Algebraic expressions involving radicals are known as radical expressions. The root of an algebraic expression makes up the radical expressions (number, variables, or combination of both).
Three conditions must be satisfied for a radical expression to be in its simplest form, that is, first, No radicands other than 1 have perfect square factors.
Secondly, there isn't a fraction in any radicand and finally, the fraction's denominator has no radicals.
A radical expression can be reduced to a simple form by first factoring it into primes, then finding the radical's index, and finally moving the group of numbers either inside or outside the radical.
The expressions both inside and outside the radical can then be made simpler by adding up all the numbers inside the radical.
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Assertion : If both zeros of the quadratic polynomial x k2 2 x 2 − + are equal in magnitude but opposite in sign then
value of k is ½.
Reason : Sum of zeros of a quadratic polynomial ax bx c 2 + + is a
-b
The assertion that both zeros of the quadratic polynomial x² - 2kx +2 in magnitude but opposite in sign then value of k is 1/2 is false.
How to obtain the sum of the roots of the quadratic function?The standard definition of a quadratic function is given as follows:
y = ax² + bx + c.
The sum of the roots of a quadratic function is given as follows:
S = -b/a.
The equation in this problem is given as follows:
x² - 2kx +2
Hence the parameters are:
a = 1, b = -2k.
If the roots are equal in magnitude but the opposite sign, the sum of the roots is of:
0.
Hence:
-b/a = 0.
b = 0.
Meaning that the value of k is obtained as follows:
-2k = 0.
k = 0.
And the statement is false.
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What is the range of f/x )= Logbx?.
The range of the function f(x) = logbx is {y | y > 0}.
The range of the function f(x) = logbx is the set of all possible output values (or "y-values") that the function can produce. In this case, the function is a logarithm in base b, where b is a positive real number, and x is the argument of the logarithm, also a positive real number.
For a logarithm function, the range is all real numbers y such that y > 0, because the logarithm of a positive number is always positive. The domain of the function is all positive real numbers x because the logarithm of a positive number is always defined.
So, the range of the function f(x) = logbx is {y | y > 0}.
It's worth noting that the logarithm function is defined only for positive real numbers x, hence the domain of this function is x > 0 and the range is y > 0.
Also, the base of the logarithm b is a positive real number, different than 1, this ensures that the logarithm function is defined for all x > 0, for b = 1 it would not be defined for x = 1.
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What is meant by a substitution reaction give an example with equation of the substitution reaction of an alkene?.
A substitution reaction is one in which one (or more) hydrogen atoms from a hydrocarbon are swapped out for another atom (like chlorine).
Example: Substitution reaction of methane with chlorine:
Methane reacts with chlorine in the presence of sunlight to form chloromethane and hydrogen chloride.
[tex]CH_{4} + Cl_{2}[/tex] ---------> [tex]CH_{3} Cl + HCl[/tex]
When one functional group in a chemical molecule is replaced by another functional group, the chemical reaction is known as a substitution reaction. In organic chemistry, substitution reactions are of utmost significance.
An alkene is a hydrocarbon with a carbon-carbon double bond in organic chemistry. Alkene, which refers to any hydrocarbon with one or more double bonds, is frequently used as a synonym for olefin.
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How do you find the height of a 45 45 90 triangle?.
The height of a right triangle can be calculated, given the length of the base, and the height of a right triangle formula can be calculated using the Pythagoras theorem as, (Hypotenuse)² = (Height)² + (Base)².
A triangle is a polygon with three edges and 3 vertices. The terminology for categorizing triangles is more than two thousand years vintage, having been defined on the first actual page of Euclid's elements.
In Euclidean geometry, any three factors, while non-collinear, decide a completely unique triangle and simultaneously, a unique plane (i.e. a -dimensional Euclidean space). In different words, there's best one plane that consists of that triangle, and every triangle is contained in some plane. If the entire geometry is only the Euclidean plane, there is the best one plane and all triangles are contained in it; however, in higher-dimensional Euclidean spaces, this is not real. this text is ready triangles in Euclidean geometry, and in particular, the Euclidean plane, except wherein otherwise stated.
