Tüm alıştırma soruları

Soru 5021Soru →

Match each algebraic expression on the left with its fully simplified equivalent form on the right. Assume all variables represent non-zero real numbers.

Soldaki öğeye tıklayın, sonra eşleşen sağdaki öğeye tıklayın

Cevabı ve açıklamayı göster

Cevap

The quotient matches (3a^6)/b^5, the squared term matches (9b^8)/a^4, the product matches 8a^2b^2, and the fraction raised to a negative power matches (8b^6)/a^6.

Each expression is thoroughly simplified by applying the fundamental laws of exponents: adding exponents for multiplication, subtracting exponents for division, multiplying exponents for a power to a power, and reciprocating terms to handle negative exponents.

Adım Adım Çözüm

1

Simplify the first expression: (15a^4b^-3) / (5a^-2b^2).

3a^6b^-5, which becomes (3a^6)/b^5.

When dividing terms with the same base, divide the coefficients and subtract the exponents. Move negative exponents to the denominator.

2

Simplify the second expression: (3a^-2b^4)^2.

9a^-4b^8, which becomes (9b^8)/a^4.

When raising a power to a power, raise the coefficient to the outer power and multiply the variable exponents.

3

Simplify the third expression: (2a^3b^-1)(4a^-1b^3).

8a^2b^2.

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

4

Simplify the fourth expression: ((a^2b^-2)/2)^-3.

a^-6b^6 / 2^-3, which becomes (8b^6)/a^6.

Apply the outer negative exponent to all factors inside the parentheses, including the denominator, then move negative powers across the fraction bar to make them positive.

Anahtar Kavram

Exponent rules: Product of powers, quotient of powers, power of a product, and negative exponents.

Soru 5022Soru →

In a specific algorithm, the output value is determined by the algebraic rule $2m^2n - 3n^2$ . Which of the following is the resulting output when the algorithm processes the inputs $m = -4$ and $n = -3$ ?

-123

-69

69

-219

Cevabı ve açıklamayı göster

Cevap: -123

Cevap

The correct evaluated result is -123.

The correct answer is found by substituting the numerical values for the variables and strictly adhering to the order of operations. First, squaring the negative integers results in positive values, transforming the expression to 2(16)(-3) - 3(9). Multiplying the terms gives -96 and 27, respectively. Finally, subtracting 27 from -96 produces a final value of -123.

Adım Adım Çözüm

1

Substitute the given values

m = -4

and

n = -3

into the expression

2m^2n - 3n^2

.

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

Direct substitution of variables with their numerical values is the required first step to evaluate any algebraic expression.

2

Evaluate the exponents according to the order of operations (PEMDAS).

(-4)^2 = 16

and

(-3)^2 = 9

. The expression becomes

2(16)(-3) - 3(9)

.

Exponents must be calculated before multiplication to maintain mathematical accuracy. A negative number raised to an even power results in a positive number.

3

Perform the multiplication operations from left to right.

For the first term:

2(16) = 32

, and

32(-3) = -96

. For the second term:

3(9) = 27

. The expression is now

-96 - 27

.

Multiplication precedes subtraction in the order of operations.

4

Perform the final subtraction to find the total value.

-96 - 27 = -123

Combining the remaining terms yields the final evaluated numerical result of the polynomial.

Anahtar Kavram

Evaluating algebraic expressions with negative integers using substitution and the order of operations.

Soru 5023Soru →

A portable generator's fuel tank holds $8\frac{1}{4}$ gallons of diesel. If the generator consumes fuel at a rate of $1\frac{2}{3}$ gallons per hour, how many gallons of fuel remain in the tank after exactly $2$ hours of continuous operation?

4\frac{11}{12}

13\frac{1}{6}

5\frac{1}{12}

1\frac{11}{12}

Cevabı ve açıklamayı göster

Cevap: $4\frac{11}{12}$

Cevap

The correct answer is

4\frac{11}{12}

gallons.

The correct answer accurately calculates the total fuel used by multiplying the hourly consumption rate by the operating time (

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

). It then correctly subtracts this total from the initial tank capacity by finding a common denominator and borrowing appropriately (

8\frac{1}{4} - 3\frac{1}{3} = 8\frac{3}{12} - 3\frac{4}{12} = 7\frac{15}{12} - 3\frac{4}{12} = 4\frac{11}{12}

).

