Tüm alıştırma soruları

Calculate the value of

x

that satisfies the given equation:

7x - 14 = 21

Which of the following represents the correct value?

1

-3

5

-5

Cevabı ve açıklamayı göster

Cevap: 5

Cevap

5

The correct answer is 5. To solve the equation, the first step is to add 14 to both sides, which balances the equation and gives 7x = 35. The final step is dividing both sides by 7 to isolate the variable, resulting in x = 5.

Adım Adım Çözüm

1

Add 14 to both sides of the equation.

7x = 21 + 14

7x = 35

This isolates the term with the variable by removing the constant on the left side.

2

Divide both sides by 7.

x = 5

This solves for x by reversing the multiplication by 7.

Anahtar Kavram

Solving One-Variable Linear Equations requires isolating the variable using inverse operations.

Soru 4962Soru →

Simplify the following algebraic expression, assuming all variables are non-zero:

\frac{(3m^2n^4)^2}{3m^5n^3}

Which of the following represents the fully simplified form?

\frac{3n^5}{m}

\frac{n^5}{m}

3m^9n^{11}

\frac{3m}{n^5}

Cevabı ve açıklamayı göster

Cevap: $\frac{3n^5}{m}$

Cevap

The fully simplified expression is

\frac{3n^5}{m}

.

The correct expression is derived by properly applying the power rules. First, squaring the numerator

(3m^2n^4)^2

yields

9m^4n^8

. Then, dividing this by the denominator

3m^5n^3

requires dividing the coefficients (

9/3 = 3

) and subtracting the exponents of corresponding variables (

4-5 = -1

for

m

, and

8-3 = 5

for

n

). Finally, rewriting

3m^{-1}n^5

with only positive exponents gives

\frac{3n^5}{m}

.

Adım Adım Çözüm

1

Simplify the numerator using the power of a product rule.

(3m^2n^4)^2 = 3^2 \cdot (m^2)^2 \cdot (n^4)^2 = 9m^4n^8

When a product is raised to a power, each factor inside the parentheses must be raised to that power.

2

Substitute the simplified numerator back into the fraction.

\frac{9m^4n^8}{3m^5n^3}

This sets up the expression for applying the quotient rules.

3

Divide the numerical coefficients.

\frac{9}{3} = 3

Constants are simplified through standard division.

4

Apply the quotient rule for exponents to the variables

m

and

n

.

m^{4-5} = m^{-1}

and

n^{8-3} = n^5

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

5

Combine the results and rewrite without negative exponents.

3m^{-1}n^5 = \frac{3n^5}{m}

A negative exponent indicates the reciprocal of the base, moving the term to the denominator.

Anahtar Kavram

Exponent rules including power of a power, power of a product, and quotient of powers.

Soru 4963Soru →

In a piece of heavy machinery, two interlocking gears are rotating together. The drive gear has $28$ teeth, and the driven gear has $42$ teeth. At a certain moment, a marked tooth on the drive gear aligns exactly with a marked tooth on the driven gear. How many teeth must pass the alignment point before these two specific marked teeth align with each other again?

84

14

1176

70

Cevabı ve açıklamayı göster

Cevap: $84$

Cevap

The gears will align again after

84

teeth pass the alignment point.

To find the moment when two rotating cycles synchronize, the Least Common Multiple (LCM) of their cycle lengths (the number of teeth) must be determined. By finding the prime factorizations (

28 = 2^2 \times 7

and

42 = 2 \times 3 \times 7

) and multiplying the highest powers of all unique prime factors present (

2^2 \times 3 \times 7

), the LCM is calculated as

84

. Thus, exactly

84

teeth must pass for the marked teeth to realign.

Adım Adım Çözüm

1

Identify the mathematical concept required to solve the synchronization problem.

The problem asks for the next time repeating cycles coincide, which requires finding the Least Common Multiple (LCM) of

28

and

42

.

The LCM determines the smallest positive integer that is a multiple of both numbers, representing the first point where both gear cycles overlap.

2

Find the prime factorization of the number of teeth on each gear.

For the drive gear:

28 = 2^2 \times 7

. For the driven gear:

42 = 2 \times 3 \times 7

.

Prime factorization breaks down each number into its core primes, allowing for an accurate calculation of the LCM.

3

Calculate the LCM using the identified prime factors.

LCM =

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

.

