Tüm alıştırma soruları

Find the value of $t$ that satisfies the linear equation below.

Aşağıdaki boşlukları doldurun

\frac{t}{3} + 8 = 15

The value of

t

is .

Cevabı ve açıklamayı göster

Cevap

21

To find the value of

t

, you must apply inverse operations to isolate the variable. First, subtract 8 from both sides to get

t/3 = 7

. Then, multiply both sides by 3 to eliminate the fraction, which gives the final solution

t = 21

.

Adım Adım Çözüm

1

Isolate the fraction term by subtracting 8 from both sides of the equation.

\frac{t}{3} = 15 - 8

\frac{t}{3} = 7

To solve for a variable, you must reverse the order of operations. Addition is reversed first using subtraction to isolate the term containing the variable.

2

Solve for

t

by multiplying both sides of the equation by 3.

t = 7 \times 3

t = 21

Multiplying by the denominator cancels out the fraction, leaving the variable

t

by itself.

Anahtar Kavram

Solving two-step linear equations using inverse operations.

Soru 5002Soru →

Solve the linear equation $10 - 4p = -18$ . Which of the following is the value of $p$ ?

7

-7

2

-3

Cevabı ve açıklamayı göster

Cevap: 7

Cevap

The correct value for p is 7.

The correct answer is 7 because applying inverse operations accurately isolates the variable. Subtracting 10 from both sides yields

-4p = -28

, and dividing both sides by

-4

results in

p = 7

.

Adım Adım Çözüm

1

Subtract 10 from both sides of the equation.

-4p = -18 - 10

which simplifies to

-4p = -28

This isolates the term containing the variable

p

on one side of the equation.

2

Divide both sides by -4.

p = \frac{-28}{-4}

which simplifies to

p = 7

This removes the coefficient from

p

, giving the final solution.

Anahtar Kavram

Solving a one-variable linear equation requires isolating the variable by performing inverse operations equally on both sides.

Soru 5003Soru →

Read the scenario below and fill in the missing values to complete the corresponding algebraic equation.

A network administrator calculates the total required bandwidth, $B$ , as seventy-five less than quadruple the sum of the base traffic, $t$ , and twenty.

Aşağıdaki boşlukları doldurun

B =

(t +

) -

Cevabı ve açıklamayı göster

Cevap

The completed equation is

B = 4(t + 20) - 75

. Therefore, blank 1 is 4, blank 2 is 20, and blank 3 is 75.

The scenario requires piecing together three mathematical operations. First, 'the sum of the base traffic, t, and twenty' creates the grouped expression

(t + 20)

. Next, 'quadruple' means multiplying that entire sum by 4, resulting in

4(t + 20)

. Finally, 'seventy-five less than' indicates that 75 must be subtracted from that product. Putting it all together gives

4(t + 20) - 75

, which fits perfectly into the provided blanks as 4, 20, and 75.

Adım Adım Çözüm

1

Identify the core grouped expression.

The phrase 'the sum of the base traffic, t, and twenty' translates to

(t + 20)

.

The word 'sum' indicates addition, and it groups the terms together.

2

Apply the multiplication factor.

The phrase 'quadruple the sum' translates to

4(t + 20)

.

The word 'quadruple' means to multiply a quantity by 4.

3

Apply the subtraction.

The phrase 'seventy-five less than' means subtracting 75 from the entire previous expression, yielding

4(t + 20) - 75

.

The phrase 'less than' is a turnaround phrase, meaning the value being subtracted goes at the end of the expression.

4

Map the components to the provided equation structure.

Comparing

4(t + 20) - 75

to the template

B =

blank_1

(t +

blank_2

) -

blank_3, we find blank_1 is 4, blank_2 is 20, and blank_3 is 75.

This completes the algebraic translation in the required format.

Anahtar Kavram

Translating verbal descriptions of arithmetic operations and their structural order into an algebraic equation.

Tahmini Süre:45s

Soru 5004Soru →

When the algebraic expression

\frac{(3x^{-2} y^4)^3}{9x^{-5} y^2}

is fully simplified to the form

C x^a y^b

(where

C

,

a

, and

b

are constants), what is the value of

b

?