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A mattress store is having a sale. All mattresses are 30% off. Nate wants to know the sale price of a mattress that is regularly $1,000.
How much is the discount? Enter the amount in the table.
Answer:
$700
Step-by-step explanation:
1. We see the original price is $1,000.
2. Change 30% to a decimal
3. 30/100=0.30
4. Multiply the original cost of the item(mattress) by the percentage.
5. 0.3×1,000=300
6. $300 is the amount discounted.
7. For final price with discount: Take its original price($1,000) and subtract the discount from the original price
8. $1,000-$300=$700
Answer:
$300
Step-by-step explanation:
The Stock Market Alexander is a stockbroker. He earns 14% commission each week. Last week, he sold $7,100 worth of stocks. How much did he make last week in commission? If he averages that same amount each week, how much did he make in commission in 2011?
The amount he earned last week in commission is $994.
The commission he earned in 2011 is $51,688
What is the commission?The commission earned is a fraction of the total sales of stocks.
Commission earned = amount of sales x percentage commission
14% x $7,100
= 14/100 x $7,100
= 0.14 x $7,100
= $994
Amount she earned in commission in a year = number of weeks in a year x commission earned per week
= $994 x 52 = $51,688
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If a rectangle' perimeter i 999 ft, give at leat one poible meaurement of it length and width
The possible measurement of Length and Width will be 249.75 feet each.
What is Rectangle?
The internal angles of a rectangle, which has four sides, are all exactly 90 degrees. At each corner or vertex, the two sides come together at a straight angle. The rectangle differs from a square because its two opposite sides are of equal length.
The whole distance that a rectangle's boundaries, or its sides, the cover is known as its perimeter. Given that a rectangle has four sides, the perimeter of the rectangle will be equal to the sum of its four sides. Since a rectangle's perimeter is a linear measurement, the unit will be in meters, centimeters, inches, feet, etc.
Perimeter of a rectangle = 2(Length + Width) square units
Area of a rectangle = Length × Width = a × b
In our question, it is given that the Perimeter is 999 ft.
Putting this in our Perimeter formula.
2(Length + width) = 999
Length + width = 499.5
Therefore our possible measurement of Length and Width will be 249.75 feet each.
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Jona i given a tet quetion aking him to how that the um of two odd integer i alway an even integer. Which choice i the bet proof of hi conjecture?
B. Let (2a + 1) and (2b + 1) are different odd numbers. sum of these odd number is 2(a+ b +1) is a multiple of 2.
Whole numbers in mathematics can be divided into even and odd numbers. There isn't a single number that can be both even and odd, making odd and even numbers distinct sets of numbers. A number can be even or odd. Odd numbers are ones that are not entirely divisible by two, whereas even numbers are entirely divisible by two.
Let (2a + 1) and (2b + 1) are different odd numbers. the sum of these odd numbers is 2(a+b+1) is a multiple of 2. furthermore, let 2a and 2b be two even numbers so the sum of those two will also be an even number. Therefore the summation of two odd numbers is always even.
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The complete question is:
Jona is given a test question asking him how that the um of two odd integers is always an even integer. Which choice is the best proof of his conjecture?
A) Every time you add two odd numbers, the sum is an even number.
B) Let a and b both represent odd numbers and a+b be an even number. Therefore, a+b=b+a which shows that the sum of two odd numbers is even.
C) Look at these different examples: 5+3=8, 15+7=22, 21+13=34. So the sum of two odd numbers must be even.
D) Let (2a+1) be one odd number and let (2b+1) be the other odd number. (2a+1)+ (2b+1) = 2a+2b+2 = 2(a+b+1). Because 2(a+b+1) is a multiple of 2,it is an even number.