The LCM is calculated by taking the highest power of each prime factor that appears in any of the original factorizations.

Anahtar Kavram

Factors, Multiples, and Primes

Soru 4964Soru →

What is the expanded form of the product $(a - 3)(a + 7)$ ?

a^2 - 21

a^2 + 4a - 21

a^2 - 4a - 21

a^2 - 17a

Cevabı ve açıklamayı göster

Cevap: $a^2 + 4a - 21$

Cevap

a^2 + 4a - 21

The expression is properly expanded by distributing each term in the first binomial to each term in the second binomial, resulting in

a^2 + 7a - 3a - 21

. Combining the like terms (

7a - 3a

) yields the final simplified polynomial

a^2 + 4a - 21

.

Adım Adım Çözüm

1

Multiply the first terms in each binomial.

a \times a = a^2

To begin distributing the first term of the first binomial.

2

Multiply the outer terms.

a \times 7 = 7a

To complete the distribution of the first term.

3

Multiply the inner terms.

-3 \times a = -3a

To begin distributing the second term of the first binomial.

4

Multiply the last terms in each binomial.

-3 \times 7 = -21

To complete the distribution of the second term.

5

Combine the like middle terms to simplify the expression.

7a + (-3a) = 4a

To write the polynomial in its fully simplified standard form, resulting in

a^2 + 4a - 21

.

Anahtar Kavram

Multiplying Binomials

Soru 4965Soru →

Solve for $n$ in the equation below:

8n - 11 = 37

Enter the exact value of $n$ .

Cevabı ve açıklamayı göster

Cevap: 6

Cevap

6

To find the value of

n

, first perform the inverse operation of subtraction by adding 11 to both sides, which simplifies the equation to

8n = 48

. Then, perform the inverse operation of multiplication by dividing both sides by 8, yielding

n = 6

.

Adım Adım Çözüm

1

Isolate the variable term.

8n = 48

Adding 11 to both sides neutralizes the -11 on the left side of the equation.

2

Solve for the variable.

n = 6

Dividing both sides by 8 isolates n completely.

Anahtar Kavram

Solving two-step linear equations using inverse operations.

Soru 4966Soru →

What is the sum of all distinct prime factors of the integer $546$ ?

Cevabı ve açıklamayı göster

Cevap: 25

Cevap

25

The integer

546

factors completely into the prime numbers

2

,

3

,

7

, and

13

. Since the question asks for the sum of all distinct prime factors, adding these together yields

2 + 3 + 7 + 13 = 25

.

Adım Adım Çözüm

1

Find the prime factorization of

546

by continuously dividing by the smallest possible prime numbers.

546 = 2 \times 273 = 2 \times 3 \times 91 = 2 \times 3 \times 7 \times 13

To identify all prime factors, we must break the number down into its foundational prime components.

2

Identify the distinct prime factors from the factorization.

The distinct prime factors are

2

,

3

,

7

, and

13

.

The question specifically asks for the sum of the 'distinct' prime factors.

3

Calculate the sum of these distinct prime factors.

2 + 3 + 7 + 13 = 25

Adding the identified distinct prime factors yields the final requested value.

Anahtar Kavram

Prime Factorization and Properties of Integers

Soru 4967Soru →

Calculate the value of each decimal expression below. Arrange the results in ascending order (from smallest to largest).

Öğeleri doğru sıraya koymak için sürükleyin

Cevabı ve açıklamayı göster

Cevap

The correct order from smallest to largest is:

1.5 \times 2

, followed by

1.2 + 2.4

, then

8.4 \div 2

, and finally

5.8 - 1.3

.

Evaluating each mathematical expression individually yields 4.5, 3.6, 4.2, and 3.0 respectively. Arranging these numerical results in ascending order gives 3.0, 3.6, 4.2, and 4.5. This directly maps to the expressions:

1.5 \times 2

, then

1.2 + 2.4

, then

8.4 \div 2

, and finally

5.8 - 1.3

.

Adım Adım Çözüm

1

Calculate the value of the first expression.

5.8 - 1.3 = 4.5

To determine the numerical value for comparison.

2

Calculate the value of the second expression.

1.2 + 2.4 = 3.6

To determine the numerical value for comparison.

3

Calculate the value of the third expression.

8.4 \div 2 = 4.2

To determine the numerical value for comparison.

4

Calculate the value of the fourth expression.