Cevabı ve açıklamayı göster

Cevap: 10; ten

Cevap

10

By distributing the outer exponent of 3 in the numerator using multiplication (

4 \cdot 3

) to get

y^{12}

, and then subtracting the denominator's exponent of 2 using the quotient rule (

12 - 2

), the final exponent on the variable

y

is determined to be 10.

Adım Adım Çözüm

1

Apply the power of a product and power of a power rules to the numerator.

(3x^{-2}y^4)^3 = 3^3 \cdot (x^{-2})^3 \cdot (y^4)^3 = 27x^{-6}y^{12}

When raising a product to a power, the outer power applies to each individual factor inside the parentheses. Exponents are multiplied when raising a power to another power.

2

Substitute the expanded numerator back into the fractional expression.

\frac{27x^{-6}y^{12}}{9x^{-5}y^2}

This sets up the expression for applying the quotient rule between the numerator and denominator.

3

Apply the quotient rule by dividing the coefficients and subtracting the exponents of like bases.

Coefficient:

27 \div 9 = 3

Base

x

:

x^{-6 - (-5)} = x^{-6 + 5} = x^{-1}

Base

y

:

y^{12 - 2} = y^{10}

When dividing exponential terms with the same base, subtract the denominator's exponent from the numerator's exponent.

4

Combine the simplified terms and identify the exponent

b

.

The simplified expression is

3x^{-1}y^{10}

. Comparing this to the target form

Cx^a y^b

, the value of

b

is

10

.

This isolates the specific value requested by the question stem.

Anahtar Kavram

Power of a Power and Quotient of Powers Rules

Soru 5005Soru →

An integer $k$ is defined as the Least Common Multiple (LCM) of $12$ and $28$ . What is the Greatest Common Factor (GCF) of $k$ and $40$ ?

2

4

40

840

Cevabı ve açıklamayı göster

Cevap: 4

Cevap

The Greatest Common Factor of

k

and

40

is 4.

The problem requires two distinct mathematical steps. First, calculate the Least Common Multiple of

12

and

28

. Using prime factorizations (

12 = 2^2 \times 3

and

28 = 2^2 \times 7

), the LCM is the product of the highest powers of all prime factors:

2^2 \times 3 \times 7 = 84

. Therefore,

k = 84

. The second step asks for the Greatest Common Factor of

k

(

84

) and

40

. The prime factorization of

40

is

2^3 \times 5

. The only shared prime factor between

84

and

40

is

2

, and the lowest power they share is

2^2

. Thus, the GCF is

4

.

Adım Adım Çözüm

1

Find the prime factorizations of 12 and 28.

12 = 2^2 \times 3

and

28 = 2^2 \times 7

.

Prime factorization is the most reliable method for finding the Least Common Multiple.

2

Calculate

k

, which is the LCM of 12 and 28.

k = 2^2 \times 3 \times 7 = 4 \times 21 = 84

.

The LCM is found by multiplying the highest power of every prime factor present in either number.

3

Find the prime factorization of 40.

40 = 2^3 \times 5

.

This is needed to compare against the prime factors of

k

(

84

) to find the Greatest Common Factor.

4

Calculate the GCF of 84 and 40.

Comparing

84

(

2^2 \times 3 \times 7

) and

40

(

2^3 \times 5

), the only shared prime base is 2. The lowest exponent for this base is 2. Thus, the GCF is

2^2 = 4

.

The GCF is the product of the lowest powers of all common prime factors shared by both integers.

Anahtar Kavram

Applying prime factorization to sequentially determine Least Common Multiples (LCM) and Greatest Common Factors (GCF).

Soru 5006Soru →

An industrial bakery has a large container holding $10\frac{3}{8}$ gallons of vegetable oil. On Monday morning, a baker uses $3\frac{1}{2}$ gallons for a batch of muffins. In the afternoon, they use an additional $2\frac{3}{4}$ gallons for bread dough. How many gallons of vegetable oil remain in the container? (Enter your answer as a decimal.)

