HW Inverse Functions
Name:
1. Let p be the price of an item and q be the number of items sold at that price. Assume q= f(p). Explain what the
following quantities mean in terms of prices and quantities sold.
A. f(25) b. f-¹ (30)

16. The length of each side of the original garden is 12 meters. The answer is (C) 12m.

17The value of c that makes x^2-11x+c a perfect-square trinomial is (B) 121/4..

18.The answer is (D) -1. 1, -9. 1.

Step by step explanation

16. Let s be the length of each side of the original garden. Then the area of the original garden is s^2. If each side is increased by 3 meters, then the new length of each side is s+3, and the area of the expanded garden is (s+3)^2. We are given that the area of the expanded garden is 225 square meters. Therefore, we can write the equation:

(s+3)^2 = 225

Taking the square root of both sides, we get:

s+3 = 15 or s+3 = -15

The second equation has no solution, since the length of a side cannot be negative. Therefore, we have:

s+3 = 15

Subtracting 3 from both sides, we get:

s = 12

17. To make x^2-11x+c a perfect-square trinomial, we need to add and subtract a constant term to make it a square of a binomial. Specifically, we want to add and subtract (11/2)^2 = 121/4 to get:

x^2 - 11x + c + 121/4 - 121/4

= (x - 11/2)^2 + (4c - 121)/4

For this to be a perfect-square trinomial, we need (4c - 121)/4 to be equal to 0. Therefore, we have:

4c - 121 = 0

Solving for c, we get:

4c = 121

c = 121/4

18. To solve the equation x^2 + 8x = 10 by completing the square, we first move the constant term to the right-hand side:

x^2 + 8x - 10 = 0

Next, we add and subtract the square of half the coefficient of x, which is (8/2)^2 = 16:

x^2 + 8x + 16 - 16 - 10 = 0

We can then write the left-hand side as a perfect-square trinomial:

(x + 4)^2 - 26 = 0

Adding 26 to both sides, we get:

(x + 4)^2 = 26

Taking the square root of both sides, we get:

x + 4 = ±√26

Subtracting 4 from both sides, we get:

x = -4 ±√26

Rounding to the nearest tenth, the solutions are approximately:

x ≈ -7.1 and x ≈ -0.9

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The age of the specimen with half-life 5760 years, to the nearest hundred years is 3184 years.

To find the age of the specimen, we can use the formula:

A = ([tex]A_{1}[/tex])[tex]2^{(-t/5760)}[/tex]

Where [tex]A_{1}[/tex] is the original amount of carbon-14 (120 milligrams), A is the current amount of carbon-14 (100 milligrams), t is the time elapsed since the organism died, and 5760 is the half-life of carbon-14.

Substituting the given values, we get:

100 = (120)[tex]2^{(-t/5760)}[/tex]

Taking the natural logarithm of both sides, we get:

ln(100) = ln(120) - t/5760 * ln(2)

Solving for t, we get:

t = -5760 * ln(100/120) / ln(2)

t ≈ 3183.7 years

Therefore, the age of the specimen is approximately 3184 years, rounded to the nearest hundred years.

It's worth noting that radiocarbon dating is only accurate up to a certain point, as the amount of carbon-14 in a specimen eventually becomes too low to measure accurately. The maximum age that can be reliably determined through radiocarbon dating is around 50,000 to 60,000 years. Beyond that, other methods such as dendrochronology (tree-ring dating) or uranium-thorium dating may be used.

Correct Question :

An archaeologist can determine the approximate age of certain ancient specimens by measuring the amount of carbon-14, a radioactive substance, contained in the specimen. The formula used to determine the age of a specimen is A = ([tex]A_{1}[/tex])[tex]2^{(-t/5760)}[/tex] where A is the amount of carbon-14 that a specimen contains, [tex]A_{1}[/tex] is the original amount of carbon-14, t is time, in years, and 5760 is the half-life of carbon-14. A specimen that originally contained 120 milligrams of carbon-14 now contains 100 milligrams of this substance. What is the age of the specimen, to the nearest hundred years?

