adding rational expressions with unlike denominators

[tex]\log _6 \sqrt[3]{6} \quad[/tex]=0.3 i.e 1/3

[tex]\log _2 64[/tex] =6

[tex]-3 \log _5 25 \quad[/tex]=-6

[tex]\log _2 \sqrt[4]{8} \quad[/tex] = 0.75

[tex]\log _3 \frac{1}{81}[/tex]=-4

Generally, A logarithm is a mathematical function that determines the power to which a given number (called the base) must be raised to obtain a certain value. In other words, a logarithm is the inverse operation of exponentiation.

For example, if we take the base 2 and the value 8, the logarithm of 8 to the base 2 (written as log2 8) is 3, because 2 raised to the power of 3 (i.e., 2^3) equals 8.

Logarithms are widely used in many fields of science, engineering, and mathematics, particularly in areas where large numbers or very small numbers are involved. They are also used in data analysis, signal processing, and many other applications.

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Answer: 4cos(pi/6x)+3

Step-by-step explanation:

The equation of the nth term is 4 + n.

The 75th term is 79.

It is a sequence where the difference between each consecutive term is the same.

Example:

5, 7, 9, 11, 13 is an arithmetic sequence.

We have,

5, 6, 7, 8, ,,,,,a(75),,,,,,,,

This is an arithmetic sequence.

a = 5

d = 1

Now,

The nth term of arithmetic is a + (n - 1)d.

So,

= 5 + (n - 1)1

= 5 + n - 1

= 4 + n

The nth term is 4 + n.

The 75th term.

i,e

For n = 75,

= 4 + n

= 4 + 75

= 79

Thus,

The equation of the nth term is 4 + n.

The 75th term is 79.

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The friend that had the least portion of action games would be = Thomas with 20 action games in total.

The number of action games that Thomas has = 5

The number of sports video games for every action game = 6

Therefore the total number of sports game he has = 5×6 = 30.

The number of action games that Franklin has = 9

The number of sports video games for every action game = 6

Therefore the total number of sports game he has = 9×6 = 54.

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Since the fish was share evenly amongst the 3 dolphins, and each one got 6, the equation will be:

f = d x n

where

f represents the fish

d represents the dolphin

n represents the number of fish each got.

Let's assume that the number of fish in the bucket before Colleen feeds the dolphins is "f". To find out how many fish there were in the bucket, you can use the following equation:

f = 3 * 6

where 3 is the number of dolphins and 6 is the number of fish each dolphin gets. Solving for "f", you get:

f = 3 * 6

f = 18

So there were 18 fish in the bucket before Colleen fed the dolphins.

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The area of the shape is given as 24.52 cm2

We have two triangles and 1 rectangle

we have to find the area of the triangles first and then solve for the area of the rectangle.

The area of a triangle = 1/2bh

for first triangle

area = 1/2 * 2.7 * 3

= 4.05

for second triangle

height = 6.8 - 2.7

= 4.1

area = 1/2 * 3.4 * 4.1

= 6.97

For the rectangle

area = l * w

= 2.7 * 5 = 13.5

The area = 13. 5 + 6.97 + 4.05

= 24.52 cm2

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Answer:

Step-by-step explanation:

There are 5 cans left over after painting the walls.

Let x represent the maximum number of trims that can be painted

Since each trim requires 1/2 a can, x trims will take x/2 cans

This cannot exceed 5 cans so we get the inequality:
x/2 ≤  5

Multiplying by 2 both sides of inequality:
x  ≤ 10

In words:
Maximum number of trims that can be painted with 5 cans is 10

Number line attached

The number of new phone sold is 1 and the number of old phone sold is 5, if the sales for the day totals $625.

It is given that

A new phone sells for $250

and a used one sells for $75

Let number of new phone is sold = x

and number of old phone is sold = y

Here the total profit for the day is $625

So, 250x + 75y = 625

or 10x + 3y = 25

Now the value of x  and y satisfy the above equation, if we take x = 2 and more than 2, it is clear that it not satisfy the above equation. So we made a conclusion here x must be equal to 1 here.