Cevabı ve açıklamayı göster

Cevap: 4.125

Cevap

4.125

Converting all fractions to a common denominator of 8 yields

10\frac{3}{8}

total,

3\frac{4}{8}

used first, and

2\frac{6}{8}

used second. Total used is

6\frac{2}{8}

. Subtracting this from

10\frac{3}{8}

leaves

4\frac{1}{8}

. Converting

\frac{1}{8}

to a decimal yields 0.125, making the final answer 4.125.

Adım Adım Çözüm

1

Find a common denominator for all the fractions involved.

The common denominator for 8, 2, and 4 is 8.

Fractions must have the same denominator before they can be added or subtracted.

2

Convert the amounts of oil used into mixed numbers with the common denominator of 8.

3\frac{1}{2} = 3\frac{4}{8}

and

2\frac{3}{4} = 2\frac{6}{8}

.

This prepares the values for addition.

3

Add the amounts used together to find the total amount of oil removed.

3\frac{4}{8} + 2\frac{6}{8} = 5\frac{10}{8} = 6\frac{2}{8}

We must consolidate the used portions before subtracting from the total.

4

Subtract the total oil used from the original amount in the container.

10\frac{3}{8} - 6\frac{2}{8} = 4\frac{1}{8}

Subtracting the total used gives the remaining oil.

5

Convert the final mixed number to a decimal format.

4\frac{1}{8} = 4 + (1 \div 8) = 4.125

The question explicitly requires the answer to be entered as a decimal.

Anahtar Kavram

Adding and subtracting mixed numbers with unlike denominators

Soru 5007Soru →

An algebraic equation is shown below.

4(x - 3) - 2x = 5x - 27

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

5

13

8

-15

Cevabı ve açıklamayı göster

Cevap: 5

Cevap

5

The correct answer is derived by first distributing the 4 to get

4x - 12

. Combining

4x

and

-2x

yields

2x - 12

. Moving the variables to one side (

5x - 2x = 3x

) and the constants to the other (

-12 + 27 = 15

) results in

15 = 3x

. Dividing by 3 gives the final result of

x = 5

.

Adım Adım Çözüm

1

Distribute the coefficient 4 into the parentheses.

4x - 12 - 2x = 5x - 27

According to the distributive property, the number outside the parentheses must multiply each term inside.

2

Combine the like terms on the left side of the equation.

2x - 12 = 5x - 27

The terms

4x

and

-2x

are like terms and subtract to form

2x

.

3

Isolate the variable terms on one side by subtracting

2x

from both sides.

-12 = 3x - 27

Subtracting

2x

from both sides keeps the equation balanced and groups all terms containing

x

on the right side.

4

Isolate the variable

x

by adding 27 to both sides.

15 = 3x

Adding 27 cancels out the

-27

on the right, leaving the variable term by itself.

5

Solve for

x

by dividing both sides by 3.

x = 5

Dividing by the coefficient of

x

provides the final value of the unknown variable.

Anahtar Kavram

Solving linear equations requires correctly applying the distributive property, combining like terms accurately, and using inverse operations equally on both sides to isolate the variable.

Soru 5008Soru →

Calculate the precise value of $k$ that makes the following algebraic equation true:

5(2k - 3) = 3k + 13

Cevabı ve açıklamayı göster

Cevap: 4

Cevap

4

The correct value is found by distributing the 5, gathering all variable terms on one side and constants on the other, and finally dividing by the variable's coefficient.

Adım Adım Çözüm

1

Apply the distributive property on the left side of the equation.

10k - 15 = 3k + 13

To eliminate the parentheses and expose the variable terms for grouping.

2

Subtract

3k

from both sides.

7k - 15 = 13

To collect all terms containing the variable

k

on one side of the equation.

3

Add

15

to both sides.

7k = 28

To isolate the variable term by moving constants to the opposite side.

4

Divide both sides by

7

.

k = 4

To solve for the final value of

k

.

Anahtar Kavram

Solving linear equations with variables on both sides using the distributive property.