1.5 \times 2 = 3.0

To determine the numerical value for comparison.

5

Compare the four calculated values and order them from smallest to largest.

3.0 < 3.6 < 4.2 < 4.5

To establish the final ascending sequence required by the question.

Anahtar Kavram

Performing basic operations (addition, subtraction, multiplication, division) with decimals and comparing their magnitudes.

Tahmini Süre:45s

Soru 4968Soru →

Translate the following verbal statement into algebraic expressions representing each side of the equation:

'Eleven less than three times a number, $q$ , is equal to five times the sum of the number and two.'

Aşağıdaki boşlukları doldurun

The expression for 'Eleven less than three times a number,

q

' is:

The expression for 'five times the sum of the number and two' is:

Cevabı ve açıklamayı göster

Cevap

The left expression is 3q - 11 and the right expression is 5(q + 2).

The verbal phrase 'less than' requires the subtracted amount to follow the term it modifies, making the first expression 3q - 11. The phrase 'times the sum of' explicitly requires multiplying by a grouped addition, making the second expression 5(q + 2).

Adım Adım Çözüm

1

Translate the phrase representing the left side of the equation: 'Eleven less than three times a number,

q

'.

3q - 11

The phrase 'three times a number,

q

' translates directly to 3q. The phrase 'less than' dictates that 11 is subtracted from 3q, reversing the apparent reading order of the words.

2

Translate the phrase representing the right side of the equation: 'five times the sum of the number and two'.

5(q + 2)

The phrase 'sum of the number and two' translates to (q + 2). Grouping symbols (parentheses) are required because the multiplication by five applies to the entire sum, not just the first term.

Anahtar Kavram

Translating complex algebraic phrases by correctly applying order-reversing terminology and grouping indicators.

Tahmini Süre:1m 0s

Soru 4969Soru →

If the equation

\frac{(3x^4 y^k)^2}{x^3 y^5} = 9x^5 y^{11}

is true for all non-zero values of

x

and

y

, what is the value of

k

?

Cevabı ve açıklamayı göster

Cevap: 8; k=8; k = 8

Cevap

8

By applying the power rule to expand the numerator, the expression becomes

9x^8 y^{2k}

. Using the quotient rule to divide these terms by the denominator

x^3 y^5

, the expression simplifies to

9x^5 y^{2k-5}

. Equating this final simplified form to the given result

9x^5 y^{11}

, we can isolate the exponent of

y

to set up the equation

2k - 5 = 11

. Solving this equation gives

k = 8

.

Adım Adım Çözüm

1

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

(3x^4 y^k)^2 = 3^2 (x^4)^2 (y^k)^2 = 9x^8 y^{2k}

When a product is raised to a power, each factor inside the parentheses must be individually raised to that power by multiplying the exponents.

2

Substitute the expanded numerator back into the fraction and apply the quotient rule.

\frac{9x^8 y^{2k}}{x^3 y^5} = 9 x^{8-3} y^{2k-5} = 9x^5 y^{2k-5}

When dividing algebraic terms with the same base, subtract the exponent of the denominator from the exponent of the numerator.

3

Equate the simplified exponent of

y

to the exponent of

y

on the right side of the original equation.

2k - 5 = 11

Because the expressions are equivalent and their coefficients and

x

-terms already match, their

y

-exponents must be strictly equal.

4

Solve the resulting linear equation for

k

.

2k = 16 \rightarrow k = 8

Adding 5 to both sides and then dividing by 2 isolates the variable

k

.

Anahtar Kavram

Operations with Variable Exponents

Tahmini Süre:1m 30s

Soru 4970Soru →

A sporting goods store stocks basketballs and footballs in a ratio of $3:4$ . After receiving a new shipment of $10$ additional basketballs, the ratio of basketballs to footballs changes to $5:6$ . How many footballs are currently in stock?

20

120

100

12

Cevabı ve açıklamayı göster

Cevap: 120

Cevap

There are 120 footballs currently in stock.

By scaling the initial ratio (

3:4

) to

9:12

and the new ratio (

5:6

) to

10:12

, we can see that the

10

added basketballs represent exactly

1

part (

10 - 9 = 1

). Since the footballs represent

12

parts in this scaled system, there are

12 \times 10 = 120

footballs in stock.

Adım Adım Çözüm

1

Identify the quantity that remains unchanged.

