Find the derivative: since the function is f(x) = (1-3x)^(1/3), the derivative is f'(x)=1/3(1-3x)^(-2/3) *(-3) =-(1-3x)^(-2/3) =evenutally we get an answer but thats not what im confused about where did the *(-3) come from? what does it mean? it made me get a different answer and i don't understand where it came from? please help asap

Answers

Answer 1

The derivative of f(x) = (1 - 3x)^(1/3) is  f'(x) = - (1 - 3x)^(-2/3). The  *(-3) comes from the derivative of (1 - 3x) with respect to x, due to the chain rule.

Chain rule:  dy/dx = dy/du. du/dx

Suppose we have f(g(x)), then its derivative is: f'(g(x)) . g'(x)

In the problem:

f(x) = (1 - 3x)^(1/3)

Let u = 1 - 3x

then f(u) = u^(1/3)

and f'(u) = 1/3 u^(-2/3)

That part only the derivative of f with respect to u. We still need to take the derivative of u with respect to x. That is:

u(x) = 1 - 3x

u'(x) = -3

Hence, the complete derivative is:

f'(x) = f'(u) . u'(x)

     = 1/3 u^(-2/3) . (-3)

     = -  u^(-2/3)    ( substitute back u = 1 - 3x )

     = - (1 - 3x)^(-2/3)

Hence the *(-3) comes from the derivative of (1 - 3x) with respect to x, due to the chain rule.

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Related Questions

A cuboid is of dimensions 60 cm × 54 cm × 30 cm. How many small cubes with side 12 cm can be placed in the given cuboid?

Answers

56 small cubes can be placed in the given cuboid that has the given dimensions.

What is the Volume of a Cuboid?

A cuboid's volume can be calculated using the formula below:

Volume of a cuboid = (length)(width)(height).

How to Find the Volume of a Cube?

A cube has equal height, length, and width. Therefore, the volume of a cube is calculated as:

Volume of a cube = (side length)³.

The number of small cubes that can be placed in the given cuboid = volume of the cuboid/volume of one small cube.

Find the volume of the cuboid:

Length = 60 cm

Height = 54 cm

Width = 30 cm

Volume of the cuboid = (60)(54)(30)

Volume of the cuboid = 97,200 cm³

Find the volume of one cube:

Side length of one small cube = 12 cm

Volume of one small cube = (12)³

Volume of one small cube = 1,728 cm³.

Number of small cubes that will be placed in the cuboid = 97,200/1,728 ≈ 56 cubes.

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The perimeter of an equilateral triangle is 17 inches more than the perimeter of a square, and the side of the triangle is 8 inches longer than the side of the square. Find the side of the triangle ( Hint: An equilateral triangle has three sides the same length.)​

Answers

The side of the triangle is 15 inches.

How to find the perimeter of a triangle and square?

The perimeter of an equilateral triangle is 17 inches more than the perimeter of a square, and the side of the triangle is 8 inches longer than the side of the square.

Hence,

let

perimeter of the square = x inches

perimeter of an equilateral triangle = x + 17

Let

side of the square = y

side of the triangle = 8 + y

Therefore,

Perimeter of the square

x = y + y + y + y

x = 4y

Perimeter of the triangle

x + 17 = 8 + y + 8 + y + 8 + y

x + 17 = 24 + 3y

x - 3y = 7

Combine the equation,

x = 4y

x - 3y = 7

Therefore,

4y - 3y = 7

y = 7

x = 4(7)

x = 28

Therefore,

side of the triangle = 8 + y = 8 + 7 = 15 inches

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What is the value of the expression −42 − 19 − (−24)?

−85
−61
−47
−37

Answers

Answer:

–37

Step-by-step explanation:

–42 – 19 – (–24)

–42 – 19 + 24

–61 + 24

= –37

Please help me with my trig
Question 28&29

Answers

The expressions related to the roots of each quadratic equation are listed below:

Case 1 - x² + 3 · x - 9 = 0

Roots: α = - 3 / 2 - 3√5 / 2, β = - 3 / 2 + 3√5 / 2

a) α + β = - 3

b) α² + β² = 27

c) α³ + β³ = - 108

Case 2 - x² + 4 · x - 2 = 0

Roots: α = - 2 - √6, β = - 2 + √6

a) α + β = - 4

b) α² + β² = 20

c) α³ + β³ = - 88

How to determine the roots of quadratic equations and calculate the values of equations based on these roots

This question consists in two parts, find the roots by factoring the quadratic equations by algebra properties and determine the expressions based on the two roots of each expression. A root is a value of x such that the polynomial is equal to zero.