Soru 5009Soru →

Calculate the unknown value in the given equation:

5n - 15 = 20

What is the value of $n$ ?

1

-2

7

-7

Cevabı ve açıklamayı göster

Cevap: 7

Cevap

7

To solve for n, the constant -15 must be canceled out by adding 15 to both sides, which results in 5n = 35. Then, dividing both sides by the coefficient 5 isolates n, giving the correct value of 7.

Adım Adım Çözüm

1

Add 15 to both sides of the equation.

5n - 15 + 15 = 20 + 15

5n = 35

This isolates the variable term on one side of the equation by using the inverse operation of subtraction.

2

Divide both sides of the equation by 5.

\frac{5n}{5} = \frac{35}{5}

n = 7

This isolates the variable n completely, yielding the final numerical value.

Anahtar Kavram

Solving one-variable linear equations requires using inverse operations to isolate the unknown variable while maintaining balance across the equals sign.

Soru 5010Soru →

A biomedical researcher is cataloging the diameters of four microscopic specimens. To properly calibrate the imaging equipment, arrange the specimens in descending order based on their diameter, from the largest to the smallest.

Öğeleri doğru sıraya koymak için sürükleyin

Cevabı ve açıklamayı göster

Cevap

The correct descending order from largest to smallest is Specimen Beta, Specimen Delta, Specimen Gamma, and Specimen Alpha.

By converting all measurements into standard scientific notation (

a \times 10^n

), their magnitudes can be compared directly. Specimen Beta has the largest exponent (

-5

). The remaining three specimens all reduce to an exponent of

-6

. Ordering those three by their coefficients (

9.0

,

8.5

, and

6.5

) yields the final sequence.

Adım Adım Çözüm

1

Convert Specimen Beta from standard decimal form to scientific notation.

0.000072

meters becomes

7.2 \times 10^{-5}

meters.

A uniform mathematical format is necessary to accurately compare the magnitudes of all specimens.

2

Normalize the coefficient for Specimen Gamma.

850 \times 10^{-8}

becomes

8.5 \times 10^{-6}

.

Proper scientific notation requires the coefficient to be greater than or equal to 1 and less than 10. Moving the decimal two places left increases the exponent by 2.

3

Normalize the coefficient for Specimen Delta.

0.9 \times 10^{-5}

becomes

9.0 \times 10^{-6}

.

Moving the decimal one place right to create a valid coefficient decreases the exponent by 1.

4

Compare the exponents first, then the coefficients to determine the descending order.

7.2 \times 10^{-5} > 9.0 \times 10^{-6} > 8.5 \times 10^{-6} > 6.5 \times 10^{-6}

.

A larger exponent (

-5

vs

-6

) dictates a larger overall number. For values sharing the identical

-6

exponent, the descending order is determined directly by their coefficients (

9.0 > 8.5 > 6.5

).

Anahtar Kavram

Normalizing Coefficients and Comparing Magnitudes in Scientific Notation

Tahmini Süre:1m 30s

Soru 5011Soru →

An algebraic expression is given as:

\frac{(3a^4b^{-2})^2 (2a^{-3}b^5)}{6a^2b}

What is the fully simplified form of this expression for all non-zero values of $a$ and $b$ ?

3a^3

a^3

3ab^4

3a^3b^8

Cevabı ve açıklamayı göster

Cevap: $3a^3$

Cevap

The simplified form of the expression is

3a^3

.

The expression is correctly simplified by first applying the power to all terms inside the parentheses (resulting in

9a^8b^{-4}

), then multiplying by the adjacent term in the numerator (resulting in

18a^5b

), and finally dividing by the denominator (giving

3a^3

).

Adım Adım Çözüm

1

Simplify the squared term in the numerator using the power of a product and power of a power rules.

9a^8b^{-4}

When a product is raised to a power, raise each factor to that power (

3^2 = 9

). For variables, multiply the inner and outer exponents:

(a^4)^2 = a^8

and

(b^{-2})^2 = b^{-4}

.

2

Multiply the simplified term by the second factor in the numerator,

(2a^{-3}b^5)

.