Adım Adım Çözüm

1

Calculate the total amount of fuel consumed in 2 hours.

2 \times 1\frac{2}{3} = 2 \times \frac{5}{3} = \frac{10}{3} = 3\frac{1}{3}

gallons

Total consumption is the hourly rate multiplied by the number of hours operated.

2

Set up the subtraction to find the remaining fuel.

8\frac{1}{4} - 3\frac{1}{3}

Remaining fuel is the initial capacity minus the total consumed fuel.

3

Find a common denominator to subtract the mixed numbers.

The least common multiple of

4

and

3

is

12

. The expression becomes

8\frac{3}{12} - 3\frac{4}{12}

.

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

4

Borrow from the whole number to subtract the fractions.

8\frac{3}{12}

becomes

7\frac{15}{12}

. Then,

7\frac{15}{12} - 3\frac{4}{12} = 4\frac{11}{12}

gallons.

Since

\frac{3}{12}

is less than

\frac{4}{12}

, one whole unit must be converted into twelfths (

\frac{12}{12}

) and added to the fraction.

Anahtar Kavram

Fraction Operations

Soru 5024Soru →

A hiker plans to complete a trail that is $15\frac{3}{4}$ miles long. If she hikes at a consistent average speed of $3\frac{1}{2}$ miles per hour, how many hours will it take her to complete the entire trail?

Cevabı ve açıklamayı göster

Cevap: 4.5

Cevap

4.5

The correct answer is found by converting the total distance (

15\frac{3}{4}

) and the speed (

3\frac{1}{2}

) into improper fractions (

\frac{63}{4}

and

\frac{7}{2}

), and then dividing the distance by the speed. This translates to

\frac{63}{4} \times \frac{2}{7}

, which cross-simplifies perfectly to

\frac{9}{2}

or

4.5

hours.

Adım Adım Çözüm

1

Convert the mixed numbers into improper fractions.

15\frac{3}{4} = \frac{63}{4}

and

3\frac{1}{2} = \frac{7}{2}

Fraction arithmetic requires mixed numbers to be in improper fraction form before dividing.

2

Set up the division equation to solve for time.

Time =

\frac{63}{4} \div \frac{7}{2}

Time is calculated by dividing total distance by the average speed.

3

Change the division to multiplication by using the reciprocal of the divisor.

\frac{63}{4} \times \frac{2}{7}

Dividing by a fraction is mathematically equivalent to multiplying by its reciprocal.

4

Multiply the fractions and simplify the result.

\frac{63 \times 2}{4 \times 7} = \frac{126}{28} = \frac{9}{2} = 4.5

Simplifying the fraction yields the exact number of hours.

Anahtar Kavram

Dividing mixed numbers by converting to improper fractions and multiplying by the reciprocal.

Tahmini Süre:1m 0s

Soru 5025Soru →

If $y = -3$ , which of the following represents all possible values for $x$ in the equation $-2 - 2|x - y| = -16$ ?

10

and

-4

1

and

-7

4

and

-10

4

Cevabı ve açıklamayı göster

Cevap: $4$ and $-10$

Cevap

The possible values for

x

are

4

and

-10

.

The correct answer accurately substitutes the negative value, isolates the absolute value term to yield

|x + 3| = 7

, and evaluates both the positive and negative branches to find the two valid solutions for

x

.

Adım Adım Çözüm

1

Substitute the value

y = -3

into the original equation.

-2 - 2|x - (-3)| = -16 \Rightarrow -2 - 2|x + 3| = -16

To reduce the equation to a single variable so it can be solved.

2

Begin isolating the absolute value expression by adding

2

to both sides.

-2|x + 3| = -16 + 2 \Rightarrow -2|x + 3| = -14

To undo the constant subtraction outside the absolute value, following reverse order of operations.

3

Divide both sides of the equation by

-2

.

|x + 3| = 7

To completely isolate the absolute value expression.

4

Set up two separate equations for the positive and negative pathways.

x + 3 = 7

and

x + 3 = -7

The expression inside the absolute value can evaluate to either

7

or

-7

and still have an absolute distance of

7

from zero.