Now, if x = 1

10(1) + 3y = 25

3y = 15

y = 5

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Answer:

Step-by-step explanation:

The factors of 96 are the integers that divide evenly into 96. To find the factors of 96, you can divide 96 by integers starting from 1 and increasing until you reach 96.

The factors of 96 are:

1, 2, 3, 4, 6, 8, 12, 16, 24, 32, 48, and 96.

These are all the integers that divide evenly into 96.

Answer:

In a box-and-whisker plot, the box represents the middle 50% of the data, with the median value marked by a vertical line inside the box. The "whiskers" extend from the box to the minimum and maximum values in the data set that are not considered outliers. Outliers are marked by individual points outside the whiskers.

To determine the percentage of data values that are greater than a given value, we need to look at the box-and-whisker plot to see where that value falls in the data set. If the value is outside the box, we know that at least 50% of the data values are either greater or less than that value, depending on whether the value is greater or less than the median. If the value is inside the box, we know that exactly 50% of the data values are greater and 50% are less than that value.

In the case of the value 72, we cannot determine the percentage of data values that are greater than it without knowing where it falls in the box-and-whisker plot. If it falls outside the box, we only know that at least 50% of the data values are greater than it. If it falls inside the box, we only know that exactly 50% of the data values are greater than it.

Step-by-step explanation:

Answer: Hello! I'm JK!.....

[tex]\frac{13}{75}[/tex]

Step-by-step explanation:

The range of the data is from 10 to 85.

[tex]\frac{85-72}{85-10}[/tex]  = [tex]\frac{13}{75}[/tex]

So your answer is [tex]\frac{13}{75}[/tex]

I really hope this helps you.  XoXoGoldenMaknae<3

The statement that best describes Jace's work is D. Jace is correct. The phrase more than suggests addition, so Jace showed that 14 plus a variable equals 56.

The concept that will be used here is word problem solution , then we need to interprete it then perform the neccessary simplification. . We were told that  the  sentence as an equation is 56 is 14 more than a number.

14+ = 56

then we can represent the number as X

then input into the equation as:

14+X = 56

x=56-14

=42

Therefore, option D is correct.

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complete question:

Jace wrote a sentence as an equation.

56 is 14 more than a number.

14+ = 56

Which statement best describes Jace's work?

Jace is not correct. The phrase more than suggests using the symbol > and Jace did not use that symbol.

Jace is not correct. He was correct to use addition, but the equation should be 56+ p = 14

Jace is not correct. The first number in the sentence is 56, so the equation should start with 56.

Jace is correct. The phrase more than suggests addition, so Jace showed that 14 plus a variable equals 56.

The numbers required in the word problem are

Let's call the numbers x and y

From the first equation, we have:

2x + y = 18

From the second equation, we have:

3x - y = 7

Solving simultaneously by substitution method

3x - y = 7

3x = 7 + y

x = (7 + y)/3

substitute this expression for x into the first equation:

2x + y = 18

2((7 + y)/3) + y = 18

14/3 + 3y/3 = 18

14/3 = 18 - 3y/3

14/3 = 18 - y

collecting like terms

y = 18 - 14/3

y = 18 - 4 2/3

y = 13 1/3

substituting to solve for y

x = (7 + y)/3

x = (7 + 13 1/3)/3

x = 10 4/3/3

x = 3 4/3

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Answer:

9.8

Step-by-step explanation:

two angles are given, 35 and 90. subtract these from 180 to get 55, the corresponding angle of x. we can then use law of sin to solve.

sin90/12 = sin55/x

sin55 * 12 = 9.8 , 9.8 = sin90x  , 9.8/sin90

=9.8

The limit as x approaches -2 is equal to 0. The correct option is B.

The limit of the function f(x) = x² - 4 / x + 2 as x approaches 2 is equal to 0. This can be evaluated by plugging in x = -2 into the expression for f(x) and simplifying it:-

The limit of the function f(x) = x² - 4 / x + 2 as x approaches -2 is equal to 4. This can be evaluated by plugging in x = -2 into the expression for f(x) and simplifying it:

f(-2) = (-2)² - 4 / -2 + 2 = 4 - 4 / 0 = 0

So, the limit as x approaches -2 is equal to 0.