18a^5b

Multiply the coefficients (

9 \times 2 = 18

). For variables with the same base, add their exponents:

a^{8+(-3)} = a^5

and

b^{-4+5} = b^1 = b

.

3

Divide the resulting numerator by the denominator,

6a^2b

.

3a^3

Divide the coefficients (

18 \div 6 = 3

). For variables with the same base, subtract the denominator's exponent from the numerator's exponent:

a^{5-2} = a^3

. The terms

b^1

and

b^1

cancel out since

b^{1-1} = b^0 = 1

.

Anahtar Kavram

Operations with Variable Exponents

Soru 5012Soru →

A mechanic is draining old oil from a vehicle into a waste oil catch pan that has a maximum capacity of $15$ quarts. The catch pan already contains $4\frac{2}{3}$ quarts of oil from a previous job. If the vehicle's engine holds $7\frac{3}{4}$ quarts of oil, and all of it is drained into the catch pan, how much more room, in quarts, is left in the catch pan?

3\frac{5}{12}

3\frac{2}{7}

2\frac{7}{12}

7\frac{1}{4}

Cevabı ve açıklamayı göster

Cevap: $2\frac{7}{12}$

Cevap

The correct answer is

2\frac{7}{12}

.

To find the remaining capacity, first add the two amounts of oil using a common denominator of 12 (

4\frac{8}{12} + 7\frac{9}{12} = 11\frac{17}{12} = 12\frac{5}{12}

quarts). Then, subtract this total from the pan's capacity of 15. By borrowing, 15 becomes

14\frac{12}{12}

. Subtracting

12\frac{5}{12}

leaves

2\frac{7}{12}

quarts.

Adım Adım Çözüm

1

Determine the total amount of oil that will be in the catch pan by adding the existing oil and the vehicle's oil.

4\frac{2}{3} + 7\frac{3}{4}

The pan's occupied space is the sum of what was already there and what is being added.

2

Find a common denominator to add the fractions.

The least common multiple of 3 and 4 is 12. Convert the fractions:

4\frac{8}{12} + 7\frac{9}{12}

.

Fractions must have the same denominator before they can be added.

3

Add the whole numbers and the numerators, then simplify.

4 + 7 = 11

and

\frac{8}{12} + \frac{9}{12} = \frac{17}{12}

. Since

\frac{17}{12} = 1\frac{5}{12}

, the total is

11 + 1\frac{5}{12} = 12\frac{5}{12}

quarts.

Combines the mixed numbers to find the total volume of oil.

4

Subtract the total volume of oil from the catch pan's maximum capacity.

15 - 12\frac{5}{12}

The remaining room is the difference between the total capacity and the occupied space.

5

Borrow 1 from the whole number 15 to perform the subtraction.

Rewrite

15

as

14\frac{12}{12}

. Subtract:

14\frac{12}{12} - 12\frac{5}{12} = (14 - 12) + (\frac{12}{12} - \frac{5}{12}) = 2\frac{7}{12}

quarts.

Borrowing allows subtraction of the fractional part.

Anahtar Kavram

Adding mixed numbers with unlike denominators and subtracting mixed numbers from whole numbers by borrowing.

Soru 5013Soru →

Solve the following linear equation for the variable $w$ :

7 - 3(2w - 4) = w + 5

What is the value of $w$ ?

Aşağıdaki boşlukları doldurun

The value of

w

is .

Cevabı ve açıklamayı göster

Cevap

2

The equation is solved by correctly applying the distributive property to get

7 - 6w + 12 = w + 5

, combining like terms to get

19 - 6w = w + 5

, and then using inverse operations to isolate

w

, which yields

w = 2

.

Adım Adım Çözüm

1

Distribute the -3 across the terms inside the parentheses.

7 - 6w + 12 = w + 5

The distributive property is required to expand the expression

-3(2w - 4)

before combining like terms. Remember that multiplying a negative by a negative results in a positive.

2

Combine the constant terms on the left side of the equation.

19 - 6w = w + 5

Adding

7

and

12

simplifies the left side.