Case 1

x² + 3 · x - 9 = 0

x² + 3 · x = 9

x² + 3 · x + 9 / 4 = 9 + 9 / 4

(x + 3 / 2)² = 45 / 4

x + 3 / 2 = ± 3√5 / 2

x = - 3 / 2 ± 3√5 / 2

The roots are α = - 3 / 2 - 3√5 / 2 and β = - 3 / 2 + 3√5 / 2 and the following expressions are evaluated:

a) α + β = - 3

b) α² + β² = (- 3 / 2 - 3√5 / 2)² + (- 3 / 2 + 3√5 / 2)² = 27

c) α³ + β³ = (- 3 / 2 - 3√5 / 2)³ + (- 3 / 2 + 3√5 / 2)³ = - 108

Case 2

x² + 4 · x - 2 = 0

x² + 4 · x + 4 = 6

(x + 2)² = 6

x + 2 = ± √6

x = - 2 ± √6

The roots are α = - 2 - √6 and β = - 2 + √6 and the following expressions are evaluated:

a) α + β = - 4

b) α² + β² = (- 2 - √6)² + (- 2 + √6)² = 20

c) α³ + β³ = (- 2 - √6)³ + (- 2 + √6)³ = - 88

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you are given two reports estimating the mean increase in monthly household spending on gasoline over the past six months: report 1: based on a simple random sample of 14 households, we are 98% confident that the mean increase in monthly household spending on gasoline over the past six months is between $64.35 and $83.59. report 2: based on a simple random sample of 53 households, we are 95% confident that the mean increase in monthly household spending on gasoline over the past six months is between $58.37 and $71.93 determine which report you find most convincing, and explain your reasoning why.

Answers

In my opinion, Report 1 is more persuasive than Report 2 due to the fact that the margin of error in Report 1 is narrower than in Report 2.

This is further explained below.

What is the margin of error.?

Generally,  we consider report one and report two

For Report

at 98% confidence interval = $64.35 to $83.59

Therefore, the Margin of error

M.E= $83.59 - $64.35

M.E = $19.24

For Report 2:

95% confidence interval = $58.37 to $71.93

Margin of error

M.E= $71.93 - $58.37

M.E= $13.56

Since Report 1 has a smaller margin of error than Report 2, I find Report 1 to be more convincing.

In conclusion, A report with a lower margin of error, in my opinion, is often more credible than one with a bigger one.

This is because more exact data indicates a report with a reduced margin of error.

To put it another way, information in a report with a lower margin of error is more likely to be correct than information in a report with a higher margin of error.

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Lacy earned $900 last week before taxes, if she paid 19 % in taxes;
a) How much did she pay in taxes?
b) What is her net pay?

Answers

Answer: a) $171          b) $729

Step-by-step explanation:

a) 0.19 * $900 = $171

b) net pay = the money one earns after all deductions

$900 (what she earned) - $171 (taxes she had to pay) = $729

PLEASE HELP ME
Solve for x. Show each step of the solution.
2.54-x+12=21-42.5x+7

Answers

The solution of the equation 2.54-x+12= 21-42.5x+7 is x = 0.324

The equation is

2.54-x+12 = 21-42.5x+7

Rearrange the terms are do the suitable arithmetic operation in the equation

2.54-x+12= 21-42.5x+7

-x+12+2.54 = -42.5x+7+21

Add the like terms together

-x+14.54 = -42.5x+28

Here we have to rearrange the term and make the all the like terms in each sides

Move 14.54 to the right hand side and move -42.5x to the left hand side

Then the equation will become

-x+42.5x = 28-14.54

41.5x = 13.46

x = 13.46/41.5

x = 0.324

Hence, the solution of the equation 2.54-x+12= 21-42.5x+7 is x = 0.324

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Randy planted a small pine tree in his yard. When he bought the tree it was 6 inches tall and grew at a rate of 2 inches weekly. Write a function of height, h, in terms of weeks, w.Using the equation from problem #1 What will the height of the tree be in 13 weeks?

