What is the 10th term of the geometric sequence where a1 = 384 and a7 = 6?
0.75

3

6

1.5

Answer:

[tex](x - 3)^2 + (y + 2)^2 = 16[/tex]

Step-by-step explanation:

The equation of a circle with a center at (h, k) and radius r is given by:

[tex](x - h)^2 + (y - k)^2 = r^2[/tex]

We are given that the points G(5,-2) and H(1, −2) lie on the circle. We can use the distance formula to find the distance between these two points.

[tex]d = \sqrt{(x_1 - x_2)^2 + (y_1 - y_2)^2}[/tex]

[tex]d = \sqrt{(5 - 1)^2 + ((-2) - (-2))^2} = \sqrt{16} = 4[/tex]

Therefore, the radius of the circle is 4.

We can now find the center of the circle by taking the average of the x-coordinates and y-coordinates of the points G and H.

[tex](h, k) = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right) = \left(\frac{5 + 1}{2}, \frac{-2 - 2}{2}\right) = (3, -2)[/tex]

Therefore, the equation of the circle is:

[tex](x - 3)^2 + (y + 2)^2 = 4^2[/tex]

Simplifying the equation, we get:

[tex]\bold{(x - 3)^2 + (y + 2)^2 = 16}[/tex] is a required equation.

Answer:

[tex]\textsf{B)} \quad f(x) = 5 \sin (16x) + 7}[/tex]

Step-by-step explanation:

The sine function is periodic, meaning it repeats forever.

[tex]\boxed{f(x) = A \sin (B(x + C)) + D}[/tex]

where:

Given values:

Calculate the value of B:

[tex]\dfrac{2\pi}{B}=\dfrac{\pi}{8}\implies 16\pi=B\pi\implies B=16[/tex]

Substitute the values of A, B C and D into the standard formula:

[tex]f(x) = 5 \sin (16(x + 0)) + 7[/tex]

[tex]f(x) = 5 \sin (16x) + 7[/tex]

Therefore, the sine function with an amplitude of 5, a period of π/8, and a midline at y = 7 is:

[tex]\Large\boxed{\boxed{f(x) = 5 \sin (16x) + 7}}[/tex]

The Domain of the function would be the interval [0, 6].The increasing interval represents the point where the company is pricing the pens appropriately, resulting in increasing profit.Average rate of change = (f(5) - 120) / (5 - 3)

Part A:

The x-intercepts of the graph, located at the ordered pairs (0, 0) and (6, 0), represent the points where the profit function intersects the x-axis. In the context of the problem, these x-intercepts indicate the price values at which the company sells pens, resulting in zero profit. When the price is $0, the company does not make any profit, and when the price is $6, the company also does not make any profit. The x-intercepts represent the break-even points where the company covers its costs but does not generate any profit.

The maximum value of the graph, located at the vertex (3, 120), represents the peak profit achieved by the company. The vertex is the highest point on the graph and indicates the optimal price at which the company can maximize its profit. In this case, when the price of pens is $3, the company achieves a maximum profit of $120.

Regarding the intervals of increasing and decreasing profit, the function is decreasing to the left of the vertex (3, 120) and increasing to the right of the vertex. This means that for prices less than $3, the profit decreases, and for prices greater than $3, the profit increases. The decreasing interval represents the point where the company is pricing the pens too low, resulting in decreasing profit. The increasing interval represents the point where the company is pricing the pens appropriately, resulting in increasing profit.

Part B:

To find the approximate average rate of change of the graph from x = 3 to x = 5, we need to calculate the slope of the line segment connecting the two points. The average rate of change represents the average amount by which the profit changes per unit increase in price over the given interval.

Using the coordinates (3, 120) and (5, f(5)) on the graph, we can determine the change in profit and price:

Change in profit = f(5) - 120

Change in price = 5 - 3

Substituting the values of the points into the equation, we can calculate the approximate average rate of change:

Average rate of change = (f(5) - 120) / (5 - 3)

Part C:

The constraints of the domain refer to the limitations or restrictions on the values that x (the price of pens) can take. Based on the given information, the x-intercepts of the graph are at (0, 0) and (6, 0), indicating that the price cannot be negative or exceed $6. Therefore, the domain of the function would be the interval [0, 6]. The constraint is that the price of the pens must be within this range for the profit function to be applicable.