The number of footballs does not change when the new shipment of basketballs arrives.

This provides an anchor point to compare the before and after ratios.

2

Find a common multiple for the football ratio parts.

The initial ratio has

4

parts for footballs and the new ratio has

6

parts. The least common multiple (LCM) of

4

and

6

is

12

.

Creating a common denominator for the unchanging item allows direct comparison of the changed item's parts.

3

Scale both ratios so the football parts equal

12

.

Initial ratio: Multiply

3:4

by

3

to get

9:12

. New ratio: Multiply

5:6

by

2

to get

10:12

.

This standardizes the 'size' of one part across both ratios.

4

Determine the change in basketball parts.

The scaled basketball parts increased from

9

to

10

, a difference of exactly

1

part.

This

1

-part increase corresponds directly to the physical items added.

5

Calculate the actual number of footballs.

Since

1

part equals the

10

basketballs added, the

12

parts for footballs equal

12 \times 10 = 120

footballs.

This yields the final answer by multiplying the scaled parts by the unit value.

Anahtar Kavram

Scaling Ratios and Proportions

Alternatif Yöntem

Using algebra, let the initial basketballs be

3x

and footballs be

4x

. After adding

10

basketballs, the new ratio yields the equation

\frac{3x + 10}{4x} = \frac{5}{6}

. Cross-multiplying gives

6(3x + 10) = 5(4x)

, which simplifies to

18x + 60 = 20x

. Solving for

x

gives

2x = 60

, so

x = 30

. The number of footballs is

4x = 4(30) = 120

.

Tahmini Süre:1m 0s

Soru 4971Soru →

A contractor is installing a security perimeter around a rectangular warehouse facility. The facility is $40$ yards long and $25$ yards wide. The fence will include three $10$ -foot gates that do not require fencing material. If the fencing costs $\$18$ per yard, what is the total cost of the fencing material required?

$990

$1,800

$2,160

$18,000

Cevabı ve açıklamayı göster

Cevap: $2,160

Cevap

The correct total cost of the fencing material is $2,160.

The total perimeter is found using the formula

2 \times (L + W)

, yielding

130

yards. The three

10

-foot gates have a combined width of

30

feet, which converts to

10

yards (

30 \div 3

). Subtracting the gate width leaves

120

yards of actual fencing needed. Multiplying this by the cost of

\$18

per yard yields the total of

\$2,160

.

Adım Adım Çözüm

1

Calculate the total perimeter of the rectangular facility in yards.

2 \times (40 + 25) = 130

yards

The fencing will go around the entire boundary of the rectangle, which requires the perimeter formula

P = 2L + 2W

.

2

Convert the total width of the gates from feet to yards.

3 \text{ gates} \times 10 \text{ feet/gate} = 30 \text{ feet}

. Since

3 \text{ feet} = 1 \text{ yard}

,

30 \text{ feet} = 10 \text{ yards}

.

The perimeter and the fencing cost are both in yards, so the gate measurements must match the primary unit.

3

Subtract the gate width from the total perimeter to find the required fencing.

130 \text{ yards} - 10 \text{ yards} = 120 \text{ yards}

The sections where the gates are located do not require any fencing material.

4

Calculate the total cost of the fencing material.

120 \text{ yards} \times \$18/\text{yard} = \$2,160

Multiplying the total required yards by the price per yard provides the final monetary cost.

Anahtar Kavram

Calculating perimeter of a rectangle, performing unit conversions (feet to yards), and subtracting openings to find total material cost.

Soru 4972Soru →

Follow the steps to evaluate the algebraic expression given the variable assignments. What are the intermediate and final values?

Aşağıdaki boşlukları doldurun

For the expression

p^3 - 6q

, given that

p = -2

and

q = 3

:

The value of the first term,

p^3

, is .
The value of the second term,

6q

, is .
The final evaluated result of the entire expression is .

Cevabı ve açıklamayı göster

Cevap

The intermediate values are -8 and 18, which subtract to a final result of -26.

By accurately substituting

p = -2

and

q = 3

into the given expression, applying the exponent correctly, and performing the final subtraction, we arrive at the correct values for each step and the final answer of -26.

Adım Adım Çözüm

1

Substitute the given value for

p

into the first term.

(-2)^3 = -8

Replacing the variable

p

with

-2

evaluates the exponent. A negative base raised to an odd power remains negative.