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Answer:

≈ 12,78 m^2

Step-by-step explanation:

The surface area is equal to the sum of the areas of all the sides

This figure has sides of 2 triangles (bases) and 3 rectangles (lateral surface)

h (triangle) = 1m

We can find the base of the triangle by using the Pythagorean theorem (multiply by 2, because the triangle's base contains two of these identicals sides)

[tex](( {1.5})^{2} - {1}^{2} ) \times 2=( 2.25 - 1 ) \times 2= 1.25 \times 2 = 2.5> 0[/tex]

The triangle's base is equal to:

[tex] \sqrt{2.5} = \frac{ \sqrt{10} }{2} [/tex]

First, let's find the area of 2 bases (triangles):

[tex]a(bases) = 2 \times \frac{1}{2} \times 1 \times \frac{ \sqrt{10} }{2} = \frac{ \sqrt{10} }{2} [/tex]

Now, we can find the whole surface area by adding the areas of the rectangles to the bases' areas:

[tex]a(surface) = \frac{ \sqrt{10} }{2} + 2.4 \times 2 + 1.5 \times 2 + 1.7 \times 2 = \frac{ \sqrt{10} }{2} + \frac{56}{5} ≈12.78[/tex]

Answer: 6

Step-by-step explanation:

So, he wears the shirt 10 out of 20 days.

10 days is half of 20 days.

This means he wears the shirt approximately half of the time, by the logic of 10/20 days.

So, now we apply this to 12 days.

What's half of 12? 6.

This means that he most likely wears the green shirt on 6 out of the 12 days.

The volume of the unit cube is 1 cubic unit, and since it is divided into identical rectangular prisms, each of those prisms has the same volume. Therefore, the volume of one of the identical prisms is also 1 cubic unit.

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Using substitution method, the cost of a chicken is $44.56 and the cost of a rabbit is $36.78.

Let's use the substitution method to solve the problem.

Let's assume that the cost of a chicken is "x" dollars and the cost of a rabbit is "y" dollars.

From the first statement, we know that:

5x + 8y = 517 --- equation (1)

And from the second statement, we know that:

8x + 11y = 761 --- equation (2)

Now we can solve for one of the variables in terms of the other from one of the equations, and then substitute that expression into the other equation to solve for the remaining variable.

Solving equation (1) for "y", we get:

y = (517 - 5x) / 8

Now we can substitute this expression for "y" into equation (2), and solve for "x":

8x + 11[(517 - 5x) / 8] = 761

Multiplying both sides by 8 to eliminate the fraction, we get:

64x + 11(517 - 5x) = 6088

Expanding the brackets, we get:

64x + 5687 - 55x = 6088

Combining like terms, we get:

9x = 401

Dividing both sides by 9, we get:

x = 44.56

So the cost of a chicken is $44.56.

Now we can substitute this value for "x" into equation (1) to solve for "y":

5(44.56) + 8y = 517

Simplifying, we get:

222.80 + 8y = 517

Subtracting 222.80 from both sides, we get:

8y = 294.20

Dividing both sides by 8, we get:

y = 36.78

So the cost of a rabbit is $36.78.

Therefore, the cost of a chicken is $44.56 and the cost of a rabbit is $36.78.

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To find the length of BD in a parallelogram where BA = 5x + 5 and AD = 10x - 20, we use the fact that opposite sides of a parallelogram are equal in length. Therefore, BD = BA = 30.

Since it is a parallelogram, we know that opposite sides are equal. So, BD = BA = 5x + 5. To find the value of x, we can use the fact that AD is also equal to BD. So, we can set the two expressions for BD equal to each other

5x + 5 = 10x - 20

Simplifying and solving for x, we get

5x = 25

x = 5

Now we can substitute x back into the expression for BD to get the final answer

BD = 5x + 5 = 5(5) + 5 = 30

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Answer:

It's ASA, AAS,

Step-by-step explanation:

AAS- If two angles and a non-included side in one triangle are congruent to two angles and the corresponding non-included side in another triangle, then the triangles are congruent.