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Note the full question may be :

A bakery sells cupcakes for $2 each. The bakery's total revenue from selling cupcakes is given by the function R(x) = 2x, where x represents the number of cupcakes sold.

Part A: What does the x-intercept of the graph of R(x) represent? Explain its significance in the context of the bakery's revenue.

Part B: What is the slope of the graph of R(x)? Interpret the meaning of this slope in terms of the bakery's revenue and the price of cupcakes.

Part C: If the bakery wants to maximize its revenue, what should be the ideal number of cupcakes to sell? Explain your answer based on the graph of R(x).

Answer: A = 83.75

Step-by-step explanation:

A=a+b

2h=12.5+21

2·5=83.75

The solution to the given simultaneous equations using Cramer's rule is x = -1, y = -4.92, z = -1.03.

To solve the given simultaneous equations using Cramer's rule, we need to find the determinants of various matrices.

Given the set of equation:

3x-3y+ 5z = 5

9x - 8y + 13z = 14

-3x+4y- 7z=-7

We start by finding the determinant of the coefficient matrix, which is denoted as D.

D = [tex]\left[\begin{array}{ccc}3&-3&5\\9&-8&13\\-3&4&-7\end{array}\right][/tex]

To calculate D, we use the formula:

D = (3 * (-8) * (-7) + (-3) * 13 * (-3) + 5 * 9 * 4) - ((-3) * (-8) * 5 + 3 * 13 * 4 + (-3) * 9 * (-7))

D = (-168 + 117 + 180) - (120 - 156 + 189)

D = 129 - 165

D = -36

Next, we need to find the determinants of the matrices obtained by replacing the columns of the coefficient matrix with the constant terms. These determinants are denoted as Dx, Dy, and Dz, respectively.

Dx = [tex]\left[\begin{array}{ccc}5&-3&5\\14&-8&13\\-7&4&-7\end{array}\right][/tex]

Dx = (-40 - 65 + 20) - (-70 - 60 + 189) = -85 - (-121) = -85 + 121

Dx= 36

Dy = [tex]\left[\begin{array}{ccc}3&5&5\\9&14&13\\-3&-7&-7\end{array}\right][/tex]

Dy = (21 + 75 + 35) - (-21 - 130 + 105) = 131 - (-46) = 131 + 46

Dy = 177

Dz = [tex]\left[\begin{array}{ccc}3&-3&5\\9&-8&14\\-3&4&-7\end{array}\right][/tex]

Dz = (39 - 39 + 0) - (-27 + 84 + 20) = 0 - (-37) = 0 + 37

Dz = 37

Finally, we can find the values of x, y, and z using the formulas:

x = Dx / D

y = Dy / D

z = Dz / D

Plugging in the values, we have:

x = 36 / -36 = -1

y = 177 / -36 ≈ -4.92

z = 37 / -36 ≈ -1.03

Therefore, the solution to the given simultaneous equations using Cramer's rule is x = -1, y ≈ -4.92, z ≈ -1.03.

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Answer:

150

Step-by-step explanation:

A. A system of equations in standard form that can be solved to find the number of adults and children who attended the performance is:

x + y = 282

3x + 5y = 1046

B. The number of adults who attended the performance is 182 adults.

The number of children who attended the performance is 100 children.

In order to write a system of linear equations to describe this situation, we would assign variables to the number of adult tickets sold and number of children tickets sold, and then translate the word problem into an algebraic equation as follows:

Since 282 people attended the recent performance by Cinderella, a linear equation that models the situation is given by:

x + y = 282     ....equation 1.

Additionally, adult tickets sold for $5 while children tickets sold for $3 each with a total revenue was $1046, a linear equation that models the situation is given by:

3x + 5y = 1046   .......equation 2.

Part B.

By multiplying equation 1 by 3, we have:

3x + 3y = 846     .......equation 3.

By subtracting equation 3 from equation 2, we have:

2y = 200

y = 100 children.

For the x-value, we have:

x = 282 - y

x = 282 - 100

x = 182 adults.