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Answer:

Step-by-step explanation:

x=12wb+rb

probability of rb=5/11

12+11=23

5/23

rb=5/23

The required population of the city is approximately 44,44.44.

The percentage is the ratio of the composition of matter to the overall composition of matter multiplied by 100.

Here,
Let's use algebra to solve for the population of the city.

Let P be the population of the city. We know that 27% of the population is from 50 to 64 years old.

The number of people from 50 to 64 years old is 27% of the whole population, which can be expressed as,

0.27P

We can set up an equation to relate the number of people from 50 to 64 years old to the number of eligible voters,

0.27P = x

where x is the number of people from 50 to 64 years old who voted.

We can solve for P by multiplying both sides of the equation by the reciprocal of 0.27,

P = x / 0.27

Substituting x with 1200,

P = 12,00 / 0.27 = 4444.44

Therefore, the population of the city is approximately 44,44.44.

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Complete quesiton,

'if 12,00 people voted in the election how many were from 50 to 64 years which is 27% of the whole population, calculate the population of the city?

The constant 0.9965 is the rate of change per hour. Specifically, it represents the factor by which the quantity changes every hour. In this case, the factor is less than 1, indicating that the quantity is decreasing over time.

To see how this rate of change works, consider an example:

If the initial quantity is 930, after one hour the quantity will be:

f(1) = 930(0.9965) ≈ 926.20

That is, the quantity has decreased by a factor of 0.9965, or about 0.35%.

After two hours, the quantity will be:

f(2) = 930(0.9965)^2 ≈ 922.42

That is, the quantity has decreased by a factor of 0.9965 twice, or about 0.70%.

So the constant 0.9965 reveals the rate of change of the quantity, which is a decrease of about 0.35% per hour.

Answer:

[tex]y-8=3(x+2)[/tex]

Step-by-step explanation:

The equation of the line with slope [tex]m[/tex] passing through [tex](x_1, y_1)[/tex] in point-slope form is [tex]y-y_1=m(x-x_1)[/tex].

Reason:

-2 * 2 = -4

-2 + 2 = 0

These values are found through trial and error. The order of -2 and 2 doesn't matter.

We write the decimal as a fraction with the decimal value as the numerator and a power of 10 as the denominator, we get 1.1 as a fraction is 11/10.

To convert a decimal to a fraction, we write the decimal as a fraction with the decimal value as the numerator and a power of 10 as the denominator.

For 1.1, we can write it as:

1.1 = 11/10

Therefore, 1.1 as a fraction is 11/10.

Decimal numbers are a way to represent numbers that are not whole numbers or integers. A decimal number consists of two parts: a whole number part and a fractional part, separated by a decimal point. To convert a decimal number to a fraction, we need to write the decimal as a fraction with the decimal value as the numerator and a power of 10 as the denominator.

In the case of 1.1, we can write it as a fraction with a numerator of 11 (the decimal value) and a denominator of 10 (the power of 10 that corresponds to the first decimal place). We can simplify this fraction by dividing both the numerator and denominator by their greatest common factor, which is 1 in this case. So the final fraction is 11/10.

Converting decimals to fractions is a useful skill in many areas of math, science, and engineering. It allows us to represent numbers in different ways and to perform calculations that involve fractions and decimals. Knowing how to convert between these two types of numbers can also help us understand the relationship between them and how they are used in real-world applications.

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The simplified form of the expression 2( ³√64 + √25 ) is 18.

Given the expression in the question;

2( ³√64 + √25 )

First, simplify each term.

2( ³√64 + √25 )

Rewrite 64 as 4³

2( ³√( 4³ ) + √25 )

Also, rewrite 25 as 5²

2( ³√( 4³ ) + √( 5² ) )

Now, pull the terms out of the radical, assuming positive real numbers.

Take the cube root of 4³

2( 4 + √( 5² ) )

Take the square root of 5²

2( 4 +  5  )

Add 4 and 5

2( 9 )

Multiply 2 and 9

18

Therefore, the simplified form is 18.