ASA-The ASA criterion for triangle congruence states that if two triangles have two pairs of congruent angles and the common side of the angles in one triangle is congruent to the corresponding side in the other triangle, then the triangles are congruent.

The missing values can be found by setting up proportions for each of the ratios whose values are given. The completed table is shown below:

x              17       1/3       11

y               5.67     3.67     1.21

Ratio y/x   3.67          1/3      0.11

In order to find the missing values of y, we can use the given ratios to set up proportions:

For the first ratio:

y/x = 5.67/17

y = (5.67/17) * x

y = (5.67/17) * 11

y = 3.67

So the first missing value of y is 3.67.

For the second ratio:

y/x = 1/3

y = (1/3) * x

y = (1/3) * 1/3

y = 0.11

Therefore, the second missing value of y is  found as 0.11.

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There are numerous expressions that have the same value as 10^6 using exponents and multiplication. These include expressions using the prime factorization of 10^6, as well as other equivalent forms of the number.

Write expressions that have the same value as 10^6, focusing on using exponents and multiplication. Here are a few examples:
1. (10^3) * (10^3): Using the exponent rule for multiplication, we add the exponents since the bases are the same (10^(3+3)) which simplifies to 10^6.
2. (10^2) * (10^2) * (10^2): Similar to the previous example, we add the exponents of the same base (10^(2+2+2)) which simplifies to 10^6.
3. (10^4) * (10^1) * (10^1): Again, we add the exponents of the same base (10^(4+1+1)) which simplifies to 10^6.
4. (10^5) * (10^1): Using the exponent rule for multiplication, we add the exponents of the same base (10^(5+1)) which simplifies to 10^6.

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The 88% confidence interval for the mean number of hours spent on work-related activities in a typical week is approximately (46.9, 50.1).

To construct an 88% confidence interval for the mean number of hours spent on work-related activities in a typical week, we will use the sample mean (48.5 hours) and sample standard deviation (7.5 hours) from the sample of size 51.

First, we need to find the critical value (z-score) corresponding to the 88% confidence level. Since the confidence level is symmetric around the mean, we will look for the z-score corresponding to (1 - 0.88)/2 = 0.06 in each tail.

Using a standard normal table, we find that the z-score is approximately 1.56.

Now, we will calculate the margin of error using the formula:

Margin of error = z-score * (sample standard deviation / sqrt(sample size))
Margin of error = 1.56 * (7.5 / sqrt(51))
Margin of error ≈ 1.63

Next, we will calculate the confidence interval as follows:

Lower limit = sample mean - margin of error
Lower limit = 48.5 - 1.63
Lower limit ≈ 46.9

Upper limit = sample mean + margin of error
Upper limit = 48.5 + 1.63
Upper limit ≈ 50.1

So, the 88% confidence interval for the mean number of hours spent on work-related activities in a typical week is approximately (46.9, 50.1).

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An equation of the line that passes through the pair of points (0,4) and (6, −3) in slope-intercept form is y = -7x/6 + 4.

In Mathematics and Geometry, the point-slope form of a straight line can be calculated by using the following mathematical expression:

y - y₁ = m(x - x₁)

Where:

First of all, we would determine the slope of this line;

Slope (m) = (y₂ - y₁)/(x₂ - x₁)

Slope (m) = (-3 - 4)/(6 - 0)

Slope (m) = -7/6

At data point (0, 4) and a slope of -7/6, a linear equation for this line can be calculated by using the point-slope form as follows:

y - y₁ = m(x - x₁)

y - 4 = -7/6(x - 0)  

y = -7x/6 + 4

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The driver could have traveled up to 5 miles for the delivery.

Let's denote the number of miles the driver traveled by "m".

According to the problem, the delivery service charged $1.25 for the delivery itself, and $0.75 for each mile traveled. This can be written as:

Total cost = $1.25 + $0.75 * m

We know that the service charged the customer $5.75, so we can set up an equation:

$5.75 = $1.25 + $0.75 * m

Solving for m, we get:

[tex]m = ($5.75 - $1.25) / $0.75 = 5[/tex]

To represent this on a number line, we can draw a line labeled from 0 to 5, with tick marks at each integer value.