2

Substitute the given value for

q

into the second term.

6(3) = 18

Replacing the variable

q

with

3

completes the multiplication required for the second term.

3

Combine the evaluated terms according to the original expression.

-8 - 18 = -26

Following the original expression

p^3 - 6q

, we subtract the value of the second term from the first term.

Anahtar Kavram

Evaluating algebraic expressions involving exponents and negative numbers

Tahmini Süre:45s

Soru 4973Soru →

What is the coefficient of the $k$ term in the expanded product of $(3k - 2)$ and $(k + 5)$ ?

Cevabı ve açıklamayı göster

Cevap: 13

Cevap

13

When expanding

(3k - 2)(k + 5)

, the

k

terms are generated by multiplying the outer terms

(3k \times 5 = 15k)

and the inner terms

(-2 \times k = -2k)

. Combining these gives

15k - 2k = 13k

. Therefore, the coefficient of the

k

term is 13.

Adım Adım Çözüm

1

Identify the terms that will produce a

k

term when multiplied.

The outer terms are

3k

and

5

. The inner terms are

-2

and

k

.

Only the product of a variable term and a constant term will result in a term with degree 1 (

k

).

2

Multiply the outer terms.

(3k)(5) = 15k

This is the 'Outer' multiplication step in expanding two binomials.

3

Multiply the inner terms.

(-2)(k) = -2k

This is the 'Inner' multiplication step in expanding two binomials.

4

Combine the resulting like terms.

15k + (-2k) = 13k

Combining like terms gives the final

k

term of the expanded polynomial.

5

Identify the coefficient.

13

The coefficient is the numerical factor of the

k

term.

Anahtar Kavram

Multiplying polynomials and combining like terms

Soru 4974Soru →

A supply convoy is distributing medical kits and rations to various outposts in a ratio of $2:7$ . If a particular outpost receives $56$ rations, how many medical kits should they receive?

Cevabı ve açıklamayı göster

Cevap: 16; 16 kits; 16 medical kits; sixteen

Cevap

16

The correct amount is found by maintaining the ratio of

2

medical kits for every

7

rations. Since

56

is

8

times as large as

7

(

56 \div 7 = 8

), the number of medical kits must also be

8

times as large as

2

. Multiplying the kit ratio component (

2

) by this scale factor (

8

) gives

16

medical kits.

Adım Adım Çözüm

1

Identify the given ratio and the known quantity.

Ratio of medical kits to rations is

2:7

. The known quantity is

56

rations.

Extracting the correct values and their relationship is required to set up the proportion.

2

Set up a proportion.

\frac{2}{7} = \frac{x}{56}

The ratio of kits to rations (

2/7

) must equal the ratio of actual kits (

x

) to actual rations (

56

).

3

Solve for the unknown value

x

.

x = \frac{2 \times 56}{7} = 16

Multiplying both sides by

56

isolates

x

, revealing the required number of medical kits.

Anahtar Kavram

Solving direct proportions using a scale factor or cross-multiplication

Alternatif Yöntem

Find the scale factor by dividing the actual number of rations by the ratio number for rations:

56 \div 7 = 8

. Then multiply the ratio number for medical kits by this scale factor:

2 \times 8 = 16

.

Tahmini Süre:45s

Soru 4975Soru →

If $x = -3$ and $y = -4$ , what does the expression $5x - 2y$ equal?

-7

-23

23

7

Cevabı ve açıklamayı göster

Cevap: -7

Cevap

-7

Substituting the given values into the expression yields

5(-3) - 2(-4)

. Multiplying the terms results in

-15 - (-8)

, which simplifies to

-15 + 8

. Adding these numbers correctly produces

-7

.

Adım Adım Çözüm

1

Substitute

x = -3

and

y = -4

into the algebraic expression.

5(-3) - 2(-4)

Variable substitution replaces each variable with its given numerical value using parentheses.

2

Multiply the terms according to the order of operations.

-15 - (-8)

Multiplication is performed before subtraction:

5 \times -3 = -15

and

2 \times -4 = -8

.

3

Simplify the double negative and add the integers.

-15 + 8 = -7

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

Anahtar Kavram

Evaluating Algebraic Expressions

Tahmini Süre:45s

Soru 4976Soru →

Find the solution to the following algebraic equation:

4(3r - 6) = 48

Which of the following is the correct value of

r

?