Simplify this expression:
13x+2y-10x-7y
Answer:
3x-5y
Step-by-step explanation:
⇒13x+2y-10x-7y
arrange the like terms together
⇒13x-10x-7y+2y
⇒3x-5y ...answer
hope that helps...
To make 4 cups of rice, Iain needs 5 cups of water. To make 16 cups of rice, he needs 20 cups of water. Write this as a proportion. 4/?=?/?
Answer: Don't use this if you are taking a test, please!
Step-by-step explanation:
Hello! This is actually easier than you think! All you have to do is make what is already written as a proportion. You don't have to solve anything! You are already given one number which represents the first thing you were told, 4 cups of rice. For 4 cups of rice, 5 cups of water are needed. As a proportion, you would write 4/5. For the second proportion you would write the 16 cups of rice per 20 cups of water as 16/20. The answer is 4/5 = 16/20. Hope this helps with your homework!
Avery is a songwriter who collects royalties on her songs whenever they are played in a commercial or a movie. Avery will earn $40 every time one of her songs is played in a commercial and she will earn $140 every time one of her songs is played in a movie. Avery earned a total of $480 in royalties on 7 commercials and movies. Determine the number of commercials and the number of movies on which Avery's songs were played.
The number of commercials in which Avery's songs were played is 5.
The number of movies in which Avery's songs were played is 5.
What is an equation?An equation is a mathematical statement that is made up of two expressions connected by an equal sign.
Example:
2x + 3 = 9 is an equation.
We have,
The number of times the song played in commercials = x
The number of times the song played in a movie = y
Now,
Avery will earn $40 every time one of her songs is played in a commercial and earn $140 every time one of her songs is played in a movie.
We can make two equations.
x + y = 7 ____(1)
x = 7 - y _____(2)
40x + 140y = 480 _____(3)
Substituting (2) in (3) we get,
40(7 - y) + 140y = 480
280 - 40y + 140y = 480
100y = 480 - 280
100y = 200
y = 2
Now,
x = 7 - 2
x = 5
Thus,'
The number of commercials and the number of movies in which Avery's songs were played are 5 and 2.
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A giant balloon i dropped from a plane landing on tahir and zach at point w(42,-15) and ending two friend flying off an equal ditance in oppoite direction. Of tahir end up at coordinate t(-18,36) where would we expect to find zack
Zack is expected to be at (-12,-10.5) by finding the midpoint of line segment WT and reflecting T over it.
What is coordinate ?
A coordinate is a set of numerical values that are used to identify the position of a point in a specific space, such as a two-dimensional plane or a three-dimensional space. In two-dimensional space, coordinates are commonly represented by an ordered pair (x, y) or (x, y, z) in three-dimensional space.
The problem states that the balloon lands at point W(42,-15) and Tahir ends up at point T(-18,36). We know that Tahir and Zack are flying off in opposite directions from point W at equal distances. To find the point where Zack is expected to land, we can use the midpoint formula which states that the midpoint of a line segment is the average of the x-coordinates and the average of the y-coordinates.
We can find the midpoint of line segment WT by taking the average of the x-coordinates and y-coordinates:
x_midpoint = (x1 + x2) / 2 = (42 + -18) / 2 = 12
y_midpoint = (y1 + y2) / 2 = (-15 + 36) / 2 = 10.5
So the midpoint of line segment WT is (12,10.5). Since Zack is flying off in the opposite direction from Tahir at the same distance, we can expect Zack to end up at the point which is the reflection of point T over the midpoint of line segment WT.
This point would be (-12,-10.5)
Therefore , Zack is expected to be at (-12,-10.5) by finding the midpoint of line segment WT and reflecting T over it.
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How many three digit numbers are divisible by 7?.
There are 128 three digit numbers divisible by 7
We know that the divisibility rule of 7 is:
If the difference between twice the unit digit of the given number and the remaining part of the given number is multiple of 7 or the difference is 0, then the number is divisible by 7.
the range of three digit numbers is 100 to 999
The first three-digit number divisible by 7 is 105 and the last three-digit number divisible by 7 is 994.