Answers

Answer:6+2w=h

The height of the tree will be 32 inches on week 13

Step-by-step explanation:

ok

Simplify each expression. Which expression is not equivalent to the other two?
A. 15j + 30
B. 5(3j - 6)
C. 3(5j - 10)

Answers

Answer: A is not equivalent to the other 2

Step-by-step explanation:

A. 15j+30

B. 5(3j-6) = 15j-30

C 3(5j-10) = 15j-30

multiply the stuff inside paranthesis for the last 2

So a is the outsider

6. three different distributions there are 559 full-service restaurants in delaware. the mean number of seats per restaurant is 99.2. [source: data based on the 2002 economic census from the us census bureau.] suppose that the true population mean µ

Answers

The standard deviation of the distribution of sample means (that is, the standard error, sigma_m) is; 2.9698

How to find the standard error of mean?

From the complete question written below, we are given that;

Mean Number; M = 103.4

Standard deviation; σ = 21

Sample size; n = 50

Now, in statistics, the formula for the standard error of mean is;

σM = σ/√n

Plugging in the relevant values gives us;

σM = 21/√50

σM = 2.9698

The z score typical average difference between the mean number of seats is;

z = (103.4 - 99.2)/( 2.9698 )

z = 1.41

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Complete Question is;

There are 559 full-service restaurants in Delaware. The mean number of seats per restaurant is 99.2. [Source: Data based on the 2002 Economic Census from U.S. Census Bureau.]

Suppose that the true population mean mu = 99.2 and standard deviation sigma = 21 are unknown to the Delaware tourism board. They select a simple random sample of 50 full-service restaurants located within the state to estimate mu. The mean number of seats per restaurant in the sample is M = 103.4, with a sample standard deviation of s = 18.2. The standard deviation of the distribution of sample means (that is, the standard error, sigma_m) is_.

a²-ab-ac+bc / a² +ab-ac-bc Simplying Algebraic fractions by factorizing​

Answers

Answer:

(a - b)/(a + b).

Step-by-step explanation:

a²-ab-ac+bc / a² +ab-ac-bc

= a(a - b) - c(a - b) / a(a + b) - c(a + b)

= (a - c)(a - b) / (a - c)(a + b)

The (a - c) is common to top and bottom, so

= (a - b)/(a + b).

will give 15 points if the question is answered

Answers

The surface area of a cylinder is equal to 208π square units.How to calculate the surface area of right cylinder?The surface area is the sum of the areas of all faces of the solid. In the case of a right cylinder, the surface area is the sum of the two areas of the circular base and the lateral area (A), in square units, whose formula is described below:A = 2π · r² + 2π · r · h           (1)Where:r - Base radiush - HeightIf we know that r = 8 and h = 5, then the surface area of the right cylinder is:A = 2π · 8² + 2π · 8 · 5A = 208πThe surface area of a cylinder is equal to 208π square units.

Ron is renting an apartment and his landlord just raised his rent.he now pays 1500 and his rent will go up 10%.what will be the new payment

Answers

Answer:

it would be 150 more, so 1650

Step-by-step explanation:

divide 1500 by 10 to get 10%

help please I rlly need help ​

Answers

Answer:

9. 160/2

10. 270/9

Step-by-step explanation:

To find unit rates, divide the top number (numerator) by the bottom number (denominator).

9.

160/2 = 80 mph

210/3 = 70 mph

10.

270/9 = 30 ft/min

180/9 = 20 ft/min

Identify the correct justification for each step, given that m∠1+m∠2=90° and m∠2+m∠3=90°.

Answers

The two column proof to justify the complementary angles are;

Statement 1. ∠1 and ∠2 are complementary

∠2 and ∠3 are complementary

Reason 1: Given

Statement 2: m∠1 + m∠2 = 90°

Reason 2: Definition of Complementary Angles

Statement 3. m∠2 + m∠3 = 90°

Reason 3: Definition of Complementary Angles

Statement 4: m∠1 + m∠2 = m∠2 + m∠3

Reason 4: Transitive property of equality

Statement 5: m∠1 = m∠3

Reason 5: Substitution Property of Equality

What are the Complementary angles?

Complementary angles are defined as angles that sum up to 90 degrees.