We can label the tick mark at 0 as "0 miles" and the tick mark at 5 as "5 miles".

We can also indicate that the cost of the delivery increases as we move to the right by drawing an arrow pointing to the right, and labeling it "increasing cost".

Here's an example of what the number line might look like:

                   0         1         2         3         4         5

                    |---------|---------|---------|---------|---------|

                    0 miles                                            5 miles

                    increasing cost  ⟶

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6 gallons of gas were used to travel 210 miles.

A4. 1) To find the amount of gas used to travel 210 miles, we can rearrange the given equation m=35g to solve for g:

g = m/35

Substituting m=210, we get:

g = 210/35 = 6

6 gallons of gas were used to travel 210 miles.

A4. 2) To find the inverse function, we need to solve the given equation for g in terms of m:

m = 35g

Divide both sides by 35:

g = m/35

So the inverse function is:

g(m) = m/35

A4. 3) Similar to A4.2, we need to solve the given equation m=hg for g in terms of m:

m = hg

Divide both sides by h:

g = m/h

Now, we can write the inverse function as:

g(m) = m/h\

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The given expression, 14⁻³ · 14¹², written as a single, positive exponent is 14⁹

From the question, we are to write the given expression using a single, positive exponent

From the given information,

The given expression is

14⁻³ · 14¹²

This means

14⁻³ × 14¹²

To solve this, let us revise some of the laws of indices

Thus,

The expression

14⁻³ × 14¹²

can be written as

14⁻³ ⁺ ¹²

14⁹

Hence, the expression written as a single exponent is 14⁹

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Anissa's statement is correct.

The classification of distance, d, as either an independent or dependent

variable depends on the context in which it is used.

Distance as an independent variable:

In this case, distance, d, is considered an independent variable because it

affects the amount of time someone has traveled.

When we are interested in studying how the distance traveled affects

other variables, such as time or fuel consumption, we treat distance as the

independent variable and manipulate it to observe its impact on the

dependent variables.

For example, if we conduct an experiment to measure the time it takes to

travel a certain distance under different conditions (e.g., different speeds or

modes of transportation), we would vary the distance as the independent

variable while keeping other factors constant.

In this scenario, distance is the independent variable, and time is the

dependent variable.

Distance as a dependent variable:

On the other hand, distance, d, can also be considered a dependent

variable when it is affected by the speed traveled.

In this case, speed becomes the independent variable, and distance is

dependent on the speed at which an object or person travels.

For instance, if we investigate how the speed of a vehicle affects the

distance it can travel within a given time, we would manipulate the speed

as the independent variable and observe the corresponding changes in

distance.

Here, distance is the dependent variable, and speed is the independent

variable.

In summary, Anissa's statement is accurate because distance, d, can be

considered an independent variable when it affects other factors such as

time, and it can also be a dependent variable when it is influenced by

factors like speed.

The designation of distance as independent or dependent depends on the

specific context and the relationship it shares with other variables in the

given situation.

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Answer:

y = |x - 1|

Step-by-step explanation:

This is an absolute value function in the form g(x) = |x - h| + k, where

(h, k) is the vertex, (1, 0).  Substitute these values into the function to get the equation of the graph:

y = |x - 1| + 0 = |x - 1|

a. Null hypothesis (H0): μ1 = μ2 and Alternative hypothesis (H1): μ1 < μ2. b. The test statistic is -2.57 and p-value is  0.005 for the hypothesis test. c. At α = 0.05 it can be concluded that CPAs in states with flat state income tax rates work fewer hours per week during tax season compared to the average U.S. CPAs.

a. First, let's formulate the hypotheses:

Null hypothesis (H0): μ1 = μ2, which means that the mean hours worked per week during tax season by CPAs in states with flat state income tax rates is equal to the mean hours worked per week by all U.S. CPAs during tax season.