3

Add

6w

to both sides of the equation.

19 = 7w + 5

This step groups all terms containing the variable

w

onto the right side.

4

Subtract 5 from both sides.

14 = 7w

This isolates the variable term

7w

on one side of the equation.

5

Divide both sides by 7.

w = 2

Dividing by the coefficient of

w

solves for the final value.

Anahtar Kavram

Solving linear equations involving the distributive property with negative coefficients and variables on both sides.

Tahmini Süre:1m 0s

Soru 5014Soru →

An equation is given as:

2(3h - 5) = 4(h + 1) - 6

What is the value of $h$ ?

-6

10

4

0

Cevabı ve açıklamayı göster

Cevap: 4

Cevap

The correct value of

h

is 4.

The correct answer is found by properly distributing the values into both sets of parentheses, combining the constants on the right side correctly (+4 - 6 = -2), and then maintaining the balance of the equation while isolating the variable using inverse operations.

Adım Adım Çözüm

1

Distribute the coefficients to the terms inside the parentheses on both sides of the equation.

6h - 10 = 4h + 4 - 6

To eliminate the parentheses according to the distributive property.

2

Combine the constant terms on the right side of the equation.

6h - 10 = 4h - 2

To simplify the right side by calculating

+4 - 6

.

3

Subtract

4h

from both sides of the equation.

2h - 10 = -2

To group all variable terms containing

h

onto the left side.

4

Add 10 to both sides of the equation.

2h = 8

To isolate the variable term by canceling out the

-10

on the left side.

5

Divide both sides by 2.

h = 4

To solve for the final value of

h

.

Anahtar Kavram

Solving One-Variable Linear Equations

Soru 5015Soru →

The length of a rectangular solar panel is given by the expression $7a^4 b^{-2}$ and its width is given by $6a^2 b^5$ . Which of the following expressions represents the area of the solar panel?

42a^6 b^3

42a^8 b^{-10}

42a^6 b^{-7}

13a^6 b^3

Cevabı ve açıklamayı göster

Cevap: $42a^6 b^3$

Cevap

The area is represented by the expression

42a^6 b^3

.

The correct answer is

42a^6 b^3

. To find the area of the rectangle, you multiply its length and width:

(7a^4 b^{-2})(6a^2 b^5)

. The coefficients are multiplied together (

7 \times 6 = 42

). For the variables, the product rule states that when multiplying terms with the same base, you add their exponents. Thus, for base

a

,

4 + 2 = 6

to get

a^6

, and for base

b

,

-2 + 5 = 3

to get

b^3

. Combining these yields

42a^6 b^3

.

Adım Adım Çözüm

1

Identify the formula for the area of a rectangle and set up the expression.

Area = length \times width = (7a^4 b^{-2})(6a^2 b^5)

The area of a rectangle is found by taking the product of its length and width dimensions.

2

Multiply the numerical coefficients.

7 \times 6 = 42

Constants must be multiplied independently from the variable bases.

3

Apply the product rule for exponents to the variables with identical bases.

a^{4+2} = a^6 and b^{-2+5} = b^3

When multiplying identical bases, their exponents must be added.

4

Combine the calculated coefficient and variables into a single expression.

42a^6 b^3

This forms the final simplified product representing the area.

Anahtar Kavram

Operations with Variable Exponents

Tahmini Süre:1m 0s

Soru 5016Soru →

Given $x = -4$ and $y = 3$ , what is the exact value of the expression $3|x - y| - |y^2 - x^2|$ ?

Cevabı ve açıklamayı göster

Cevap: 14

Cevap

14

By correctly substituting the variables, following the order of operations, and applying the definition of absolute value, the expression simplifies to

3(7) - 7

, which evaluates precisely to

14

.

Adım Adım Çözüm

1

Substitute the given values for

x

and

y

into the expression.

3|-4 - 3| - |(3)^2 - (-4)^2|

Variable substitution is the first step in evaluating an algebraic expression.