The numbers divisible by 7 must be: 105, 112, . . ., 994
We can observe that the above series represents the arithmetic progression with common difference d = 7, the first term a = 105 and the last term a_n = 994
We know that the formula for the n-th term of AP is:
a_n = a1 + (n - 1)d
994 = 105 + (n - 1)7
7n - 7 = 889
7n = 889 + 7
7n = 896
n = 128
Thus, there are 128 three digit numbers.
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The house Trevor's family lives in has 6 66 people (including Trevor) and 3 33 bathrooms. In the past month, each person showered for an average of 480 480480 minutes and used an average 72 7272 liters of shower water (over the entire month). Water costs 0.20 0.200, point, 20 dollars per liter. How much money did Trevor's family spend, in total, on shower water in the past month? dollars
The total amount of money which Trevor's family spend, in total, on shower water in the past month is equal to $86.4.
How to calculate the required amount of money?Based on the information provided about the house Trevor's family lives in, we have the following important parameters;
The total number of persons in Trevor's family house is equal to 6The total volume of water which each person showered with during the month is equal to 72 liters of shower water (over the entire month). The cost of water is equal to $0.20 per liter.Next, we would calculate the total volume of water that is used to shower by six (6) persons as follows;
Total volume of water for 6 persons = 72 × 6
Total volume of water for 6 persons = 432 liters.
Furthermore, the total amount of money that was spent on shower water by Trevor's family in the previous month is given by;
Total amount of money = total volume of water used for shower × cost per liter of water
Total amount of money = 432 × 0.2
Total amount of money = $86.4.
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Can a graphing calculator be used as a scientific calculator?.
Yes , the graphing calculator can be used as a scientific calculator.
As given in the question,
Scientific calculator can not be used as a graphing calculator.
But graphing calculator can be used as a scientific calculator.
Calculators help us to make our calculation easy and save time.Scientific calculator help us to do the calculation of mathematics based on algebra and Engineering mathematics.Scientific calculator also help us to do statistics calculation.Graphing calculator have all the functions of scientific calculator with more addition of plotting coordinates of the given graph function.Therefore, yes the graphing calculator can be used in place of scientific calculator.
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(Theoretical Probability LC)
When rolling a fair, eight-sided number cube, determine P(number greater than 6).
A: 0.25
B: 0.50
C: 0.66
D: 0.75
Answer:0.25
Step-by-step explanation:
i passed the test
The perimeter of a train ticket is 60 centimeters. It is 17 centimeters long. How tall is it?
If the perimeter of the train ticket is 60 centimeters and it's length is 17 centimeter long it's height is 13 centimeter
What is perimeter of a rectangle?A perimeter is a closed path that encompasses, surrounds, or outlines either a two dimensional shape or a one-dimensional length.
To find the perimeter of a rectangle we add all this sides together. This means that the perimeter of a rectangle is given as ;
P = l+l+w+w
P= 2(l+w)
60 = 2( 17+w)
60 = 2( 17+w)
17+w= 60/2
17+ w = 30
w = 30-17
w = 13cm
Therefore the height of the train ticket is 13cm
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What is the inverse of the logarithmic function f/x log9 X?.
The inverse of the logarithmic function f(x) = [tex]\log_{e}x[/tex] is [tex]e^{x}[/tex].
In the given question, we have to find the inverse of the logarithmic function f(x) = [tex]\log_{e}x[/tex].
The given function is f(x) = [tex]\log_{e}x[/tex].
The features of logarithms will simply follow from our understanding of exponents if the logarithm is viewed as the inverse of the exponential function.
