Tüm alıştırma soruları

What is the value of the following expression?

$|-4 - |-9||$

Cevabı ve açıklamayı göster

Cevap: 13

Cevap

13

The correct answer is found by first evaluating the innermost absolute value,

|-9| = 9

. Substituting this value back into the expression gives

|-4 - 9|

. Simplifying the subtraction inside yields

|-13|

. Finally, the absolute value of

-13

is

13

.

Adım Adım Çözüm

1

Evaluate the inner absolute value.

|-9| = 9

According to the order of operations, we must simplify the expression from the inside out. The absolute value of a negative number is its positive counterpart.

2

Substitute and simplify the remaining expression.

|-4 - 9| = |-13|

Replacing

|-9|

with

9

gives the subtraction

-4 - 9

, which simplifies to

-13

.

3

Evaluate the outer absolute value.

|-13| = 13

The absolute value represents the distance from zero on the number line, which is always non-negative.

Anahtar Kavram

Absolute Value

Soru 4942Soru →

If $4(k - 3) = 28$ , what is the value of $k$ ?

Cevabı ve açıklamayı göster

Cevap: 10

Cevap

10

Applying inverse operations to isolate the variable results in

k = 10

. This can be verified by substituting 10 back into the original equation:

4(10 - 3) = 4(7) = 28

, which makes the statement true.

Adım Adım Çözüm

1

Divide both sides by 4.

k - 3 = 7

Dividing by the coefficient outside the parentheses simplifies the equation and isolates the binomial.

2

Add 3 to both sides of the equation.

k = 10

Using the inverse operation of subtraction isolates the variable

k

.

Anahtar Kavram

Solving One-Variable Linear Equations

Soru 4943Soru →

An aircraft's current cruising altitude is $a$ feet. The required altitude for a new training maneuver is defined as four hundred feet less than triple the sum of the current altitude and two thousand. Which of the following algebraic expressions represents the required altitude in feet?

3(a + 2000) - 400

400 - 3(a + 2000)

3a + 2000 - 400

3(2000a) - 400

Cevabı ve açıklamayı göster

Cevap: $3(a + 2000) - 400$

Cevap

The correct expression is

3(a + 2000) - 400

.

The correct answer accurately models the sequence of operations described in the text. It uses parentheses to group the addition

(a + 2000)

, multiplies that entire group by 3, and places the subtraction of 400 at the end of the expression to correctly satisfy the 'less than' condition.

Adım Adım Çözüm

1

Translate 'the sum of the current altitude and two thousand'.

(a + 2000)

The word 'sum' indicates addition between the variable

a

and the number 2000. Parentheses are required because subsequent operations apply to this entire result.

2

Apply the multiplier 'triple' to the sum.

3(a + 2000)

'Triple' means multiplying the entire grouped quantity by 3.

3

Incorporate the phrase 'four hundred feet less than'.

3(a + 2000) - 400

The phrase 'less than' is a turnaround phrase, meaning the 400 must be subtracted from the previously established quantity.

Anahtar Kavram

Translating multi-step verbal phrases into algebraic expressions requires careful attention to grouping symbols and translation order, particularly for subtraction.

Soru 4944Soru →

Calculate the product of the two algebraic terms below:

-4m^3 \times 3m^4

What is the correct result of this multiplication?

-12m^7

12m^7

-12m^{12}

12m^{12}

Cevabı ve açıklamayı göster

Cevap: $-12m^7$

Cevap

The correct answer is

-12m^7

.

To multiply these terms, first multiply the numerical coefficients:

-4 \times 3 = -12

. Then, apply the product rule for exponents by adding the exponents of the common base

m

:

3 + 4 = 7

. Combining these yields

-12m^7

.

Adım Adım Çözüm

1

Multiply the numerical coefficients.

-4 \times 3 = -12

When multiplying algebraic terms, the constant factors are multiplied together using standard integer rules.