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a. The variables are x and y which represents 2 shots and 3 shots respectively.

b. The system of equations are;

x + y = 11

2x + 3y = 29

a. Let's define the variables to write the system of equations:

Let x be the number of two-point shots Noah made.

Let y be the number of three-point shots Noah made.

b. We can write a system of equations that model this problem.

Equation 1: x + y = 11

This equation represents the total number of shots Noah made, which is 11.

Equation 2: 2x + 3y = 29

This equation represents the total number of points Noah scored, which is 29. Since each two-point shot is worth 2 points and each three-point shot is worth 3 points, we can multiply the number of two-point shots (x) by 2 and the number of three-point shots (y) by 3 and sum them up to get the total score of 29.

The system of equations that can be used to determine the number of two-point shots Noah made and the number of three-point shots he made is:

x + y = 11

2x + 3y = 29

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Answer: 36

Step-by-step explanation: 3y by 56

Answer:

3.999 millimeters.

Step-by-step explanation:

To find the height of the cylinder, we can use the formula for the volume of a cylinder:

V = πr²h

Given that the radius (r) of the cylinder is 4 millimeters and the volume (V) is 200.96 cubic millimeters, we can substitute these values into the formula and solve for the height (h).

200.96 = π(4²)h

200.96 = 16πh

To solve for h, we can divide both sides of the equation by 16π:

200.96 / (16π) = h

Using a calculator, we can calculate the approximate value of h:

h ≈ 200.96 / (16 × 3.14159)

h ≈ 3.999

Therefore, the height of the cylinder is approximately 3.999 millimeters.

Answer: 1080

Step-by-step explanation:

12x10x9=1080

Hoped this helped?!

When a student is selected at random, the solution for the various possible outcome would be given below as follows:

p(pass or did not study)=0.409

p(pass or did study)=0.312

p(fail and pass)= 0.201

p(pass or fail)= 1

The formula that is used to calculate probability = possible outcome/sample space.

The total number of scores of the statistics students = 93

For p(pass or did not study):

possible outcome = 38

sample size = 93

probability = 38/93 = 0.409

For p(pass or did study):

possible outcome = 29

sample size = 93

probability = 29/93 = 0.312

For p(fail and pass):

p(fail) = 26/93

p(pass) = 67/93

probability = 26/93× 67/93

= 1742/8649

= 0.201

For p(pass or fail);

P(pass) = 67/93

P(fail) = 26/93

Probability of pass or fail = 67/93+26/93

= 93/93 = 1

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41.

Out of the 18 parts, we shade 5 parts

Out of the 27 parts, we shade 4 parts

42.

There are 60 pieces.

43.

The fractional part for each person.

44.

10(1/2), 1/21, 2(14/15), and 18.

We have,

41.

a.

1/3 x 5/6

= 5/18

This means,

Out of the 18 parts, we shade 5 parts

b.

2/9 x 2/3

= 4/27

This means,

Out of the 27 parts, we shade 4 parts

42.

String = 15 feet

Length of each piece = 1/4 feet

Now,

The number of 1/4 feet pieces.

= 15/(1/4)

= 15 x 4

= 60 pieces

43.

Original pizza = 1

Half pizza = 1/2

Number of people = 3

Now,

The fractional part for each person.

= 1/2 ÷ 3

= 1/6

44.

a.

7/6 x 9

= 7/2 x 3

= 21/2

= 10(1/2)

b.

1/7 ÷ 3

= 1/(7 x 3)

= 1/21

c.

4/5 x 3(2/3)

= 4/5 x 11/3

= 44/15

= 2(14/15)

d.

2 ÷ 1/9

= 2 x 9/1

= 18

Thus,

41.

Out of the 18 parts, we shade 5 parts

Out of the 27 parts, we shade 4 parts

42.

There are 60 pieces.

43.

The fractional part for each person.

44.

10(1/2), 1/21, 2(14/15), and 18.

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The population reached in the year 2079 is 1,65,400 and the total population in the year 2081 would be 1,80,623.

To find the population reached in the year 2079, we need to consider the initial population and the growth rate, as well as the number of people who migrated.

The initial population in 2078 was 1,20,000. The population growth rate is 4.5%, which means the population will increase by 4.5% each year.

