$

Cevap

\frac{42}{5}

Substituting

x = -2

,

y = \frac{1}{3}

, and

z = -4

into the expression yields

(-2)^3 = -8

,

(\frac{1}{3})^{-1} = 3

, and

(-2 - (-4))^2 = 2^2 = 4

. Thus, the numerator equals

-8 \times 3 - 4 = -28

. The denominator equals

-2 + (\frac{1}{3})(-4) = -2 - \frac{4}{3} = -\frac{10}{3}

. Dividing the numerator by the denominator gives

\frac{-28}{-\frac{10}{3}} = \frac{84}{10} = \frac{42}{5}

.

Adım Adım Çözüm

1

Substitute the given values into the numerator of the expression:

x^3 y^{-1} - (x - z)^2

.

Numerator =

(-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 + y z

.

Denominator =

-2 + \left(\frac{1}{3}\right)(-4) = -2 - \frac{4}{3} = -\frac{6}{3} - \frac{4}{3} = -\frac{10}{3}

.

Multiply

y

and

z

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.

Anahtar Kavram

Evaluating Algebraic Expressions with Negative Numbers and Negative Exponents

Tahmini Süre:1m 15s

Evaluating Algebraic Expressions

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