2

Apply the product rule for exponents to the variables.

m^3 \times m^4 = m^{3+4} = m^7

The product rule states that when multiplying terms with the same base, their exponents should be added.

3

Combine the coefficient and the variable part.

-12m^7

Putting the results from the previous steps together gives the final simplified expression.

Anahtar Kavram

Product Rule for Exponents and Multiplying Monomials

Tahmini Süre:45s

Soru 4945Soru →

A commercial bakery prepares a special bread dough using flour, water, and sourdough starter in a ratio of $11:6:3$ by weight. If a single batch uses $42$ pounds of water, what is the total weight of the dough for this batch, in pounds?

Cevabı ve açıklamayı göster

Cevap: 140; 140 pounds; 140 lbs; 140 lb

Cevap

140

Since the water represents

6

parts of the ratio and weighs

42

pounds, each part weighs exactly

7

pounds (

42 \div 6 = 7

). The entire batch of dough is composed of

20

total parts (

11

for flour +

6

for water +

3

for starter). Multiplying the

20

total parts by the

7

pounds per part results in a total batch weight of

140

pounds.

Adım Adım Çözüm

1

Find the value of one part in the ratio.

42 \div 6 = 7

pounds per part.

Water corresponds to

6

parts of the

11:6:3

ratio, and its total weight is

42

pounds.

2

Calculate the total number of parts in the entire dough mixture.

11 + 6 + 3 = 20

total parts.

The total weight is the sum of the parts for flour, water, and starter.

3

Multiply the total number of parts by the weight of a single part.

20 \times 7 = 140

pounds.

Scaling the

20

total parts by the

7

pounds per part gives the total weight of the batch.

Anahtar Kavram

Solving proportions by finding the unit rate of a single part and multiplying it by the total number of parts.

Soru 4946Soru →

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

Aşağıdaki boşlukları doldurun

To evaluate the expression

5x - y^2

for

x = 3

and

y = -4

, we first substitute the given numbers. The term

5x

evaluates to . The term

y^2

evaluates to . Subtracting the second result from the first gives a final answer of .

Cevabı ve açıklamayı göster

Cevap

The intermediate values are 15 and 16, resulting in a final answer of -1.

Substituting the given values into the expression

5x - y^2

yields

5(3) - (-4)^2

. Following the order of operations, first evaluate the exponent:

(-4)^2 = 16

. Next, perform the multiplication:

5(3) = 15

. Finally, perform the subtraction:

15 - 16 = -1

.

Adım Adım Çözüm

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.

Anahtar Kavram

Evaluating Algebraic Expressions

Soru 4947Soru →

A civil engineering project requires a specific concrete mixture consisting of cement, sand, and gravel in a ratio of $2:3:6$ by weight. If a batch of this concrete mixture weighs a total of $6,600\text{ lbs}$ , how many more pounds of gravel are used than cement?

1,800

2,400

3,300

3,600

Cevabı ve açıklamayı göster

Cevap: $2,400$

Cevap

The concrete mixture requires

2,400

more pounds of gravel than cement.

The entire mixture consists of

11

equal parts (

2

cement

+ 3

sand

+ 6

gravel). Since the total mixture weighs

6,600\text{ lbs}

, one part weighs

600\text{ lbs}

(

6,600 \div 11

). The difference between the amount of gravel (

6

parts) and cement (

2

parts) is

4

parts. Multiplying this difference by the weight of one part yields

4 \times 600 = 2,400\text{ lbs}

.

Adım Adım Çözüm

1

Find the total number of parts in the ratio.

2 + 3 + 6 = 11\text{ parts}

.

The total given weight of the concrete mixture must be distributed across all the ratio parts combined.

2

Calculate the weight of a single proportional part.

6,600 \div 11 = 600\text{ lbs}

.

Dividing the total mixture weight by the total number of parts determines the exact value of one unit.

3

Determine the difference in parts between the gravel and the cement.

6 - 2 = 4\text{ parts}

.

The question asks 'how many more pounds' of gravel are used than cement, which corresponds to the difference in their ratio shares.