To calculate the population in 2079, we first need to calculate the increase in population due to the growth rate:

Population increase due to growth rate = 1,20,000 * (4.5/100) = 5,400

Then we add the number of people who migrated:

Total population in 2079 = Initial population + Population increase due to growth rate + Number of migrants

= 1,20,000 + 5,400 + 20,000

= 1,45,400 + 20,000

= 1,65,400

To calculate the total population in the year 2081, we need to consider the growth rate and the population in 2080.

The population in 2080 would be the population in 2079 plus the population increase due to the growth rate:

Population increase due to growth rate in 2080 = 1,65,400 * (4.5/100) = 7,444

Total population in 2080 = 1,65,400 + 7,444

= 1,72,844

To calculate the total population in 2081, we need to consider the growth rate and the population in 2080:

Population increase due to growth rate in 2081 = 1,72,844 * (4.5/100) = 7,779

Total population in 2081 = Population in 2080 + Population increase due to growth rate in 2081

= 1,72,844 + 7,779

= 1,80,623

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The volume of the trapezoidal prism is 420 cm³

Volume is the amount of space that is occupied by a three dimensional shape or object.

The volume of a trapezoidal prism is the product of the base area and its height.

For the trapezoidal prism:

Area of base = (1/2)(15 + 5) * 7 = 70 cm²

Volume of Prism = Area of base * height = 70 cm² * 6 cm = 420 cm³

The volume of the figure is 420 cm³

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The probability that the next customer will pay with something other than cash, rounded to the nearest Whole number, is approximately 12%.

The probability that the next customer will pay with something other than cash, we need to consider the total number of customers who paid with something other than cash and divide it by the total number of customers in the last week.

In the given information, it is stated that 69 customers paid with cash, 2 customers used a debit card, and 7 customers used a credit card. To find the total number of customers who paid with something other than cash, we add the number of customers who used a debit card and the number of customers who used a credit card:

Total number of customers who paid with something other than cash = Number of customers who used a debit card + Number of customers who used a credit card

= 2 + 7

= 9

Now, to calculate the probability, we divide the number of customers who paid with something other than cash by the total number of customers:

Probability = Number of customers who paid with something other than cash / Total number of customers

= 9 / (69 + 2 + 7)

= 9 / 78

= 0.1154 (rounded to four decimal places)

To express the probability as a percentage, we multiply the probability by 100:

Probability as a percent = 0.1154 * 100

≈ 11.54

Therefore, the probability that the next customer will pay with something other than cash, rounded to the nearest whole number, is approximately 12%.

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Answer:  A  129

Step-by-step explanation:

Because the 2 chords are the same (lines in the circle), the 2 arcs are the same  create an equation that makes them equal

3x+24 = 4x -11             >bring x to one side by subtracting both sides by 3x

24 = x -11                     > add both sides by 11

35 = x

Now that we have solved for x you need to plug that back into the equation for BC

BC= 4x-11

BC = 4(35) - 11

BC =  140 - 11

BC = 129                   >A

The solution of the given expression is,

x = 1/2.

The given expression is,

[tex]36^{3x} = 216[/tex]

Since we know that,

As the name indicates, exponents are utilized in the exponential function. An exponential function, on the other hand, has a constant as its base and a variable as its exponent, but not the other way around (if a function has a variable as its base and a constant as its exponent, it is a power function, not an exponential function).

Now we can write it as,

⇒ [tex]6^{2^{3x}} = 6^3[/tex]

⇒  [tex]6^{6x}} = 6^3[/tex]

Now equating the exponents we get,

⇒ 6x = 3

⇒   x = 3/6

⇒   x = 1/2

Hence,

Solution is, x = 1/2.

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x+77=180

x=180-77=103°

x+5y-98=180

=> 103+5y-98=180

=> 5y=180-5

=> y=175/5=35°

The area of the kite is 38 mm squared.

A kite is a quadrilateral with two pairs of adjacent sides equal. The diagonals of a kite are perpendicular. The vertex angle are bisected by the diagonals.

It has two pairs of consecutive equal sides and perpendicular diagonals.