2

Simplify the expression inside the first absolute value bars.

|-4 - 3| = |-7|

Absolute value bars act as grouping symbols, so operations inside them must be performed first.

3

Evaluate the exponents inside the second absolute value bars.

|(3)^2 - (-4)^2| = |9 - 16|

Following the order of operations, exponents are evaluated before subtraction.

4

Simplify the expression inside the second absolute value bars.

|9 - 16| = |-7|

Complete the subtraction inside the grouping symbols.

5

Apply the absolute value function to both simplified terms.

3(7) - 7

The absolute value of any non-zero real number is its positive counterpart:

|-7| = 7

.

6

Perform the final multiplication and subtraction.

21 - 7 = 14

Multiplication is performed before subtraction according to the standard order of operations.

Anahtar Kavram

Evaluating algebraic expressions involving absolute value, exponents, and signed numbers.

Tahmini Süre:1m 30s

Soru 5017Soru →

An engineer is calculating a scaling factor for a fluid dynamics simulation. The calculation reduces to the following algebraic expression:

\frac{ (3m^3n^{-4})(8m^{-1}n^2) }{ (2m^{-2}n^3)^2 }

Assuming $m$ and $n$ are non-zero variables, what is the fully simplified form of this scaling factor?

\frac{6m^6}{n^8}

\frac{6m}{n^{14}}

\frac{6n^4}{m^2}

\frac{12m^6}{n^8}

Cevabı ve açıklamayı göster

Cevap: $\frac{6m^6}{n^8}$

Cevap

The fully simplified expression is

\frac{6m^6}{n^8}

.

The correct answer is found by simplifying the numerator and denominator independently, then dividing. Multiplying the terms in the numerator gives

24m^2n^{-2}

. Squaring the entire term in the denominator (including the 2) gives

4m^{-4}n^6

. Dividing these expressions yields a coefficient of

6

. Subtracting the exponents gives

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

and

n^{-2 - 6} = n^{-8}

. Finally, converting the negative exponent into a fraction results in

\frac{6m^6}{n^8}

.

Adım Adım Çözüm

1

Simplify the numerator using the product rule for exponents.

(3 \times 8) m^{3 + (-1)} n^{-4 + 2} = 24m^2n^{-2}

When multiplying terms with the same base, add their exponents and multiply their numerical coefficients.

2

Simplify the denominator using the power of a product rule.

2^2(m^{-2 \times 2})(n^{3 \times 2}) = 4m^{-4}n^6

When raising a product to a power, raise the coefficient and multiply every exponent by the outside power.

3

Divide the simplified numerator by the simplified denominator using the quotient rule.

(\frac{24}{4}) m^{2 - (-4)} n^{-2 - 6} = 6m^6n^{-8}

When dividing terms with the same base, subtract the denominator's exponent from the numerator's exponent.

4

Rewrite the expression with only positive exponents.

\frac{6m^6}{n^8}

A negative exponent means taking the reciprocal of the base, so

n^{-8}

becomes

\frac{1}{n^8}

.

Anahtar Kavram

Operations with Variable Exponents

Soru 5018Soru →

What is the value of the following expression?

\frac{2}{3} - \frac{1}{4} \times 1\frac{1}{2} + \frac{5}{8}

1\frac{1}{4}

\frac{11}{12}

-\frac{1}{3}

\frac{3}{8}

Cevabı ve açıklamayı göster

Cevap: $\frac{11}{12}$

Cevap

The correct value of the expression is

\frac{11}{12}

.

The correct answer is found by rigorously applying the order of operations. The multiplication

\frac{1}{4} \times 1\frac{1}{2}

must be done first. After converting

1\frac{1}{2}

to

\frac{3}{2}

, the product is

\frac{3}{8}

. Substituting this back gives

\frac{2}{3} - \frac{3}{8} + \frac{5}{8}

. By combining the terms with eighths, we get

-\frac{3}{8} + \frac{5}{8} = \frac{2}{8}

, or

\frac{1}{4}

. Adding this to the initial term requires a common denominator of 12:

\frac{8}{12} + \frac{3}{12} = \frac{11}{12}

.