4

Calculate the final difference in weight.

4 \times 600 = 2,400\text{ lbs}

.

Multiplying the part difference by the weight per part gives the final amount.

Anahtar Kavram

Solving part-to-whole ratio scenarios and calculating the difference between specific components.

Tahmini Süre:1m 0s

Soru 4948Soru →

A military engineering unit is installing a security fence around a rectangular vehicle depot. The depot is $150$ yards long and $80$ yards wide. The fencing plan includes leaving a single $30$ -foot wide opening for a main access gate. If the fencing material costs $\$15$ per linear foot, what is the total cost of the fencing material needed? (Do not include commas or dollar signs in your answer)

Cevabı ve açıklamayı göster

Cevap: 20250

Cevap

20250

To find the total cost, first determine the total perimeter of the depot in yards:

2 \times (150 + 80) = 460

yards. Since the fencing cost and gate width are in feet, convert the perimeter to feet:

460 \times 3 = 1380

feet. Next, subtract the

30

-foot gate opening where no fencing is needed:

1380 - 30 = 1350

feet of fencing. Finally, multiply the required fencing by the cost per foot:

1350 \times \$15 = \$20,250

.

Adım Adım Çözüm

1

Calculate the perimeter of the rectangular depot in yards.

2(150 + 80) = 2(230) = 460

yards.

The fencing will enclose the boundary of the rectangular area.

2

Convert the perimeter from yards to feet.

460 \times 3 = 1380

feet.

The cost of the fencing is given per linear foot and the gate is measured in feet, so all measurements must be in feet.

3

Subtract the gate opening from the total perimeter.

1380 - 30 = 1350

feet.

Fencing material is not needed for the gate opening.

4

Calculate the total cost of the fencing material.

1350 \times 15 = 20250

dollars.

The total length of fencing required is multiplied by the cost per linear foot.

Anahtar Kavram

Calculating perimeter, converting units (yards to feet), and determining total cost based on a linear rate.

Soru 4949Soru →

Determine the value of $y$ that makes the following equation true:

6y - 12 = 36

Which of the following represents the correct value of $y$ ?

42

4

8

-6

Cevabı ve açıklamayı göster

Cevap: 8

Cevap

The correct value of y is 8.

The correct value is 8. To isolate the variable y, the constant 12 must first be added to both sides of the equation, which results in 6y = 48. Finally, dividing both sides by the coefficient 6 gives the solution y = 8.

Adım Adım Çözüm

1

Add 12 to both sides of the equation.

6y = 36 + 12 \Rightarrow 6y = 48

This isolates the term with the variable on one side of the equation.

2

Divide both sides by 6.

y = \frac{48}{6} \Rightarrow y = 8

This isolates the variable y and provides the final solution.

Anahtar Kavram

Solving two-step linear equations by performing inverse operations.

Soru 4950Soru →

An event coordinator is setting up a temporary security perimeter for an outdoor festival. The secured VIP zone is rectangular, measuring $30$ yards in length and $20$ yards in width. An un-fenced opening of $5$ yards must be left on one side for the main entrance gate. If the temporary fencing costs $\$4.50$ per linear foot, what is the total cost, in dollars, of the fencing required?

Cevabı ve açıklamayı göster

Cevap: 1282.5

Cevap

1282.5

The correct total cost is

1282.50. To find this, the full perimeter of the 30-by-20 yard rectangle is calculated as 100 yards. After subtracting the 5-yard opening, 95 yards of fencing are needed. Because the fencing is priced per foot, the 95 yards must be converted to 285 feet by multiplying by 3. Finally, multiplying 285 feet by the

4.50 per foot cost yields 1282.50.

Adım Adım Çözüm

1

Calculate the total perimeter of the rectangular zone in yards.

100 yards

The perimeter represents the total distance around the rectangle before accounting for any openings.

2

Subtract the gate opening from the perimeter.

95 yards

Fencing is not required for the 5-yard entrance gate.