Now, from the question, we are given that;

∠1 and ∠2 are complementary, and ∠2 and ∠3 are complementary.

From the image attached, we can get a two column proof based on the given questions as;

Statement 1. ∠1 and ∠2 are complementary

∠2 and ∠3 are complementary

Reason 1: Given

Statement 2: m∠1 + m∠2 = 90°

Reason 2: Definition of Complementary Angles

Statement 3. m∠2 + m∠3 = 90°

Reason 3: Definition of Complementary Angles

Statement 4: m∠1 + m∠2 = m∠2 + m∠3

Reason 4: Transitive property of equality

Statement 5: m∠1 = m∠3

Reason 5: Substitution Property of Equality

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Complete Question is;

Identify the correct justification for each step, given that ∠1 and ∠2 are complementary, and ∠2 and ∠3 are complementary.

1. ∠1 and ∠2 are complementary

∠2 and ∠3 are complementary

2. m∠1 + m∠2 = 90°

3. m∠2 + m∠3 = 90°

4. m∠1 + m∠2 = m∠2 + m∠3

5. m∠1 = m∠3

find the slope of the line that goes through the points (-4,1) and (8,-5)

Answers

[tex](\stackrel{x_1}{-4}~,~\stackrel{y_1}{1})\qquad (\stackrel{x_2}{8}~,~\stackrel{y_2}{-5}) ~\hfill \stackrel{slope}{m}\implies \cfrac{\stackrel{rise} {\stackrel{y_2}{-5}-\stackrel{y1}{1}}}{\underset{run} {\underset{x_2}{8}-\underset{x_1}{(-4)}}} \implies \cfrac{-6}{8 +4}\implies {\Large \begin{array}{llll} -\cfrac{1}{2} \end{array}}[/tex]

Use this table to answer the following questions.If the number of gallons of gas is known, can you find the number of gallons consumed? Explain how this value would be calculated.

Answers

The table of values indicates a proportional relationship between the number of gallons of gas and the number of miles driven, therefore;

The number of miles driven, f(x) can be found from the number of gallons of gas consumed, x, using the equation; f(x) = 31•x

What is a proportional relationship?

A proportional relationship is one in which one variable, y, can be obtained from another variable, x, by multiplying x by a factor.

Please find attached a table showing the number of gallons of gas and the miles driven obtained from a similar question online

Let x represent the number of gallons of gas and let f(x) represent the number of miles driven

The difference between the consecutive terms of the first three values in the number of gallons and miles driven are each equal to 2 and 62 respectively

4 - 2 = 2 - 0 = 2

124 - 62 = 62 - 0 = 62

Therefore, given that the first difference is constant, the first three points have a linear relation, which gives;

Slope of the line = 62/2 = 31

Intercept = 0

Therefore;

f(x) = 31•x

Checking the fourth point, gives;

x = 10

f(10) = 31 × 10 = 310

The relationship between the miles driven, f(x), and the number of gallons is therefore a proportional relationship, which is presented as follows;

f(x) = 31•x

Therefore, if the number of gallons of gas, x, is known, the number of miles driven, f(x), can be found using the equation;

f(x) = 31 × x

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Write an algebraic expression for each verbal expression

22. x more than 7
24. 5 times a number
26. f divided by 10
28. three times a number plus 16
30. k squared minus 11
32. the sum of a number and 10
34. the product of 18 and q
23. a number less 35
25. one third of a number
27. the quotient of 45 and r
29. 18 decreased by 3 times d
31. 20 divided by t to the fifth power
33. 15 less than the sum of k and 2
35. 6 more than twice m

Answers

Step-by-step explanation:

22. 7 + x

24. 5x

26. f/10

28. 3x + 16

30. k² - 11

32. x + 10

34. 18q

23. x - 35

25. x/3

27. 45/r

29. 18 - 3d

31. 20/t⁵

(although, if being superficial, it could mean (20/t)⁵)

33. (k + 2) - 15

35. 2m + 6

elizabeth gives a walking tour of a popular tourist city to one person for to increase her​ business, she would lower the price by​ $2 per person for each additional person. write the cost per person c as a function of the number of people n on the tour. how much does she make for a tour with ​people?