Therefore,

area of a kite = ab / 2

where

Therefore,

a = 9.5 mm

b = 4 mm

Therefore,

area of a kite = 9.5 × 4 / 2

area of a kite = 38 / 2

area of a kite = 19 mm²

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Answer:

The area of the kite would be 19 mm².

Step-by-step explanation:

A kite is a quadrilateral with two pairs of equal-length adjacent sides. The diagonals of a kite are perpendicular to each other and intersect at a right angle, dividing the kite into four right triangles. The area of a kite can be calculated by finding the product of the lengths of its diagonals and dividing it by 2.

If we assume that the height and width you provided are the lengths of the diagonals, we can calculate the area using the following formula:

Area = (height * width) / 2

Substituting the given values:

Area = (9.5 mm * 4 mm) / 2

= 38 mm² / 2

= 19 mm²

Therefore, if the height and width represent the lengths of the diagonals, the area of the kite would be 19 square millimeters.

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Answer: There are 16 fluid ounces in 1 pint. To determine the number of pints in the jug, we need to divide the total number of fluid ounces by 16.

Given that the jug contains 36 fluid ounces of apple juice, we divide 36 by 16:

36 fluid ounces ÷ 16 fluid ounces/pint = 2.25 pints

Therefore, the jug contains 2.25 pints of apple juice.

Answer:

13)  4.9 m

14)  0.9 m

Step-by-step explanation:

The given diagram shows the height of the same cactus plant a year apart:

We are told that the cactus continues to grow at the same percentage rate. To calculate the growth rate per year (percentage increase), use the percentage increase formula:

[tex]\begin{aligned}\sf Percentage \; increase &= \dfrac{\sf Final\; value - Initial \;value}{\sf Initial \;value}\\\\&=\dfrac{ 2-1.6}{1.6}\\\\&=\dfrac{0.4}{1.6}\\\\&=0.25\end{aligned}[/tex]

Therefore, the growth rate of the height of the cactus is 25% per year.

As the cactus grows at a constant rate, we can use the exponential growth formula to calculate its height in Year 6.

[tex]\boxed{\begin{minipage}{7.5 cm}\underline{Exponential Growth Formula}\\\\$y=a(1+r)^t$\\\\where:\\\phantom{ww}$\bullet$ $a$ is the initial value. \\ \phantom{ww}$\bullet$ $r$ is the growth factor (in decimal form).\\ \phantom{ww}$\bullet$ $t$ is the number of time periods.\\\end{minipage}}[/tex]

The initial value is the height in Year 1, so a = 1.6.

The growth factor is 25%, so r = 0.25.

As we wish to calculate its height in Year 6, the value of t is t = 5 (since there are 5 years between year 1 and year 6).

Substitute these values into the formula and solve for y (the height of the cactus):

[tex]\begin{aligned}y&=a(1+r)^t\\&=1.6(1+0.25)^5\\&=1.6(1.25)^5\\&=1.6(3.0517578125)\\&=4.8828125\\&=4.9\; \sf m\;(nearest\;tenth)\end{aligned}[/tex]

Therefore, if the cactus continues to grow at the same rate, its height in Year 6 will be 4.9 meters (to the nearest tenth).

Check by multiplying the height each year by 1.25:

[tex]\hrulefill[/tex]

The given diagram shows the height of the same snowman an hour apart:

We are told that the snowman continues to melt at the same percentage rate. To calculate the decay rate per hour (percentage decrease), use the percentage decrease formula:

[tex]\begin{aligned}\sf Percentage \; decrease&= \dfrac{\sf Initial\; value - Final\;value}{\sf Initial \;value}\\\\&=\dfrac{1.8-1.53}{1.8}\\\\&=\dfrac{0.27}{1.8}\\\\&=0.15\end{aligned}[/tex]

Therefore, the decay rate of the snowman's height is 15% per hour.

As the snowman melts at a constant rate, we can use the exponential decay formula to calculate its height after another 3 hours.

[tex]\boxed{\begin{minipage}{7.5 cm}\underline{Exponential Decay Formula}\\\\$y=a(1-r)^t$\\\\where:\\\phantom{ww}$\bullet$ $a$ is the initial value. \\ \phantom{ww}$\bullet$ $r$ is the decay factor (in decimal form).\\ \phantom{ww}$\bullet$ $t$ is the number of time periods.\\\end{minipage}}[/tex]

The initial value is the snowman's initial height, so a = 1.8.