3

Convert the required fencing length from yards to feet.

285 feet

The cost of the fencing is provided per linear foot, so the length must be in feet.

4

Calculate the total cost by multiplying the length in feet by the price per foot.

1282.5

Total cost equals the total amount of material multiplied by the unit price.

Anahtar Kavram

Calculating perimeter of a rectangle, adjusting for openings, and converting units for cost calculation.

Soru 4951Soru →

During a physics calculation, a student models a system's energy output using the following expression, where $x$ and $y$ are non-zero variables:

\frac{(2x^{-2}y^3)^3}{4x^{-4}y^5}

What is the fully simplified form of this expression using only positive exponents?

\frac{2y^4}{x^2}

\frac{2y^4}{x^{10}}

\frac{y^4}{2x^2}

\frac{3y^4}{2x^2}

Cevabı ve açıklamayı göster

Cevap: $\frac{2y^4}{x^2}$

Cevap

\frac{2y^4}{x^2}

Applying the power of a product rule to the numerator yields

8x^{-6}y^9

. Dividing this by the denominator

4x^{-4}y^5

involves dividing the coefficients (

8/4 = 2

) and subtracting the exponents of like bases. For

x

,

-6 - (-4) = -2

, and for

y

,

9 - 5 = 4

. The resulting expression is

2x^{-2}y^4

. To represent this with positive exponents, the

x^{-2}

term is moved to the denominator, giving

\frac{2y^4}{x^2}

.

Adım Adım Çözüm

1

Apply the power of a product rule to the numerator.

(2x^{-2}y^3)^3 = 2^3 \cdot (x^{-2})^3 \cdot (y^3)^3 = 8x^{-6}y^9

The exponent outside the parentheses applies to every individual factor inside.

2

Divide the numerical coefficients.

8 \div 4 = 2

Numerical coefficients are handled separately from variables with exponents.

3

Apply the quotient of powers rule to the variables.

x^{-6 - (-4)} = x^{-2}

and

y^{9 - 5} = y^4

When dividing like bases, you subtract the denominator's exponent from the numerator's exponent.

4

Combine the results and convert the negative exponent to a positive exponent.

2x^{-2}y^4 = \frac{2y^4}{x^2}

Moving a term with a negative exponent to the denominator changes its exponent to positive.

Anahtar Kavram

Simplifying rational algebraic expressions using the power of a product rule, the quotient of powers rule, and negative exponents.

Soru 4952Soru →

The prime factorizations of two integers, $X$ and $Y$ , are given below:

$X = 2^3 \times 3^2 \times 5$
$Y = 2^2 \times 3^3$

Based on this information, what is the greatest common factor (GCF) of $X$ and $Y$ ?

36

180

1080

6

Cevabı ve açıklamayı göster

Cevap: $36$

Cevap

The greatest common factor is

36

.

To determine the greatest common factor (GCF) from prime factorizations, we must identify the prime bases shared by both numbers and raise each to its lowest given exponent. The shared bases for

X

and

Y

are

2

and

3

. Taking the lowest exponent for each gives

2^2

and

3^2

. Their product,

4 \times 9

, equals

36

.

Adım Adım Çözüm

1

Identify the common prime bases in both factorizations.

The prime bases present in both

X

and

Y

are

2

and

3

. The prime base

5

is only in

X

.

The greatest common factor (GCF) only includes prime factors that are shared by all given numbers.

2

Determine the lowest exponent for each common prime base.

For base

2

, the exponents are

3

and

2

, so the lowest is

2

. For base

3

, the exponents are

2

and

3

, so the lowest is

2

.

To find the greatest common factor, we must use the smallest available power of each common prime to ensure it divides both numbers perfectly.

3

Multiply the resulting prime powers to calculate the GCF.

2^2 \times 3^2 = 4 \times 9 = 36

.

The product of these prime powers yields the greatest integer that divides both

X

and

Y

without a remainder.