6

2

-4

-6

Cevabı ve açıklamayı göster

Cevap: 6

Cevap

The correct value is 6.

The correct answer is found by first distributing the 4 to get 12r - 24 = 48. Next, adding 24 to both sides gives 12r = 72. Finally, dividing both sides by 12 yields r = 6.

Adım Adım Çözüm

1

Distribute the 4 into the terms inside the parentheses.

12r - 24 = 48

To eliminate the parentheses and simplify the left side of the equation.

2

Add 24 to both sides of the equation.

12r = 72

To isolate the term containing the variable

r

on one side of the equation using inverse operations.

3

Divide both sides of the equation by 12.

r = 6

To solve for the final, single value of the variable

r

.

Anahtar Kavram

Solving One-Variable Linear Equations

Soru 4977Soru →

A structural engineer calculates the maximum safe load, $L$ , for a support beam. The safe load is equal to fifty less than triple the sum of the beam's cross-sectional area, $A$ , and twelve. What algebraic equation models the maximum safe load?

L = 3(A + 12) - 50

L = 50 - 3(A + 12)

L = 3A + 12 - 50

L = 3(A + 12) + 50

Cevabı ve açıklamayı göster

Cevap: $L = 3(A + 12) - 50$

Cevap

The correct equation is

L = 3(A + 12) - 50

.

The correct equation methodically translates the text: 'the sum of... A, and twelve' becomes the grouped quantity (A + 12). 'Triple' multiplies this entire group by 3, giving 3(A + 12). Finally, 'fifty less than' dictates subtracting 50 from that product, yielding 3(A + 12) - 50.

Adım Adım Çözüm

1

Identify the base quantity being modified.

The phrase 'the sum of the beam's cross-sectional area,

A

, and twelve' translates algebraically to the grouped expression

(A + 12)

.

The word 'sum' implies addition, and it creates a single quantity that must be grouped in parentheses before further operations are applied.

2

Translate the multiplier.

The word 'triple' indicates multiplication by 3, resulting in the expression

3(A + 12)

.

The multiplier applies to the entire sum identified in the previous step.

3

Translate the subtraction phrase.

The phrase 'fifty less than' means subtracting 50 from the previously established quantity, yielding

3(A + 12) - 50

.

In algebraic translation, 'X less than Y' dictates an order reversal to become Y - X.

4

Form the final equation.

Set the maximum safe load,

L

, equal to the derived expression:

L = 3(A + 12) - 50

.

The problem states that the safe load is equal to the verbal expression.

Anahtar Kavram

Translating order-dependent phrases and implied grouping symbols from English into algebraic notation.

Tahmini Süre:45s

Soru 4978Soru →

In a structural modeling calculation, an engineer must simplify a given formula before evaluating it. Which of the following expressions is mathematically equivalent to the algebraic fraction below?

\frac{16a^5b^2}{(2ab^{-1})^3}

2a^2b^5

8a^2b^5

2a^2b^{-1}

8a^4b^5

Cevabı ve açıklamayı göster

Cevap: $2a^2b^5$

Cevap

The mathematically equivalent expression is

2a^2b^5

.

The correct answer is derived by first applying the power rule to the entire denominator, yielding

8a^3b^{-3}

. Next, dividing the numerical coefficients (

16 \div 8 = 2

) and applying the quotient rule for exponents (subtracting the denominator's exponents from the numerator's exponents) results in

a^{5-3} = a^2

and

b^{2-(-3)} = b^{2+3} = b^5

. Combining these distinct parts yields

2a^2b^5

.

Adım Adım Çözüm

1

Apply the power of a product rule to the denominator.

(2ab^{-1})^3 = 2^3 \cdot a^3 \cdot (b^{-1})^3

An exponent outside parentheses must be applied to every distinct factor inside the parentheses.

2

Simplify the factors in the denominator.

8a^3b^{-3}

2^3 = 8

, and for a power of a power

(b^{-1})^3

, the exponents are multiplied (

-1 \times 3 = -3

).

3

Rewrite the fraction using the expanded denominator.

\frac{16a^5b^2}{8a^3b^{-3}}

This sets up the expression for application of the quotient rule.

4

Divide the numerical coefficients.

16 \div 8 = 2

Constants are separated from variables and divided normally.