The decay factor is 15%, so r = 0.15.

As we wish to calculate the snowman's height after another 3 hours, the value of t is t = 4 (i.e. the first hour plus a further 3 hours).

Substitute these values into the formula and solve for y (the height of the snowman):

[tex]\begin{aligned}y&=a(1-r)^t\\&=1.8(1-0.15)^4\\&=1.8(0.85)^4\\&=1.8(0.5220065)\\&=0.93961125\\&=0.9\; \sf m\;(nearest\;tenth)\end{aligned}[/tex]

Therefore, if the snowman continues to melt at the same rate, its height after another 3 hours will be 0.9 meters (to the nearest tenth).

Check by multiplying the height each hour by 0.85:

[tex] \begin{aligned}3 ^{3k} &= \frac{1}{81} \\3^{3k} &= \frac{1}{3^{4} } \\ 3^{ \blue{3k}} &= 3^{ - \blue{4}} \\ 3k&= - 4 \\ k&= \frac{ - 4}{3} \\ k&= \bold{ - \frac{4}{3} } \end{aligned}[/tex]

[tex]\rm{The\:value\:of\:\mathit{k}\:is\:\bold{ - \frac{4}{3}}} \\ \\ \small{ \blue{ \mathfrak{That's\:it\: :)}}}[/tex]

Answer: The figure you are describing seems to be a rectangle with a length of 6 yards and a width of 4 yards. The area of a rectangle is calculated by multiplying its length by its width. So, the area of this rectangle would be 6 yd * 4 yd = 24 square yards.

Answer: Total Area = 36yd

Step-by-step explanation:

You have to find the area of both square and triangle.

Area Triangle = .5x5x8= 20yd
Area Square = 4x4 = 16yd

Triangle + Square - 20yd+16yd = 36yd

Answer:

4.816

Step-by-step explanation:

5.66sin45 = 4.816
sin = opposite over hypotenuse

Answer:

Not a right triangle

Step-by-step explanation:

[tex]10^2+15^2\stackrel{?}{=}18^2\\100+225\stackrel{?}{=}324\\325\neq324[/tex]

As the sides of the triangle do not follow the Pythagorean Theorem, then the triangle is not a right triangle.

The error in the problem is given by the equation in step(3). and the value is given by g ( x ) = 2x² + 4x - 4

Given data ,

Let the equation be represented as g ( x )

Now , the value of g ( x ) is

g ( x ) = 2 ( x + 1 )² - 6

On simplifying the equation , we get

From the distributive law of multiplication , we get

g ( x ) = 2 ( x + 1 ) ( x + 1 ) - 6

g ( x ) = 2 ( x² + 2x + 1 ) - 6

On further simplification , we get

g ( x ) = 2x² + 4x + 2 - 6

g ( x ) = 2x² + 4x - 4

Therefore, the value of g ( x ) is g ( x ) = 2x² + 4x - 4

Hence , the equation is g ( x ) = 2x² + 4x - 4

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A surveyor wants to determine the width of a river is 43.34 m.

From given figure, angle ACB = angle ECD  (Vertically opposite angles are equal)

Angle BAC = Angle EDC = 90

So, triangle ABC and triangle ECD are similar.

AB/DE = AC/CD

AB/18.6 = 79.6/34.2

AB/18.6 = 2.33

AB = 2.33×18.6

AB = 43.34 m

Therefore, a surveyor wants to determine the width of a river is 43.34 m.

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The statements that are true regarding the expression with the exponents -1, -3, and 1 are;

An exponent is a quantity or power to which a value, expression or number is to be raised.

The specified expression can be presented as follows;

8⁻¹ × 8⁻³ × 8

Therefore; 8⁻¹ × 8⁻³ × 8 = 8⁽⁻¹ ⁺ ⁻³ ⁺ ¹⁾ = 8⁻³ = 1/8³

1/8³ = 1/512

Similarly, we get; 8⁷ × 8⁻¹⁰ = 8⁽⁷ ⁻ ¹⁰⁾ = 8⁻³ = 1/8³

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