5

Solve both resulting linear equations for

x

.

x = 4

and

x = -10

To find all possible values that satisfy the original equation.

Anahtar Kavram

Solving linear absolute value equations requires accurately substituting variables, strictly following the order of operations to isolate the absolute value, and creating two distinct pathways (positive and negative) to find all solutions.

Tahmini Süre:1m 30s

Soru 5026Soru →

The algebraic expression

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

can be simplified to the form

kx^a y^b

, where

k

is the numerical coefficient, and

a

and

b

are the exponents. What is the value of

a + b

?

Cevabı ve açıklamayı göster

Cevap: 18; 18.0

Cevap

18

By applying the power rule, the numerator becomes

8x^{12}y^9

. Dividing this by the denominator

4x^5y^{-2}

yields a numerical coefficient of

8/4=2

. For the

x

terms, applying the quotient rule gives

12 - 5 = 7

. For the

y

terms,

9 - (-2) = 11

. The simplified expression is

2x^7y^{11}

. The exponents are

a=7

and

b=11

, so their sum is

7 + 11 = 18

.

Adım Adım Çözüm

1

Apply the power of a product rule to the numerator.

(2^3)(x^{4 \times 3})(y^{3 \times 3}) = 8x^{12}y^9

To distribute the outer exponent 3 to all factors inside the parentheses.

2

Divide the numerical coefficients.

\frac{8}{4} = 2

To simplify the constants to find

k

.

3

Apply the quotient rule for the variable

x

.

x^{12 - 5} = x^7

When dividing like bases, subtract the exponent in the denominator from the exponent in the numerator. This gives

a = 7

.

4

Apply the quotient rule for the variable

y

, accounting for the negative exponent.

y^{9 - (-2)} = y^{9 + 2} = y^{11}

Subtracting a negative exponent is equivalent to adding. This gives

b = 11

.

5

Calculate the sum of the exponents

a

and

b

.

7 + 11 = 18

To answer the specific question asked in the prompt.

Anahtar Kavram

Applying the power of a product rule and the quotient rule for exponents, paying special attention to subtracting negative exponents.

Soru 5027Soru →

Consider the following equation:

4(v - 2) + 6 = 7v - 2(v + 5)

What is the value of $v$ that satisfies this equation?

12

-12

8

-4

Cevabı ve açıklamayı göster

Cevap: 8

Cevap

8

Expanding the terms by distributing gives 4v - 8 + 6 = 7v - 2v - 10. Simplifying both sides yields 4v - 2 = 5v - 10. By subtracting 4v from both sides and adding 10 to both sides, the variable v is isolated to yield 8.

Adım Adım Çözüm

1

Distribute the constants into the parentheses on both sides of the equation.

4v - 8 + 6 = 7v - 2v - 10

This removes the parentheses and expands the expressions so like terms can be combined.

2

Combine the constant terms on the left side and the variable terms on the right side.

4v - 2 = 5v - 10

Simplifying both sides makes it easier to isolate the variable.

3

Subtract 4v from both sides to group the variable terms on the right side.

-2 = v - 10

This step groups the variables together while keeping the coefficient of v positive.

4

Add 10 to both sides to solve for v.

8 = v

This step fully isolates v, giving the final solution.

Anahtar Kavram

Solving linear equations involves distributing constants to expand expressions, combining like terms, and performing identical operations on both sides to isolate the unknown variable.

Soru 5028Soru →

Solve for $t$ in the equation below:

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

Aşağıdaki boşlukları doldurun

The value of

t

is .

Cevabı ve açıklamayı göster

Cevap

The correct value for

t

is 8.

By properly distributing values into the parentheses, combining like terms, and isolating the variable using inverse operations, the equation simplifies to

2t = 16

, which yields

t = 8

.

Adım Adım Çözüm

1

Distribute the 3 into the parentheses on the left side, and distribute the negative sign into the parentheses on the right side.

6t - 12 + 5 = 5t - t + 9

To eliminate the parentheses so that all terms can be properly combined.

2

Combine the constant terms on the left side and the like variable terms on the right side.

6t - 7 = 4t + 9

To simplify both sides of the equation independently before moving terms across the equals sign.