Adım Adım Çözüm

1

Convert the mixed number to an improper fraction.

1\frac{1}{2} = \frac{3}{2}

Fraction multiplication requires all terms to be expressed as proper or improper fractions.

2

Perform the multiplication operation.

\frac{1}{4} \times \frac{3}{2} = \frac{3}{8}

According to the order of operations (PEMDAS), multiplication must be performed before addition and subtraction.

3

Substitute the multiplication result back into the expression.

\frac{2}{3} - \frac{3}{8} + \frac{5}{8}

This updates the expression for the remaining addition and subtraction operations.

4

Combine the fractions that already share a common denominator.

-\frac{3}{8} + \frac{5}{8} = \frac{2}{8}

, which simplifies to

\frac{1}{4}

.

Grouping terms with the same denominator makes the calculation faster and simpler.

5

Find a common denominator for the remaining terms and add.

\frac{2}{3} + \frac{1}{4} = \frac{8}{12} + \frac{3}{12} = \frac{11}{12}

Fractions must have a common denominator (in this case, 12) before their numerators can be added.

Anahtar Kavram

Order of Operations with Fractions

Tahmini Süre:1m 0s

Soru 5019Soru →

Determine the value of the variable $y$ that satisfies the given linear equation.

Aşağıdaki boşlukları doldurun

Given the equation

5(y - 2) + 3y = 4y + 26

, the value of

y

is .

Cevabı ve açıklamayı göster

Cevap

9

By distributing the

5

to obtain

5y - 10

, combining the

y

terms on the left to get

8y

, and subsequently isolating

y

through basic inverse operations (

8y - 4y = 26 + 10

), we determine that

4y = 36

, making

y = 9

.

Adım Adım Çözüm

1

Distribute the 5 to the terms inside the parentheses.

5y - 10 + 3y = 4y + 26

Applying the distributive property removes the parentheses, allowing like terms to be combined.

2

Combine the like terms on the left side of the equation.

8y - 10 = 4y + 26

Simplifying each side individually makes it easier to isolate the variable.

3

Subtract

4y

from both sides of the equation.

4y - 10 = 26

This groups all terms containing the variable

y

on one side of the equation.

4

Add 10 to both sides of the equation.

4y = 36

This isolates the variable term by removing the constant from the left side.

5

Divide both sides by 4.

y = 9

Dividing by the coefficient of

y

yields the final value.

Anahtar Kavram

Solving linear equations with variables on both sides using the distributive property

Soru 5020Soru →

A technician is testing two different electrical circuits. The voltage in Circuit A is given by the expression $5v - 12$ , where $v$ is a variable setting on the power supply. The voltage in Circuit B is given by $2(v + 9)$ . Both circuits are adjusted to have the exact same voltage. Fill in the blanks with the correct numerical values.

Aşağıdaki boşlukları doldurun

The required setting

v

is and the resulting voltage is .

Cevabı ve açıklamayı göster

Cevap

The setting

v

is 10 and the resulting voltage is 38.

By setting the two expressions equal, expanding the parentheses, and systematically isolating

v

, we find the setting must be 10. Plugging 10 back into the expression evaluates the voltage to 38 for both circuits.

Adım Adım Çözüm

1

Set the expressions for Circuit A and Circuit B equal to each other.

5v - 12 = 2(v + 9)

The problem states that both circuits are adjusted to have the exact same voltage.

2

Distribute the 2 on the right side of the equation.

5v - 12 = 2v + 18

The distributive property is required to eliminate parentheses and expose the variable term.

3

Subtract

2v

from both sides of the equation.

3v - 12 = 18

To consolidate all variable terms on one side of the equation.

4

Add 12 to both sides of the equation.

3v = 30

To isolate the variable term by moving constants to the other side.

5

Divide both sides by 3.

v = 10

To solve for the variable

v

.

6

Substitute

v = 10

back into either original expression.

5(10) - 12 = 50 - 12 = 38

To find the resulting voltage as requested by the second blank.

Anahtar Kavram

Solving linear equations with variables on both sides and distributive property

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