Now let, y = [tex]\log_{e}x[/tex]
Now x = log y
Then, y = [tex]e^{x}[/tex]
So, f(x) = [tex]e^{x}[/tex]
As a result, [tex]e^{x}[/tex] is the logarithmic function's inverse, where f(x) = [tex]\log_{e}x[/tex].
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The complete question is:
What is the inverse of the logarithmic function f(x) = [tex]\log_{e}x[/tex]?
Select the number below that is irrational.
O 5462.263
O √19
O 13/24
O 3.33333
The number that is irrational is √19.
An irrational number is a number that cannot be expressed as a ratio of two integers (a fraction) and it cannot be represented as a repeating or terminating decimal. The square root of 19 is an irrational number because it cannot be expressed as a ratio of two integers. It is non-repeating, non-terminating decimal that goes on forever, so it cannot be represented as a decimal.
Please Help me with this question ASAP
The credit utilisation ratio to the nearest percentage is found as 17%.
Define the term credit utilisation ratio?The proportion of a borrower's total credit available that is presently being used is known as the credit utilisation ratio. When determining a borrower's credit score, credit reporting companies take into account the credit utilisation ratio.Steps:
You must collect all of the credit cards in order to determine your credit utilisation ratio. Add the credit limits after adding up all the remaining balance. Your credit usage ratio is expressed as a percentage by taking the entire amounts, dividing them but by total credit limit, and thereafter multiplying the result by 100.For the stated question:
Balance = $180 + $230 = $410.
credit limits = $1500 + $1000 = $2500.
Credit utilisation ratio = Balance / credit limits
Credit utilisation ratio = 410 / 2500 * 100 = 16.4%
Thus, the credit utilisation ratio to the nearest percentage is found as 17%.
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The lengths of salamanders have a normal distribution with mean 15cm, and standard deviation 2cm.
In a collection of 200 salamanders, about how many have length greater than 16.5cm?
Answer:
2cm
Step-by-step explanation:
103. 68 in. 2
2cmo salamanders
What is the difference between a frequency histogram and a relative frequency histogram?
The vertical axis of a relative frequency histogram employs relative or proportional frequency rather than plain frequency, which is the only difference between it and a frequency histogram.
Explain the difference between frequency and relative frequency?Frequency can be expressed most simply in absolute terms. The number of times a specific value for a variable (data item) has been seen to occur in relation to the overall number of values for that variable is referred to as the relative frequency.Histogram is a term from the quality glossary. The frequency distribution of a collection of data reveals how frequently each unique value appears. The most typical graph for displaying frequency distributions is a histogram. Although it closely resembles a bar chart, there are significant variances.Relative frequencies, on the other hand, do not employ raw counts. Instead, they use percentages, proportions, or fractions to compare the count for one type of event to the entire number of events. The word "relative" refers to a specific tally in relation to the overall number, which is where it is used.Learn more about frequency histogram refer to :
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which of the following number lies between 5595 and 5610 a=5509 b=5590 c=5605
Answer:
The answer is c which is=5605
Answer:
C.
Step-by-step explanation:
Since there's no sequence of numbers, we assume that the number that lies between 5,595 and 5,610 is the mean of the 2 numbers.
However, if we calculate its mean, we can have 5,602.5 and it doesn't appear on any options. Thus, we will choose the nearest value from those options.
What is the solution to the system of linear equations y =- 5x 3?.
The Solution to the system of linear equations given as "y = -5x+3" and "y= 1" is that x = 2/5 and y = 1 .
Linear equations are the equations that are made up of Variables and Constants which are joined by mathematical operators .
The system of linear equations is given to us is: "y = -5x+3" and "y= 1"
Substituting the values of y=1 in y = -5x+3 ,
we get ;
⇒ 1 = -5x + 3 ;
⇒ -5x = 1 - 3 ;
⇒ -5x = -2 ;
⇒ x = (-2)/(-5) ⇒ x = 2/5 .
Therefore , solution can be written as : x = 2/5 and y = 1 .