3

Subtract

4t

from both sides of the equation.

2t - 7 = 9

To gather all the variable terms onto the left side.

4

Add 7 to both sides of the equation.

2t = 16

To isolate the term containing the variable.

5

Divide both sides by 2.

t = 8

To find the final numerical value of

t

.

Anahtar Kavram

Solving One-Variable Linear Equations

Tahmini Süre:1m 30s

Soru 5029Soru →

A plumber is connecting three sections of PVC pipe to bridge a gap that is $12\frac{1}{4}$ feet wide. The first section of pipe measures $4\frac{2}{3}$ feet in length, and the second section measures $3\frac{3}{4}$ feet. Assuming there is no overlap between the connected sections, what must be the length of the third section of pipe to exactly bridge the gap?

3\frac{5}{6}

feet

4\frac{1}{6}

feet

4\frac{5}{6}

feet

20\frac{2}{3}

feet

Cevabı ve açıklamayı göster

Cevap: $3\frac{5}{6}$ feet

Cevap

The third section of pipe must be

3\frac{5}{6}

feet long.

The correct answer accurately adds the two pipe lengths by finding a common denominator of 12, resulting in

8\frac{5}{12}

. It then subtracts this sum from the total gap width of

12\frac{3}{12}

by properly borrowing from the whole number to get

11\frac{15}{12} - 8\frac{5}{12} = 3\frac{10}{12}

, which simplifies to

3\frac{5}{6}

.

Adım Adım Çözüm

1

Find the combined length of the first two pipe sections.

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

To determine how much of the gap has already been covered.

2

Convert the fractions to a common denominator and add.

4\frac{8}{12} + 3\frac{9}{12} = 7\frac{17}{12}

The least common multiple of 3 and 4 is 12.

3

Simplify the improper fraction.

7\frac{17}{12} = 7 + 1\frac{5}{12} = 8\frac{5}{12}

feet

Converting to a proper mixed number makes the final subtraction step accurate.

4

Subtract the combined length from the total gap width.

12\frac{1}{4} - 8\frac{5}{12} = 12\frac{3}{12} - 8\frac{5}{12}

The remaining distance requires subtracting the covered distance from the total.

5

Borrow from the whole number to perform the subtraction.

11\frac{15}{12} - 8\frac{5}{12} = 3\frac{10}{12} = 3\frac{5}{6}

feet

Since

\frac{3}{12}

is smaller than

\frac{5}{12}

, you must borrow 1 whole (or

\frac{12}{12}

) from 12.

Anahtar Kavram

Fraction Operations with Mixed Numbers

Soru 5030Soru →

Assume

x

and

y

are non-zero real numbers. If the algebraic expression

\frac{(3x^2y^4)^3}{9x^ky^7}

simplifies completely to

3xy^5

, what is the value of the exponent

k

?

Cevabı ve açıklamayı göster

Cevap: 5; k=5; k = 5

Cevap

5

The correct value is 5. By expanding the numerator using the power of a product rule,

(3x^2y^4)^3

becomes

27x^6y^{12}

. Dividing this by the denominator

9x^ky^7

yields

\left(\frac{27}{9}\right) \cdot x^{6-k} \cdot y^{12-7}

, which simplifies to

3x^{6-k}y^5

. Equating this result to the given simplified expression

3x^1y^5

means the exponent of

x

must be 1. Solving the simple equation

6 - k = 1

gives

k = 5

.

Adım Adım Çözüm

1

Expand the numerator using the power of a product and power of a power rules.

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

The outer exponent must be applied to the numerical coefficient and multiplied by the inner exponents of each variable.

2

Substitute the expanded numerator back into the fractional expression.

\frac{27x^6y^{12}}{9x^ky^7}

This sets up the expression for division.

3

Simplify the numerical coefficients and apply the quotient rule for exponents (

x^a / x^b = x^{a-b}

).

\left(\frac{27}{9}\right) \cdot x^{6-k} \cdot y^{12-7} = 3x^{6-k}y^5

To combine the numerical terms and the variable terms with identical bases.

4

Set the simplified expression equal to the target expression provided in the problem.

3x^{6-k}y^5 = 3x^1y^5

The problem states that the original expression simplifies completely to

3xy^5

.