Answers

Thus cost per person c as a function of the number of people n is

c(n) = 43 + (n-1)(43-(n-1)2) and amount she makes is  $229

In the problem it is given that Elizabeth gives a walking tour of a popular tourist city to one person for 43 to increase her​ business, she would lower the price by​ $2 per person for each additional person. we have to find the cost per person c as a function of the number of people n on the tour. Also the amount she makes for a tour with 7 people.

So we can write it as

c(1) = 43

c(2) = 43 + 41

c(3) = 43 + 2(43-2*2)

c(4) = 43 + 3(43-3*2)

c(n) = 43 + (n-1)(43-(n-1)2)

To find the amount she makes we write c as a function of n where n=7

c(7) = 43 + 6(43-6^2) = 43 + 6*31 = 43 + 186 = $229

Thus cost per person c as a function of the number of people n is c(n) = 43 + (n-1)(43-(n-1)2) and amount she makes is  $229.

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help me im having trouble solving this

Answers

Step-by-step explanation:

[tex]sa = 2\pi \: r {}^{2} + 2\pi \: rh \\ = 2 \times 3.14 \times 3 { }^{2} + 2 \times 3.14 \times 11[/tex]

For a party, Bert gets 4 different kinds of soda. He buys a 6-pack of each kind. At home, he divides the sodas evenly among 3 coolers. How many sodas are in each cooler?

Answers

The number of sodas that can fit in inside each cooler is 8

How to determine the number of sodas in each cooler?

From the question, the given parameters are:

Number of packs = 6

Kinds of soda = 4

Number of cooler = 3

The number of sodas in each cooler is calculated as

Sodas = Number of packs x Kinds of soda/Number of cooler

Substitute the known values in the above equation

So, we have

Sodas = 6 x 4/3

Evaluate

Sodas = 8

Hence, there are 8 sodas in each cooler

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Solve the inequality. (NESTED absolute value inequality)

[tex]||x-5| -4| \leq 1[/tex]

Answers

The result of the Nested absolute inequality  when resolved gives:
x = -10, 0. See the solution below.

What is a Nested absolute inequality?

It is to be noted that when one absolute-value expression is inside another one, this is called a "nested" expression. Absolute expressions are expressions that values that can NEVER be negative.

The solution hence is given as follows:

Given: ||x-5|4|≤1

One absolute-value expression is stacked inside another in this equation. I'll work from the outside in, sorting items into their respective cases. Assume the argument of the outer bars is negative (do the "minus" case):


Step 1 - first splitting:

|x-5| -4 ≤ -1
|x-5|  ≤ 3

I have to stop here since the last statement above states that an absolute value equals a negative number. Absolute values, on the other hand, are never negative! As a result, this step is invalid; it yields no viable solutions to the original problem.

Now I'll look at the scenario where the argument of the outside bars is positive (the "plus" case):

|x-5| -4 ≤ 1

|x-5|  ≤ 1 + 4

Recall that absolute values are never negative. This equation is correct, and I've isolated the leftover absolute value, so I'll go over the two situations again.

Step 2 - First, consider the "minus" case:

x+5  ≤ -5

x ≤ -5-5
x≤ -10

The variable can have a "minus" value; the sole constraint is that an absolute-value expression cannot have a "minus" value. Now consider the "plus" case:

⇒ x + 5 ≤ 5 again recall that this is an absolute expression hence the laws of polarity while cross the inequality sign is suspended.

x ≤ 5-5

x≤0

Hence the answer which is complete is:

x = -10, 0

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Jack mails 10 packages that each weigh the same amount. If the combined weigh of all 10 package is 67 1/2 pounds, how much does one package weigh? Show your work.​

Answers

One package weighs 6.75 pounds.

In this question, we have been given Jack mails 10 packages that each weigh the same amount.

We need to find the weight of one packet.

Let each package weighs m pounds.

The combined weigh of all 10 package is 67 1/2 pounds.

So, we get an equation,

10m = 67.5

We solve above equation.

m = 67.5 / 10

m = 6.75 pounds

Therefore, one package weighs 6.75 pounds.

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What value is equivalent to 2 × 3 + 4 − 62?

Answers

Answer:

-52

Step-by-step explanation:

I hope this helps!

Answer: -52

Step-by-step explanation: Using PEMDAS

(2x3)+4-62

6+4-62

10-62

= -52

Find the midpoint of points (-4,3) and (4,-3)

Answers

Answer:
The mid point would be (-4,0).
As shown in the image…..