The given question is incomplete , the complete question is
What is the solution to the system of linear equations y = -5x+3 and y = 1 ?
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Taylor and Raimi each earn an hourly wage. Taylor earn $308 for working 14 hour. Raimi earn $288 for working 12 hour. Which tatement i true?
The statement that is true is that Taylor earned more than Raimi for the hours worked.
Taylor earned $308 for working 14 hours, which is equivalent to $22 per hour. Raimi earned $288 for working 12 hours, which is equivalent to $24 per hour. Therefore, Taylor earned a higher hourly wage than Raimi.
This statement can be further verified by computing the total wages earned by both Taylor and Raimi for the same amount of hours: if both Taylor and Raimi worked 14 hours, Taylor would have earned more than Raimi by $20. To demonstrate, Taylor would have earned $308 for 14 hours and Raimi would have earned $336 for 14 hours.
Therefore, Taylor would have earned the higher wage.
In conclusion, Taylor earned more than Raimi for the hours worked, verifying the statement that Taylor earned more than Raimi.
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Write each calculation as a single power. 8 to the power of 5 Multiply by 8 to the power of 4 REMEMBER TO WRITE THE ANSWER AS A SINGLE POWER PLS HELP
Answer:
[tex]8^{9}[/tex]
Step-by-step explanation:
[tex]8^{5}[/tex] x [tex]8^{4}[/tex]
In expanded form this would be
8·8·8·8·8·8·8·8·8
The product of power principle tells us that when we are multiplying powers with the same bases, we add the exponents.
please help me i dont know this
When the coin is tossed and then dice is rolled, the possible outcomes are 12.
Define the term sample space/possible outcome?All potential results make up as sample space of such a random experiment.A subset of such sample space is an event connected to a random experiment.Any outcome's probability is a value between 0 and 1. The sum of all the outcomes' probabilities is one.The total likelihood of all outcomes in an event A is the probability of that event.For the stated question.
Sample space for tossing coin. { H, T}.
Sample space for rolling dice. { 1, 2, 3, 4, 5, 6 }
Then,
Total sample = { H1, H2, H3,H4, H5, H6, T1, T2, T3, T4, T5, T6}
Thus, the total possible outcomes for tossing coin and rolling dice is 12.
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Three equivalent ratios are shown on the graph. On a coordinate plane, points (1, 0.5), (2, 1), and (5, 2.5) are plotted. Starting on one of the plotted points, how can you plot more equivalent ratio points? Move up one and right one and one-half. Move up one and one-half and right three. Move up two and right three. Move up one and right two
To plot more equivalent ratios on the coordinate plane, move up one and right one and half.
What are equivalent ratios?Equivalent ratios are the ratios that are the same when we compare them. Two or more ratios can be compared with each other to check whether they are equivalent or not.
Given that, Three equivalent ratios are shown on the graph. On a coordinate plane, points (1, 0.5), (2, 1), and (5, 2.5) are plotted.
As we see, the x-coordinates are 1, 2, 5 and y-coordinates are 0.5, 1, 2.5
We see that, x-coordinates are increasing by 1, and y-coordinates are increasing by a half,
Therefore, to plot more equivalent coordinates, we will move up 1 on y-axis and a half right on x-axis,
Hence, to plot more equivalent ratios on the coordinate plane, move up one and right one and half.
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Doug etimate that hi occer team will win 7 game thi year. The team actually win 10 game. What i the percent error of Doug' etimate? Round the anwer to the nearet tenth percent if neceary
Doug's estimate was incorrect by 30% because probability his team won 10 games instead of the 7 he predicted.
Doug estimated 7 wins
Actual number of wins was 10
Error = (Actual - Estimate) / Estimate x 100
Error = (10 - 7) / 7 x 100
Error = 3 / 7 x 100
Error = 30%
Doug estimated that his soccer team would win 7 games this year. However, the team ended up winning 10 games in total. This means that Doug's estimate was incorrect by 30%. To calculate this percent error, we used the following formula: Error = (Actual - Estimate) / Estimate x 100. In this case, the actual number of wins was 10 and the estimate was 7. Plugging these numbers into the formula, we got that the error was 3/7 x 100, which is equal to 30%. This means that Doug's estimate was off by 30%. This is a significant overestimate since the team ended up winning 3 more games than what Doug had predicted.