Alternative hypothesis (H1): μ1 < μ2, which means that the mean hours worked per week during tax season by CPAs in states with flat state income tax rates is less than the mean hours worked per week by all U.S. CPAs during tax season.

b. Now, let's compute the test statistic and p-value using the given sample data:

Sample mean (x) = 55 hours

Population mean (μ) = 60 hours

Population standard deviation (σ) = 27.4 hours

Sample size (n) = 150

We'll use the z-test for this hypothesis test:

z = (x - μ) / (σ / √n) = (55 - 60) / (27.4 / √150) ≈ -2.57

To find the p-value, we need to look up the z-value in the standard normal table, which gives us a p-value of approximately 0.005.

c. Lastly, let's draw our conclusion using α = 0.05:

Since the p-value (0.005) is less than α (0.05), we reject the null hypothesis (H0). This suggests that CPAs in states with flat state income tax rates work fewer hours per week during tax season compared to the average U.S. CPAs.

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Answer:

Step-by-step explanation:

Solution: -4

The graph gets closer and closer to the line x=-4 but never touches it.

Step-by-step explanation:

To estimate the difference between 5 1/2 and 3 5/9, we can first round the fractions to the nearest whole number or simpler fractions. In this case, we can round 1/2 to 1/2 and 5/9 to 1/2 as well. Now, we have:

5 1/2 − 3 1/2

Subtracting the whole numbers, we get:

5−3=2

Subtracting the fractions, we get:

1/2 − 1/2 = 0

So, the estimated difference is 2, which falls between 2 and 2 1/2. Therefore, Elise chose option D as the correct answer.

The function f(x, y, z) that has the constant gradient vector (3, 9, 4) is: f(x, y, z) = (3/2)x^2 + (9/2)y^2 + (2z - y - 2x)^2 + C where C is a constant of integration.

To find a function f(x, y, z) such that the gradient of f, ∇f, is the constant vector (3, 9, 4), we can use the fact that the gradient of a function points in the direction of maximum increase and that the components of the gradient give the rates of change in the corresponding directions.

Let's assume that f(x, y, z) has the form:

f(x, y, z) = ax^2 + by^2 + cz^2 + dxy + exz + fyz + gx + hy + iz + C

where a, b, c, d, e, f, g, h, i, and C are constants that we need to determine.

The gradient of f is:

∇f = (2ax + dy + ez + g, 2by + dx + fz + h, 2cz + ex + fy + i)

If ∇f is equal to the constant vector (3, 9, 4), then we can set up a system of equations:

2ax + dy + ez + g = 3

2by + dx + fz + h = 9

2cz + ex + fy + i = 4

We need to solve this system of equations for a, b, c, d, e, f, g, h, i, and C.

To make the solution simpler, we can set some of the constants to zero. Let's set d = e = f = g = h = i = 0. Then the system becomes:

2ax + ez = 3

2by + fz = 9

2cz + fy = 4

Now we can solve for a, b, and c:

a = 3/2x - 1/2z

b = 9/2y - 1/2z

c = 2z - y - 2x

Substituting these values back into the original equation for f, we get:

f(x, y, z) = (3/2)x^2 + (9/2)y^2 + (2z - y - 2x)^2 + C

So the function f(x, y, z) that has the constant gradient vector (3, 9, 4) is:

f(x, y, z) = (3/2)x^2 + (9/2)y^2 + (2z - y - 2x)^2 + C

where C is a constant of integration.

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Answer:

Step-by-step explanation:

201.87 + 3(9.82) + 15.79 + 32.80(o) < 480

247.12 + 32.80(o) < 480

32.80(o) < 480 - 247.12

32.80(o) < 232.88

o < 232.88/32.8

o < 7.1

He can purchase 7 outfits and stay within the $480 budget

The production level that will maximize profit is 80 units

To find the production level that will maximize profit given the cost function C(q) = 660 + 5q + 0.03q^2 and demand function p = 10 - q/400, follow these steps:

1. Write down the revenue function: Revenue (R) is the product of price (p) and quantity (q). So, R(q) = p * q.

2. Substitute the demand function into the revenue function: R(q) = (10 - q/400) * q

3. Simplify the revenue function: R(q) = 10q - q^2/400

4. Write down the profit function: Profit (P) is the difference between revenue and cost. So, P(q) = R(q) - C(q).

5. Substitute the revenue and cost functions into the profit function: P(q) = (10q - q^2/400) - (660 + 5q + 0.03q^2)

6. Simplify the profit function: P(q) = 10q - q^2/400 - 660 - 5q - 0.03q^2

7. Combine like terms: P(q) = 5q - q^2/400 - 0.03q^2 - 660

8. Differentiate the profit function with respect to q to find the first derivative: P'(q) = 5 - q/200 - 0.06q

9. Set the first derivative equal to 0 and solve for q: 5 - q/200 - 0.06q = 0

10. Solve for q: q ≈ 80

The production level that will maximize profit is approximately 80 units (rounded to the nearest whole number).

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The weight of the contents in the container is approximately 222 pounds when rounded to the nearest pound.

The weight of the contents in the container can be calculated using the formula:

Weight = Volume x Density

First, we need to find the volume of the container, which is simply the product of its length, width, and height:

Volume = 11.5 x 8.5 x 7 = 696.5 cubic feet

Next, we need to multiply the volume by the density of the contents:

Weight = 696.5 x 0.32 = 222.08 pounds

It's important to note that the weight may vary depending on the actual density of the contents. Additionally, it's also important to consider the weight of the container itself, which would add to the total weight of the shipment.

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The rule that best represents the dilation that was applied to quadrilateral FGHJ to create quadrilateral F'G'H'J' is option B, which is (x, y) à (1.4x, 1.4y).

A dilation is a transformation that changes the size of an object without changing its shape. It is performed by multiplying the coordinates of each point by a scale factor.

In this case, the center of dilation is the origin, which means that the coordinates of each point are multiplied by the same scale factor in both the x and y directions.

The scale factor can be found by comparing the corresponding side lengths of the two quadrilaterals. In this case, the scale factor is 1.4, which means that the lengths of the sides of F'G'H'J' are 1.4 times the lengths of the corresponding sides of FGHJ.

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To find out how much Casho would have to pay in a month in which she downloaded 30 songs, we need to consider both the monthly subscription cost and the cost per song for offline listening.

Step 1: Determine the cost of the monthly subscription, which is $7.

Step 2: Calculate the cost of downloading 30 songs for offline listening. To do this, multiply the cost per song ($1.50) by the number of songs (30).
1.50 * 30 = $45

Step 3: Add the monthly subscription cost ($7) to the cost of downloading 30 songs ($45).
7 + 45 = $52

So, Casho would have to pay $52 a month in which she downloaded 30 songs.

Now, let's find out how much Casho would have to pay if she downloaded ss songs.

Step 1: The cost of the monthly subscription remains the same at $7.

Step 2: Calculate the cost of downloading ss songs for offline listening. Multiply the cost per song ($1.50) by the number of songs (ss).
1.50 * ss = 1.50ss

Step 3: Add the monthly subscription cost ($7) to the cost of downloading ss songs (1.50ss).
7 + 1.50ss = 7 + 1.50ss

The total amount Casho would have to pay if she downloaded ss songs is 7 + 1.50ss.

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The product 5x^(2/3) can be written as the product of the whole number 5 and the unit fraction 1/x^(-2/3), which simplifies to x^(2/3)/1 or just x^(2/3). So, we have:

5x^(2/3) = 5 * (1/x^(-2/3)) = 5x^(2/3) = 5 * (x^(2/3) / 1) = 5x^(2/3) = 5x^(2/3)

To write the product 5x^(2/3) as the product of a whole number and a unit fraction, we need to express x^(2/3) as a unit fraction.

Recall that a unit fraction is a fraction with a numerator of 1, so we need to find a fraction that has 1 as the numerator and x^(2/3) as the denominator. We can do this by using the reciprocal property of exponents:

x^(2/3) = 1 / x^(-2/3)

Now we can substitute this expression into the original product:

5x^(2/3) = 5 * (1 / x^(-2/3))

Simplifying the right-hand side of the equation, we can write it as:

5 / x^(-2/3) = 5x^(2/3)

To know more about unit fraction refer to

https://brainly.com/question/30860953

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