Anahtar Kavram

Calculating the greatest common factor (GCF) using prime factorization by finding the lowest powers of all common prime bases.

Soru 4953Soru →

Match each verbal phrase or word problem to its correct algebraic equation translation.

Soldaki öğeye tıklayın, sonra eşleşen sağdaki öğeye tıklayın

Cevabı ve açıklamayı göster

Cevap

The correct matches pair each verbal description with its mathematically equivalent equation by carefully translating order of operations, rates, geometric formulas, and grouping words.

Each verbal phrase uniquely defines a single algebraic relationship. By identifying key terms such as 'less than' for reverse-order subtraction, rate contexts for variable multiplication, geometric formulas for perimeter, and grouping words for parentheses, each scenario is mapped accurately to its corresponding equation.

Adım Adım Çözüm

1

Analyze the first phrase: 'Eight less than five times a number

x

is equal to twenty.'

Translates to

5x - 8 = 20

.

'Five times a number' is

5x

. The phrase 'less than' dictates that 8 is subtracted from

5x

.

2

Analyze the second phrase: 'A service charges 8 dollars per hour plus a 5 dollar flat fee. The total cost for

x

hours is 20 dollars.'

Translates to

8x + 5 = 20

.

The variable

x

is multiplied by the hourly rate of 8, and the one-time flat fee of 5 is added to this product.

3

Analyze the third phrase: 'The perimeter of a rectangle is 20. Its length is 5 units more than its width,

x

.'

Translates to

2x + 2(x + 5) = 20

.

The perimeter formula is

2W + 2L

. Substituting

x

for width and

(x + 5)

for length provides the correct equation.

4

Analyze the fourth phrase: 'Five times the difference of a number

x

and eight results in twenty.'

Translates to

5(x - 8) = 20

.

The word 'difference' combined with 'times' indicates that the subtraction

(x - 8)

must occur first, requiring parentheses before multiplying by 5.

Anahtar Kavram

Translating complex word phrases into algebraic equations requires strict attention to operation order, grouping symbols, and contextual formulas.

Soru 4954Soru →

Determine the value of the unknown variable in the following linear equation.

Aşağıdaki boşlukları doldurun

Solve the equation for

m

:

4m - 7 = 13

The correct value for

m

is .

Cevabı ve açıklamayı göster

Cevap

The value of the variable m is 5.

Applying the inverse operation by adding 7 to both sides gives

4m = 20

. Dividing by 4 isolates the variable, resulting in the correct value of 5.

Adım Adım Çözüm

1

Add 7 to both sides of the equation.

4m = 13 + 7 \Rightarrow 4m = 20

To isolate the term with the variable (

4m

) on one side of the equation.

2

Divide both sides by 4.

m = \frac{20}{4} \Rightarrow m = 5

To solve for

m

by canceling out its coefficient.

Anahtar Kavram

Solving one-variable linear equations by applying inverse operations to isolate the variable.

Soru 4955Soru →

A military motor pool maintains a fleet of tactical trucks, transport vans, and passenger cars in a ratio of $8:5:3$ . If there are exactly $112$ total vehicles in this fleet, how many transport vans does the motor pool maintain?

Cevabı ve açıklamayı göster

Cevap: 35

Cevap

The motor pool maintains 35 transport vans.

The problem outlines a part-to-part-to-part ratio of

8:5:3

and a total quantity of

112

. To solve this, first add the parts of the ratio together (

8 + 5 + 3 = 16

). Next, divide the total quantity by the total ratio parts (

112 \div 16 = 7

) to determine that each 'part' represents

7

actual vehicles. Since the ratio for transport vans is the middle number (

5

), multiply

5

by

7

to correctly get

35

transport vans.

Adım Adım Çözüm

1

Determine the total number of parts in the given ratio.

8 + 5 + 3 = 16

parts

Since the total number of vehicles in the fleet is given, we must find the total sum of the ratio parts to determine the size of a single part.

2

Calculate the exact value of a single ratio part.