5

Equate the exponents of the base

x

to solve for

k

.

6 - k = 1 \Rightarrow k = 5

Since the expressions are equal and the bases are the same, their corresponding exponents must be equal.

Anahtar Kavram

Applying power and quotient rules of exponents to solve for an unknown variable exponent.

Soru 5031Soru →

What is the value of $k$ that makes the following equation true?

4(k - 2) - 3k = 5k - 2(k + 10)

6

4

3

-14

Cevabı ve açıklamayı göster

Cevap: 6

Cevap

The correct value for the variable is 6.

By following the correct order of operations, distributing carefully, and balancing the equation, we find that k = 6 makes both sides of the equation equal.

Adım Adım Çözüm

1

Distribute the values outside the parentheses on both sides of the equation.

4k - 8 - 3k = 5k - 2k - 20

Applying the distributive property removes the parentheses and allows for combining like terms.

2

Combine like terms on each side of the equation.

k - 8 = 3k - 20

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

3

Subtract

k

from both sides to collect the variables on the right side.

-8 = 2k - 20

Balancing the equation by performing the inverse operation isolates the variable terms on one side.

4

Add 20 to both sides to isolate the variable term.

12 = 2k

Moving the constants to the opposite side of the variable completes the isolation process.

5

Divide both sides by 2.

k = 6

Dividing by the coefficient gives the final value of the variable.

Anahtar Kavram

Solving multi-step linear equations involves distributing, combining like terms, and balancing the equation using inverse operations.

Soru 5032Soru →

Simplify the following algebraic expression:

\left( \frac{12p^4 q^{-3}}{3p^{-2} q^2} \right)^{-1}

What is the fully simplified form of this expression?

\frac{4q^5}{p^6}

\frac{4p^6}{q^5}

\frac{q^5}{4p^6}

\frac{q}{4p^2}

Cevabı ve açıklamayı göster

Cevap: $\frac{q^5}{4p^6}$

Cevap

The expression simplifies to

\frac{q^5}{4p^6}

.

The correct answer is derived by first simplifying the inner fraction using the quotient rule for exponents, resulting in

4p^6q^{-5}

. Applying the outer negative exponent of

-1

takes the reciprocal of each factor, leading to

4^{-1}p^{-6}q^5

, which is written with positive exponents as

\frac{q^5}{4p^6}

.

Adım Adım Çözüm

1

Divide the numerical coefficients.

12 \div 3 = 4

Coefficients follow standard division rules.

2

Simplify the variable

p

using the quotient rule for exponents.

p^4 \div p^{-2} = p^{4 - (-2)} = p^6

When dividing like bases, subtract the exponent in the denominator from the exponent in the numerator.

3

Simplify the variable

q

using the quotient rule for exponents.

q^{-3} \div q^2 = q^{-3 - 2} = q^{-5}

Subtracting a positive exponent is mathematically equivalent to adding its negative value.

4

Combine the simplified parts into a single expression inside the parentheses.

4p^6q^{-5}

This represents the fully simplified value of the fraction before applying the outer exponent.

5

Apply the outer exponent of

-1

to every factor in the expression.

(4p^6q^{-5})^{-1} = 4^{-1} \cdot (p^6)^{-1} \cdot (q^{-5})^{-1}

The power of a product rule states that

(abc)^x = a^x b^x c^x

.

6

Simplify the resulting exponents and write with positive exponents.

4^{-1}p^{-6}q^5 = \frac{q^5}{4p^6}

A negative exponent indicates the reciprocal of the base (

x^{-n} = 1/x^n

).

Anahtar Kavram

Simplifying algebraic fractions using the quotient, product, and negative power rules for exponents.

Soru 5033Soru →

Determine the value of the polynomial $-2a^2 + 5ab - b^3$ under the condition that $a = -2$ and $b = -3$ . Which of the following represents the correct evaluation?

-5

65

49

73

Cevabı ve açıklamayı göster

Cevap: 49

Cevap

49

The correct answer is found by substituting the variable values, properly applying the exponents to the negative numbers (where

(-2)^2 = 4

and

(-3)^3 = -27

), and adhering to the order of operations to multiply before adding and subtracting.