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z+5=55(5^z)
How to solve this.
The given equation's answer is [tex]$z+5=55\left(5^z\right):$[/tex] [tex]$z=-\frac{W_{-1}\left(-\frac{11 \ln (5)}{625}\right)+5 \ln (5)}{\ln (5)}$[/tex], [tex]$z=-\frac{\mathrm{W}_0\left(-\frac{11 \ln (5)}{625}\right)+5 \ln (5)}{\ln (5)}$[/tex]
What is the solution of the given equation ?[tex]$z+5=55\left(5^z\right):$[/tex] [tex]$z=-\frac{W_{-1}\left(-\frac{11 \ln (5)}{625}\right)+5 \ln (5)}{\ln (5)}$[/tex], [tex]$z=-\frac{\mathrm{W}_0\left(-\frac{11 \ln (5)}{625}\right)+5 \ln (5)}{\ln (5)}$[/tex]
[tex](Decimal: $z=-1.75934 \ldots, z=-4.98187 \ldots$ )[/tex]
[tex]$$z+5=55\left(5^z\right)$$[/tex]
Prepare [tex]$z+5=55\left(5^z\right)$[/tex] Lambert form: [tex]$(z+5) e^{-\ln (5) z}=55$[/tex]
the equation once more using [tex]$(-z-5) \ln (5)=u$[/tex] and [tex]z=-\frac{u+5 \ln (5)}{\ln (5)}$[/tex]
[tex]$$\left(\left(-\frac{u+5 \ln (5)}{\ln (5)}\right)+5\right) e^{-\ln (5)\left(-\frac{u+5 \ln (5)}{\ln (5)}\right)}=55$$[/tex]
[tex]$$e^{u+5 \ln (5)}\left(-\frac{u+5 \ln (5)}{\ln (5)}+5\right)=55$$[/tex]
Rewrite [tex]$e^{u+5 \ln (5)}\left(-\frac{u+5 \ln (5)}{\ln (5)}+5\right)=55$[/tex]in Lambert form: [tex]$e^u u=-\frac{11 \ln (5)}{625}$[/tex]
Solve [tex]$e^u u=-\frac{11 \ln (5)}{625}: \quad u=\mathrm{W}_{-1}\left(-\frac{11 \ln (5)}{625}\right), u=\mathrm{W}_0\left(-\frac{11 \ln (5)}{625}\right)$$$u=\mathrm{W}_{-1}\left(-\frac{11 \ln (5)}{625}\right), u=\mathrm{w}_0\left(-\frac{11 \ln (5)}{625}\right)$$[/tex]
Substitute back [tex]$u=(-z-5) \ln (5)$[/tex] , solve for z
[tex]$$\begin{aligned}& \text { Solve }(-z-5) \ln (5)=W_{-1}\left(-\frac{11 \ln (5)}{625}\right): \quad z=-\frac{W_{-1}\left(-\frac{11 \ln (5)}{625}\right)+5 \ln (5)}{\ln (5)} \\& \text { Solve }(-z-5) \ln (5)=W_0\left(-\frac{11 \ln (5)}{625}\right): z=-\frac{W_0\left(-\frac{11 \ln (5)}{625}\right)+5 \ln (5)}{\ln (5)} \\& z=-\frac{W_{-1}\left(-\frac{11 \ln (5)}{625}\right)+5 \ln (5)}{\ln (5)}, z=-\frac{W_0\left(-\frac{11 \ln (5)}{625}\right)+5 \ln (5)}{\ln (5)}\end{aligned}$$[/tex]
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