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The airline fees was $41 each

The over the weight limit of the bag is 3 pounds

Number pounds of Natalie's bag = x

Airline 1:

29 + 4x

Airline 2:

23 + 6x

Equate airlines 1 and 2

29 + 4x = 23 + 6x

collect like terms

29 - 23 = 6x - 4x

6 = 2x

divide both sides by 2

x = 6/2

x = 3 pounds

Hence,

Airline 1:

29 + 4x

= 29 + 4(3)

= 29 + 12

= $41

Airline 2:

23 + 6x

= 23 + 6(3)

= 23 + 18

= $41

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Answer: A: Yes. The input and output are both feasible.

Step-by-step explanation:

To determine whether the solution (29.5, $212.50) makes sense for the function c(x), we need to evaluate whether the input and output are feasible and make sense in the context of the problem.

The input is given as 29.5, which represents the number of students. It is unclear from the information given whether a non-integer value for the number of students is feasible or not. However, if we assume that 29.5 students means 29 students and 1 child attending the carnival, then the input of 29.5 students is feasible.

The output is given as $212.50, which represents the amount of money spent at the carnival. It is reasonable to assume that a school group on a field trip would spend a certain amount of money at a carnival, and $212.50 is not an unreasonable amount to spend.

Therefore, based on the information given, we can conclude that the solution (29.5, $212.50) makes sense for the function c(x). So the answer is A: Yes. The input and output are both feasible.

Using a system of equations, Company A charges 25 + 0.06x more than Company B.

A system of equations involves two or more equations whose solutions are found concurrently or at the same time.

A system of equations is also known as simultaneous equations.

                                   Company A     Company B    Difference

Fixed charge                  $75.50             $50.50        $25 ($75.50 - $50.50)

Variable charge per mile $0.16               $0.10         $0.06 ($0.16 - $0.10)

Let the number of miles covered = x

The total cost of Company A (Equation 1) = 75.50 + 0.16

The total cost of Company B (Equation 2) = 50.50 + 0.1

The difference in total cost for x miles      = 25 + 0.06w

Thus, for each mile of travel that the family rents the car, Company A will charge more than Company B by exactly 25 + 0.06w.

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The extraneous values of the rational function are given as follows:

x = 7 and x = -3.

The extraneous values of a function are the values of x that are outside the function's domain.

In this problem, we have a rational function, hence the extraneous values are the values that make the denominators zero, as follows:

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The Fundamental Counting Principle, choosing from 3 sizes of bottled water and from distilled, filtered, or spring water is 9 ways.

The Basic Counting Principle is a technique used in mathematics, more especially in probability theory and combinatorics, to determine how many combinations of options, items, or outcomes are possible. The Rule of Multiplication, the Product Rule, the Multiplication Rule, and the Basic Counting Rule are some of its alternate names.

It has a connection to the Sum Rule, often known as the Rule of Sum, which is a fundamental counting method used to calculate probabilities.

According to the Basic Counting Principle, if a decision or event has a potential result or set of options, and a different decision or event has b possible outcomes or choices, then the sum of all the unique combinations of outcomes for the two is ab.

Using the fundamental counting principle, if an event occurs in m ways and another event occurs in n ways then the total occurrence is given as (m)(n).

Given that,

m = 3 for choosing between sizes.

n = 3 for distilled, filtered, or spring water.

Thus, the total number of occurrence is given as:

T = (3) (3) = 9

Hence, the Fundamental Counting Principle, choosing from 3 sizes of bottled water and from distilled, filtered, or spring water is 9 ways.

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Answer: 5/8

Step-by-step explanation:

We can make the fractions easier to read by changing 1/4 to 2/8, simply by multiplying the fraction by 2/2.

2/8 : 7/8

now you can see how much more George ate.

7/8 - 2/8 = 5/8

George ate 5/8 more than Grace.

Answer:10[tex]\sqrt{7}[/tex]

Step-by-step explanation:

Answer:

Step-by-step explanation:

reflict means to put the point in the other way but in the same point.

The graph of the system is added as an attachment

From the question, we have the following parameters that can be used in our computation:

f(x) = 2x + 8 for x < -1

      -2x + 7 for x > 3

The above system is a piece wise fucntion that include

Two linear equations

Next, we plot the graph

See attachment for the graph of the equation

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