Adım Adım Çözüm

1

Substitute

-2

for

a

and

-3

for

b

into the given polynomial.

-2(-2)^2 + 5(-2)(-3) - (-3)^3

To prepare the expression for numerical evaluation.

2

Evaluate the exponential terms, noting that a negative number raised to an even power is positive, and to an odd power is negative.

-2(4) + 5(-2)(-3) - (-27)

The order of operations (PEMDAS) requires resolving exponents before multiplication.

3

Perform the multiplication operations from left to right.

-8 + 30 - (-27)

The order of operations dictates that multiplication must be completed before addition and subtraction.

4

Convert the subtraction of a negative into addition and calculate the final sum.

-8 + 30 + 27 = 22 + 27 = 49

This is the final step to find the total value of the polynomial.

Anahtar Kavram

Evaluating Algebraic Expressions

Tahmini Süre:45s

Soru 5034Soru →

When the algebraic expression $(4x - 3)(2x^2 + 5x - 1)$ is fully expanded and simplified into standard polynomial form, what is the value of the coefficient of the $x$ term?

Cevabı ve açıklamayı göster

Cevap: -19

Cevap

-19

The correct answer is

-19

because the

x

term is exclusively formed by the sum of the products

(4x)(-1)

and

(-3)(5x)

, which equals

-4x - 15x = -19x

.

Adım Adım Çözüm

1

Identify the goal.

We need to find the coefficient of the

x

term in the product of

(4x - 3)

and

(2x^2 + 5x - 1)

.

Focusing only on the terms that produce an

x

term saves time compared to expanding the entire polynomial expression.

2

Find the first partial product that results in an

x

term.

Multiply the

x

term from the binomial (

4x

) by the constant term from the trinomial (

-1

) to get

-4x

.

An

x

term multiplied by a constant results in an

x

term.

3

Find the second partial product that results in an

x

term.

Multiply the constant term from the binomial (

-3

) by the

x

term from the trinomial (

5x

) to get

-15x

.

A constant multiplied by an

x

term results in an

x

term.

4

Combine the partial products.

-4x + (-15x) = -19x

To find the total

x

term in the expanded polynomial, all like terms must be combined.

5

Identify the final coefficient.

The coefficient of

-19x

is

-19

.

The coefficient is the numerical multiplier of the variable.

Anahtar Kavram

Multiplying Polynomials and Combining Like Terms

Soru 5035Soru →

Evaluate the algebraic expression $P = 4v^3 - 3uv + 2u^2$ using the assigned values $u = -4$ and $v = -2$ . Fill in the missing numerical values for the intermediate calculation step and the final evaluated result.

Aşağıdaki boşlukları doldurun

The calculated value for the isolated term

4v^3

is , which makes the final evaluated total for the entire expression

P

equal to .

Cevabı ve açıklamayı göster

Cevap

The first term evaluates to -32, and the entire expression evaluates to -24.

The expression requires careful substitution of

u = -4

and

v = -2

. Following the order of operations (PEMDAS), we first calculate the exponents:

(-2)^3 = -8

and

(-4)^2 = 16

. Next, we perform the multiplication for each term:

4(-8) = -32

,

-3(-4)(-2) = -24

, and

2(16) = 32

. Finally, combining these intermediate values gives

-32 - 24 + 32 = -24

.

Adım Adım Çözüm

1

Substitute the value of v into the first term

4v^3

and evaluate according to the order of operations.

4(-2)^3 = 4(-8) = -32

Exponents must be evaluated before multiplication. The cube of a negative number remains negative.

2

Substitute the values of u and v into the second term

-3uv

and evaluate.

-3(-4)(-2) = -3(8) = -24

Multiplying two negative numbers yields a positive product, which is then multiplied by the negative coefficient.

3

Substitute the value of u into the third term

2u^2

and evaluate.

2(-4)^2 = 2(16) = 32

The square of a negative number is positive. This must be calculated before multiplying by the coefficient.

4

Combine the evaluated terms to find the final total for P.

-32 - 24 + 32 = -24

Adding and subtracting the values from left to right yields the final result.

Anahtar Kavram

Evaluating multi-variable algebraic expressions by correctly substituting negative integers and rigorously applying the order of operations.