5

Apply the quotient rule to the variables with the base

a

.

a^{5-3} = a^2

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

6

Apply the quotient rule to the variables with the base

b

.

b^{2 - (-3)} = b^{2 + 3} = b^5

Subtracting a negative exponent is equivalent to adding a positive number.

7

Combine the simplified coefficient and variables.

2a^2b^5

Bringing the distinct evaluated parts back together provides the final simplified term.

Anahtar Kavram

To simplify a rational expression involving exponents, first use the power rule to eliminate grouping symbols, ensuring numerical coefficients are raised to the power. Then, divide the coefficients and use the quotient rule to subtract the denominator's exponents from the numerator's exponents for matching bases.

Tahmini Süre:1m 30s

Soru 4979Soru →

An airport authority is installing a security barrier around a new rectangular radar facility. The facility measures $40$ yards in length and $32$ yards in width. The design for the security barrier requires a heavy-duty steel post at each of the four corners, with additional posts placed exactly every $8$ feet along all four sides. How many steel posts are required in total to enclose the facility?

Cevabı ve açıklamayı göster

Cevap: 54

Cevap

54 posts are required in total.

The correct answer is found by first ensuring all units match. Converting the dimensions to feet gives a length of 120 feet and a width of 96 feet. The total perimeter is 432 feet. Dividing this continuous perimeter by the 8-foot spacing requirement yields exactly 54 posts.

Adım Adım Çözüm

1

Convert the given dimensions from yards to feet.

Length = 40 * 3 = 120 feet. Width = 32 * 3 = 96 feet.

The post spacing is provided in feet, so all measurements must be converted to the same unit before calculating the perimeter.

2

Calculate the total perimeter of the rectangular facility.

Perimeter = 2 * (120 + 96) = 2 * 216 = 432 feet.

The security barrier encloses the entire outer boundary of the facility, which corresponds to its perimeter.

3

Determine the total number of posts required.

Total posts = 432 / 8 = 54.

For any closed loop, the total number of equally spaced posts equals the total perimeter divided by the distance between the posts.

Anahtar Kavram

Calculating perimeter of a rectangle, unit conversions, and dividing a perimeter into equal segments.

Soru 4980Soru →

Match each unsimplified algebraic expression on the left with its fully simplified equivalent form on the right. Assume $p$ and $q$ represent non-zero numbers.

Soldaki öğeye tıklayın, sonra eşleşen sağdaki öğeye tıklayın

Cevabı ve açıklamayı göster

Cevap

The correct matches are: (4p^5 q^-2)(2p^-1 q^6) matches 8p^4 q^4; the fraction 15p^7 q^2 over 3p^2 q^-3 matches 5p^5 q^5; (3p^3 q^2)^2 matches 9p^6 q^4; and the fraction (2p^2 q^3)^3 over 4p^4 q^7 matches 2p^2 q^2.

Each expression simplifies to exactly one of the given target forms by consistently applying the laws of exponents (product, quotient, and power rules) and handling numerical coefficients accurately.

Adım Adım Çözüm

1

Simplify the first expression using the product rule for exponents.

The coefficients multiply to 8. Adding the exponents for p gives 5 + (-1) = 4. Adding the exponents for q gives -2 + 6 = 4. The result is 8p^4 q^4.

When multiplying terms with the same base, you add their exponents.

2

Simplify the second expression using the quotient rule for exponents.

The coefficients divide to 5. Subtracting the p exponents gives 7 - 2 = 5. Subtracting the q exponents gives 2 - (-3) = 5. The result is 5p^5 q^5.

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

3

Simplify the third expression using the power of a product rule.

The coefficient squared is 3^2 = 9. Multiplying the p exponent by 2 gives 3 * 2 = 6. Multiplying the q exponent by 2 gives 2 * 2 = 4. The result is 9p^6 q^4.

When raising a power to a power, multiply the exponents. Remember to also apply the power to the numerical coefficient.

4

Simplify the fourth expression by applying the power rule first, then the quotient rule.

The numerator becomes 8p^6 q^9. Dividing this by 4p^4 q^7 gives a coefficient of 2, p exponent of 6 - 4 = 2, and q exponent of 9 - 7 = 2. The result is 2p^2 q^2.

Order of operations requires handling the exponent on the entire numerator before performing division.

Anahtar Kavram

Applying the product, quotient, and power rules of exponents to simplify algebraic expressions.

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