In Exercises 1-6, describe the transformation of f(x) = x² represented by g.
Then graph each function.
1. g(x) = x² - 2
G=
h=
M=
2. g(x) = (x - 5)²
3. g(x) = (x + 1)² + 3

Answers

The graph of the function after transformation is attached below.

The connection between several systems is defined by coordinate transformations, which offer formulas for the coordinates in one system in terms of the coordinates in another system.

The coordinate transformation from polar to Cartesian coordinates is given by the formulas x = r cos and y = r sin, for example, if the polar axis is the positive x axis on the plane and the origins of the Cartesian coordinates (x, y) and polar coordinates (r) are the same.Every bijection from the space to itself can be related to two coordinate transformations.The formulas for the new coordinates of each point's image are constructed in a way that the old coordinates of the original point's image are maintained.In particular, the change of algebraic equations, it is a typical kind of algebraic problem.

The graph of the transformed function is attached below. The red graph indicates the parent function f(x) = x² and the blue graph indicated the graph of transformation g(x) = x² - 2

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If x= -12 and y=8 then what is the value of 2/x-3/ - /2y/ over /x+y/

Answers

If x= -12 and y=8 then the value of 2/x-3/ - /2y/ over /x+y/ is: 7/2. See the explanation below.

What is the computation supporting the above answer?

First, recall that the numbers enclosed in |   | are absolute values. That is they remain in the positive domain regardless of their polarity. Recall that the absolute value is the distance between a number and Zero.

Since we know:

x = -12; and

y = 8

Hence,

[2|x-3| - |2y|] / |x+y|

= [2|-12-3| - |2 * 8| ]/ (|-12+8|)
= [2|-15| - | 16|] / |-4|

= ((2*15) - (16))/4    [Using the absolute values]

= (30 -16)/4

= 14/4

= 7/2

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Which of the equations are true identities?
\begin{aligned} \text{A. }&(5b-2)^2+4=25b^2-20b \\\\ \text{B. }&(2x^2+y^2)(2x^2-y^2)=4x^2-y^2 \end{aligned}
A.
B.


(5b−2)
2
+4=25b
2
−20b
(2x
2
+y
2
)(2x
2
−y
2
)=4x
2
−y
2

Answers

The given equations (5b - 2)² + 4 = 25b²– 20b and (2x² + y²)(2x² - y²) = 4x² – y have been found not to be true identities

What is the True Identity of the Algebra Expression?

A) We want to check if (5b - 2)² + 4 = 25b²– 20b is a true identity.

A true identity is an equality that holds true regardless of the values chosen for its variables. They are used in simplifying or rearranging algebra expressions. By definition, the two sides of an identity are interchangeable, so we can replace one with the other at any time.

(5b - 2)² + 4 when expanded gives;

25b² - 20b + 4 + 4 = 25b² - 20b + 8

That is not equal to the right hand side and as such, we can say that both equations are not true identities

B) (2x² + y²)(2x² - y²) = 4x² – y

Expand the left side to get;

4x⁴ - y⁴

This result is not equal to the right hand side and as such, we can say that both equations are not true identities.

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Correct question is;

Which of the equations are true identities?

A. (5b - 2)² + 4 = 25b²– 20b

B. (2x² + y²)(2x² - y²) = 4x² – y

domain and range puzzle answers

Answers

Can u give a example for a textbook or whatever cuz I didn’t get it

What is the radical form of the expression? (5x3y4)37

Answers

The Radical Form is [tex]185x^3y^4[/tex].

What is the Radical Form?

If n is a positive integer that is greater than 1 and 'a' is a real number then,

 [tex]n\sqrt{a} = a1n[/tex]   where n is called the index, a is called the radicand, and the symbol [tex]\sqrt{}[/tex] is called the radical. The left side of this equation is often called  The radical form and the right side is often called the exponent form.

Now, The question is :

The Expression is : (5x3y4)37

Convert to Radical form by using the formula :

[tex]a^\frac{x}{n} = \sqrt[n]{a^x}[/tex]

(5x3y4)37 = [tex]185x^3y^4[/tex]

Therefore, The Radical Form is [tex]185x^3y^4[/tex]

Learn more about Radical Form at:

https://brainly.com/question/27272065

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