Soru 5036Soru →

An algebraic equation requires simplification to isolate the unknown variable. Which of the following represents the correct value of $y$ in the equation below?

3(2y + 5) - y = 2(2y - 1) + 13

0

6

-4

26

Cevabı ve açıklamayı göster

Cevap: -4

Cevap

The correct value for

y

is

-4

.

The correct answer is found by properly applying the distributive property to both sides, combining the resulting like terms, and systematically isolating the variable. Maintaining the balance of the equation by performing inverse operations yields a final value of -4.

Adım Adım Çözüm

1

Distribute the constants into the parentheses on both sides of the equation.

6y + 15 - y = 4y - 2 + 13

To eliminate the parentheses and prepare the equation for combining terms.

2

Combine like terms on each individual side.

5y + 15 = 4y + 11

To simplify the expressions on both sides before isolating the variable.

3

Subtract

4y

from both sides.

y + 15 = 11

To group all variable terms onto the left side of the equation.

4

Subtract

15

from both sides.

y = -4

To completely isolate

y

and find the final root of the equation.

Anahtar Kavram

Solving one-variable linear equations by applying the distributive property and combining like terms.

Soru 5037Soru →

What value of $x$ satisfies the equation $\frac{1}{2}(4x - 6) + 3x = \frac{2}{3}(9x - 12)$ ?

5

-5

6

11

Cevabı ve açıklamayı göster

Cevap: 5

Cevap

5

The correct answer is

5

. By distributing the fractions, combining the like terms, and properly using inverse operations to isolate

x

, the equation simplifies step-by-step to

x = 5

.

Adım Adım Çözüm

1

Distribute the fractions into the parentheses on both sides of the equation.

2x - 3 + 3x = 6x - 8

Applying the distributive property removes the parentheses and prepares the equation for combining terms.

2

Combine the like terms on the left side.

5x - 3 = 6x - 8

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

3

Subtract

5x

from both sides to group the variable terms on the right side.

-3 = x - 8

Keeping the variable coefficient positive reduces the chance of a sign error.

4

Add

8

to both sides to isolate

x

.

5 = x

Applying the inverse operation isolates the variable and provides the final solution.

Anahtar Kavram

Solving One-Variable Linear Equations

Soru 5038Soru →

An electrician has a spool containing $25\frac{1}{6}$ yards of copper wire. They use $7\frac{1}{2}$ yards of the wire for a kitchen circuit and $8\frac{3}{4}$ yards for a living room circuit. How many yards of wire are left on the spool?

9\frac{1}{12}

9\frac{1}{2}

8\frac{11}{12}

26\frac{5}{12}

Cevabı ve açıklamayı göster

Cevap: $8\frac{11}{12}$

Cevap

The correct answer is

8\frac{11}{12}

yards.

First, the total wire used must be calculated by adding

7\frac{1}{2}

and

8\frac{3}{4}

. Using a common denominator of 4,

7\frac{2}{4} + 8\frac{3}{4} = 15\frac{5}{4}

, which simplifies to

16\frac{1}{4}

. Next, subtract this total from the initial amount on the spool:

25\frac{1}{6} - 16\frac{1}{4}

. The common denominator for 6 and 4 is 12, resulting in

25\frac{2}{12} - 16\frac{3}{12}

. Since 3 cannot be subtracted from 2, you must borrow 1 from 25, turning it into

24\frac{14}{12}

. Finally,

24\frac{14}{12} - 16\frac{3}{12} = 8\frac{11}{12}

.

Adım Adım Çözüm

1

Determine the total amount of wire used by adding

7\frac{1}{2}

and

8\frac{3}{4}

.

16\frac{1}{4}

yards

The total wire consumed is the sum of the pieces used for the kitchen and the living room.

2

Find a common denominator to subtract the total used wire from the initial amount:

25\frac{1}{6} - 16\frac{1}{4}

.

25\frac{2}{12} - 16\frac{3}{12}

A common denominator of 12 is needed to perform the subtraction between the fractional parts.

3

Borrow 1 from the whole number 25 to perform the subtraction.