112 \div 16 = 7

vehicles per part

Dividing the total quantity of vehicles by the total number of ratio parts gives the multiplier for the ratio.

3

Find the number of transport vans.

5 \times 7 = 35

transport vans

The ratio value specifically assigned to transport vans is

5

. Multiplying this by the value of one part gives the final count.

Anahtar Kavram

Solving part-to-whole ratio problems by establishing the total parts before finding specific component quantities.

Tahmini Süre:1m 0s

Soru 4956Soru →

A supply clerk is organizing copper wire. A spool initially contains $15\frac{4}{5}$ meters of wire. If a technician uses $6\frac{1}{2}$ meters of wire from the spool, how many meters of wire remain? Express your answer as a decimal.

Cevabı ve açıklamayı göster

Cevap: 9.3

Cevap

9.3

The initial length of the wire is

15\frac{4}{5}

meters and the used portion is

6\frac{1}{2}

meters. By converting these to decimals, you get

15.8

meters and

6.5

meters. Subtracting the used portion from the initial portion yields

15.8 - 6.5 = 9.3

meters. Alternatively, using a common denominator of 10 gives

15\frac{8}{10} - 6\frac{5}{10} = 9\frac{3}{10}

, which also equals 9.3.

Adım Adım Çözüm

1

Convert the mixed numbers to decimals or find a common denominator.

15\frac{4}{5} = 15.8

and

6\frac{1}{2} = 6.5

Numbers must be in a uniform format to perform subtraction easily.

2

Subtract the length of wire used from the initial length.

15.8 - 6.5 = 9.3

The remaining length is the difference between the starting amount and the used amount.

Anahtar Kavram

Fraction Operations

Tahmini Süre:45s

Soru 4957Soru →

A logistics company is securing an L-shaped equipment depot with a chain-link fence. The facility has a 120-foot southern border, a 90-foot western border, a 75-foot northern border, and a 45-foot eastern border. The remaining two inner boundary walls connect the northern and eastern borders to complete the L-shape. The entire perimeter will be fenced except for a 15-foot wide entrance gate on the southern border. If the fencing material costs $25$ per yard, what is the total cost of the fencing?

$2,625

$3,375

$10,125

$24,375

Cevabı ve açıklamayı göster

Cevap: $3,375

Cevap

The total cost of the fencing is $3,375.

The correct answer is found by first determining the missing inner dimensions of the L-shape (120 - 75 = 45 feet, and 90 - 45 = 45 feet) to find the total 6-sided perimeter (420 feet). After subtracting the 15-foot gate, 405 feet of fencing is needed. Converting this to yards by dividing by 3 gives 135 yards. Multiplying 135 yards by the

25 per yard cost results in

3,375.

Adım Adım Çözüm

1

Calculate the lengths of the two missing inner boundaries.

Inner horizontal border = 120 - 75 = 45 feet. Inner vertical border = 90 - 45 = 45 feet.

All six sides of the L-shape are needed to find the total perimeter. The missing sides are found by subtracting the known parallel outer edges.

2

Calculate the total perimeter of the equipment depot.

120 + 90 + 75 + 45 + 45 + 45 = 420 feet.

The perimeter is the total length of the continuous boundary line enclosing the facility.

3

Subtract the width of the entrance gate from the total perimeter.

420 - 15 = 405 feet.

The 15-foot gate opening does not require fencing material, so it must be removed from the total.

4

Convert the required fencing from feet to yards.

405 / 3 = 135 yards.

The fencing cost is provided in yards, so the perimeter measurement must match the pricing unit.

5

Calculate the total cost of the fencing material.

135 \times 25 = 3,375 dollars.

Multiplying the total yards of fencing by the cost per yard gives the final price.

Anahtar Kavram

Calculating the perimeter of composite rectilinear shapes, subtracting non-fenced gaps, and applying unit conversions for total cost.

Soru 4958Soru →

Arrange the following pairs of integers in ascending order based on the value of their Greatest Common Factor (GCF).