24\frac{14}{12} - 16\frac{3}{12} = 8\frac{11}{12}

yards

Since

\frac{3}{12}

is larger than

\frac{2}{12}

, you must borrow a whole (

\frac{12}{12}

) to complete the operation.

Anahtar Kavram

Adding and subtracting mixed numbers with unlike denominators, including the borrowing method.

Soru 5039Soru →

Determine the simplified form of the following product, assuming $c$ and $d$ are non-zero real numbers:

(2c^{-3}d^5)^{-2}(4c^2d^{-1})

What is the resulting fully simplified expression?

-\frac{16c^8}{d^{11}}

c^{12}d^{10}

\frac{c^8}{d^{11}}

\frac{8c^8}{d^{11}}

Cevabı ve açıklamayı göster

Cevap: $\frac{c^8}{d^{11}}$

Cevap

The fully simplified expression is

\frac{c^8}{d^{11}}

.

The correct answer is obtained by first applying the power of a power rule to

(2c^{-3}d^5)^{-2}

, yielding

\frac{1}{4}c^6d^{-10}

. Multiplying this by

4c^2d^{-1}

involves adding the exponents of like bases, which results in

c^8d^{-11}

. Moving the negative exponent to the denominator yields the fully simplified form.

Adım Adım Çözüm

1

Apply the power of a product rule to the first term, distributing the

-2

exponent to the coefficient and both variables.

2^{-2}(c^{-3})^{-2}(d^5)^{-2} = \frac{1}{4}c^6d^{-10}

The rules

(xy)^a = x^ay^a

and

(x^a)^b = x^{ab}

require every part of the base to be raised to the outer power.

2

Multiply the expanded first term by the second term by grouping the coefficients and the like variables together.

(\frac{1}{4} \cdot 4)(c^6 \cdot c^2)(d^{-10} \cdot d^{-1})

Multiplication is commutative, which allows rearranging the terms to combine matching bases.

3

Multiply the numerical coefficients and add the exponents for the variables with the same base.

1 \cdot c^{6+2} \cdot d^{-10+(-1)} = c^8d^{-11}

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

x^a \cdot x^b = x^{a+b}

).

4

Rewrite the expression so that all exponents are positive.

\frac{c^8}{d^{11}}

A fully simplified algebraic expression is typically written with positive exponents, using the rule

x^{-a} = \frac{1}{x^a}

.

Anahtar Kavram

Operations with Variable Exponents

Soru 5040Soru →

A pottery studio has a storage bin containing $15\frac{1}{4}$ pounds of dry clay. An instructor removes $4\frac{5}{8}$ pounds of the clay for a beginner class and another $3\frac{1}{2}$ pounds for an advanced class. Later that day, a new shipment arrives, and $5\frac{7}{8}$ pounds of clay are added to the bin. How many pounds of clay are in the bin now?

Cevabı ve açıklamayı göster

Cevap: 13

Cevap

13

The correct answer is found by establishing a common denominator for all fractions, correctly borrowing a whole number to subtract the first amount of clay, subtracting the second amount, and then adding the final shipment to yield exactly 13 pounds.

Adım Adım Çözüm

1

Find a common denominator for all fractions.

15\frac{1}{4} = 15\frac{2}{8}

and

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

.

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

2

Subtract the amount of clay used for the beginner class from the total.

15\frac{2}{8} - 4\frac{5}{8} = 14\frac{10}{8} - 4\frac{5}{8} = 10\frac{5}{8}

pounds.

Since

\frac{2}{8}

is smaller than

\frac{5}{8}

, we borrow 1 from 15 to properly execute the subtraction.

3

Subtract the amount of clay used for the advanced class from the new total.

10\frac{5}{8} - 3\frac{4}{8} = 7\frac{1}{8}

pounds.

This accounts for the second removal of clay.

4

Add the newly shipped clay to the remaining amount.

7\frac{1}{8} + 5\frac{7}{8} = 12\frac{8}{8}

pounds.

The new shipment increases the total inventory.

5

Simplify the final mixed number.

13

pounds.

\frac{8}{8}

is equal to 1 whole, so

12 + 1 = 13

.

Anahtar Kavram

Addition and subtraction of mixed numbers with unlike denominators, requiring whole-number borrowing.

Tahmini Süre:1m 15s

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