Öğeleri doğru sıraya koymak için sürükleyin

Cevabı ve açıklamayı göster

Cevap

The correct ascending order based on the Greatest Common Factor (GCF) is:

32

and

40

(GCF = 8),

36

and

48

(GCF = 12),

42

and

56

(GCF = 14),

45

and

60

(GCF = 15).

By evaluating the Greatest Common Factor for each pair, we find that the GCF of

32

and

40

is

8

; the GCF of

36

and

48

is

12

; the GCF of

42

and

56

is

14

; and the GCF of

45

and

60

is

15

. The correct ascending order is

8

,

12

,

14

,

15

.

Adım Adım Çözüm

1

Find the GCF for

45

and

60

.

The GCF is

15

.

The factors of

45

are

1, 3, 5, 9, 15, 45

. The factors of

60

include

1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60

. The largest shared factor is

15

.

2

Find the GCF for

32

and

40

.

The GCF is

8

.

The largest number that divides exactly into both

32

and

40

is

8

.

3

Find the GCF for

42

and

56

.

The GCF is

14

.

The largest number that divides exactly into both

42

and

56

is

14

(

42 = 14 \times 3

,

56 = 14 \times 4

).

4

Find the GCF for

36

and

48

.

The GCF is

12

.

The largest number that divides exactly into both

36

and

48

is

12

.

5

Order the pairs based on the calculated GCFs.

The sequence from smallest to largest GCF is

8

,

12

,

14

,

15

.

Arranging the values in ascending order determines the correct sequence.

Anahtar Kavram

Calculating the Greatest Common Factor (GCF) of pairs of integers.

Alternatif Yöntem

You can also use prime factorization for each number to find the GCF by multiplying the lowest powers of all common prime factors between the pairs.

Tahmini Süre:1m 30s

Soru 4959Soru →

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

Cevabı ve açıklamayı göster

Cevap: 33

Cevap

33

By correctly applying substitution and the order of operations, squaring

-5

gives

25

, and multiplying

4

by

-2

gives

-8

. Subtracting

-8

from

25

results in

33

.

Adım Adım Çözüm

1

Substitute the given variable values into the expression

(-5)^2 - 4(-2)

To replace the variables with their corresponding numerical values.

2

Evaluate the exponent

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)

Multiply 4 by -2 to find the term to be subtracted.

4

Perform the final subtraction

25 + 8 = 33

Subtracting a negative number is equivalent to adding its positive counterpart.

Anahtar Kavram

Evaluating Algebraic Expressions

Soru 4960Soru →

Multiply the binomials $(x - 5)$ and $(x + 2)$ . Fill in the missing values for the resulting expanded expression $x^2 + bx + c$ .

Aşağıdaki boşlukları doldurun

The value of

b

is .
The value of

c

is .

Cevabı ve açıklamayı göster

Cevap

The value of b is -3 and the value of c is -10.

By applying the FOIL method, you multiply the First terms (

x \times x = x^2

), Outer terms (

x \times 2 = 2x

), Inner terms (

-5 \times x = -5x

), and Last terms (

-5 \times 2 = -10

). Combining the like terms (

2x - 5x

) results in

-3x

. The fully expanded expression is

x^2 - 3x - 10

. Comparing this to the standard form

x^2 + bx + c

, we find that the coefficient for the middle term

b

is

-3

and the constant term

c

is

-10

.

Adım Adım Çözüm

1

Use the FOIL method to distribute the terms in the binomials.

(x \times x) + (x \times 2) + (-5 \times x) + (-5 \times 2)

Each term in the first binomial, including its sign, must be multiplied by each term in the second binomial.

2

Perform the multiplications for each pair.

x^2 + 2x - 5x - 10

This expands the product using the rules for multiplying positive and negative numbers.

3

Combine the like terms.

x^2 - 3x - 10

Adding the middle terms (

2x

and

-5x

) simplifies the expression into the standard form

x^2 + bx + c

.

Anahtar Kavram

Multiplying Polynomials

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