Compute The Divergence F And The Curl F Of The Vector Field. (your Instructors Prefer Angle Bracket Notation (2024)

Mathematics High School

Answers

Answer 1

The divergence of f is ∇ · f = 2x + 4y + 4z. The curl of the vector field is ∇ ✕ f = < -4yz, -2x, 4xy >.

Let's first write the vector field f in component form:

f(x,y,z) = < [tex]x^2, 2y^2, 2z^2[/tex] >

Now we can compute the divergence and curl:

Divergence:

The divergence of a vector field F = < F1, F2, F3 > is defined as:

∇ · F = (∂F1/∂x) + (∂F2/∂y) + (∂F3/∂z)

Applying this formula to our vector field f(x,y,z), we get:

∇ · f = (∂/∂x)([tex]x^2[/tex]) + (∂/∂y)(2[tex]y^2[/tex]) + (∂/∂z)(2[tex]z^2[/tex])

= 2x + 4y + 4z

So the divergence of f is:

∇ · f = 2x + 4y + 4z.

Curl:

The curl of a vector field F = < F1, F2, F3 > is defined as:

∇ ✕ F = < (∂F3/∂y) - (∂F2/∂z), (∂F1/∂z) - (∂F3/∂x), (∂F2/∂x) - (∂F1/∂y) >

Applying this formula to our vector field f(x,y,z), we get:

∇ ✕ f = < (∂/∂y)(2[tex]z^2[/tex]) - (∂/∂z)(2[tex]y^2[/tex]), (∂/∂z)([tex]x^2[/tex]) - (∂/∂x)(2[tex]z^2[/tex]), (∂/∂x)(2[tex]y^2[/tex]) - (∂/∂y)([tex]x^2[/tex]) >

= < -4yz, -2x, 4xy >

So the curl of f is:

∇ ✕ f = < -4yz, -2x, 4xy >.

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Answer 2

We have the vector field f = <x^2, 2y^2, 2z^2>. The divergence of f is given .

The curl of f is given by:

curl(f) = <(∂f_3/∂y - ∂f_2/∂z), (∂f_1/∂z - ∂f_3/∂x), (∂f_2/∂x - ∂f_1/∂y)>

= <0, -2z, 4y - 4x>

Therefore, div(f) = 2x + 4y + 4z and curl(f) = <0, -2z, 4y - 4x>.

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Related Questions

You want to prove that FHG is similar to RXS by the SSS Similarity Theorem. Complete the proportion that is needed to use this theorem.

Answers

Answer:

[tex]\frac{FH}{RX} =\frac{HG}{XS} =\frac{FG}{RS}[/tex]

Step-by-step explanation:

The order of the letters matters in any similarity theorem.

When triangle FHG is similar to triangle RXS, the lengths of FH must be similar to RX.

When 2 similar lines are divided, that number is the factor by which one triangle can be dilated to match the other.

That being said, each of these proportion statements can be read as individual division expressions that equal the next division expression.

This means [tex]FH\div RX = HG\div XS = FG\div RS[/tex].

If you look at it this way, you can see that the sides you need to fill in must be the corresponding similar sides, so that you end up with the factor for the dilation of the 2 triangles.

What is the area of the trapezoid above?
A. 188 cm²
B. 196 cm²
C. 392 cm²
D. 5,236 cm²

Answers

Answer:

B. 196cm²

Step-by-step explanation:

Area = h/2 (b_1+b_2)

[tex]\sf\cfrac{7}{2}\:(22+34)[/tex]

Add 22 and 34 = 56

[tex]\sf \cfrac{7}{2} \times 56[/tex]

Let's express 7/2 * 56 as a single fraction:-

[tex]\sf \cfrac{7\times 56}{2}[/tex]

Multiply 7 and 56 = 392

[tex]\sf \cfrac{392}{2}[/tex]

Divide

[tex]\sf 196[/tex]

Therefore, the area of the trapezoid is 196cm².

_______________________

Zhang is constructing a loading ramp (BA) that is 4-feet high (AC). The back of the base (BC) must be 12.8 ft. How long must the entire base (BD) be?

Proportion : __________________

Front of base: __________

Total base: __________

Answers

Answer:

Step-by-step explanation:

This is a geometric means problem where the height (length AC of 4) is the geometric mean between the base BC and the base DC. The proportion is

[tex]\frac{12.8}{4} =\frac{4}{CD}[/tex] . Cross multiply to get

12.8CD = 16 and divide both sides by 12.8 to get that

CD = 1.25

That means that the length of the whole thing, BD, is

12.8 + 1.25 = 14.05

Scores for a civil service exam are normally distributed with a mean of 75 and a standard deviation of 6.5. what score marks the difference between the bottom 10% and the top 90%?

Answers

The lowest score you can earn and still be eligible for employment is 85.6925.

Problems of normally distributed samples can be solved using the z-score formula.

In a set with mean and standard deviation , the zscore of a measure X is given by:

The Z-score measures how many standard deviations the measure is from the mean. After finding the Z-score, we look at the z-score table and find the p-value associated with this z-score. This p-value is the probability that the value of the measure is smaller than X, that is, the percentile of X. Subtracting 1 by the pvalue, we get the probability that the value of the measure is greater than X.

In this question, we have that:

Top 5%.

At least the 100-5 = 95th percentile.

The 95th percentile is X when Z has a pvalue of 0.95. So X when Z = 1.645.

x-75 = 1.645*6.5

x = 85.692

Thus the lowest score you can earn and still be eligible for employment is 85.6925.

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4(3) / 4(8) +.9(7.8 x 7.45)

Answers

The value is 52. 674

How to determine the value

We have;

4(3) / 4(8) +.9(7.8 x 7.45)

From BODMAS, we have that we must must open up the bracket

= [tex]\frac{12}{32} + 0. 9 (58. 11)[/tex]

Divide the fraction to get decimals

= [tex]0. 375 + 52. 299[/tex]

Add the decimals

= [tex]52. 674[/tex]

Thus, the value is 52. 674

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Yolanda paid for her movie ticket using 28 coins, all nickels and quarters. the ticket cost $4. which system of linear equations can be used to find the number of nickels, n, and the number of quarters, q, yolanda used? n q = 28 0.05n 0.25q = 4 n q = 4 5n 25q = 28 n q = 28 5n 25q = 4 n q = 4 0.05n 0.25q = 28

Answers

The system of linear equations can be used to find the number of nickels, n, and the number of quarters, q, yolanda used is n + q = 28

n + q = 280.05n + 0.25q = 4

Simultaneous equation

n + q = 28

0.05n + 0.25q = 4

n = 28 - q

substitute n = 28 - q

0.05n + 0.25q = 4

0.05(28 - q) + 0.25q = 4

1.4 - 0.05q + 0.25q = 4

- 0.05q + 0.25q = 4 - 1.4

0.20q = 2.6

q = 2.6 / 0.20

q = 13

Substitute into

n + q = 28

n + 13 = 28

n = 28 - 13

n = 15

Answers

Answer:

Step-by-step explanation:

This is the only problem that could be answered with this equation, because you would add the bracelets he made (2+4) then times that number by the days he made them, 5, then divide by the number of groups, 6. Which equals 5 x ( 2 + 4 ) divided by 6.

I hope this helps!

Larry has 6 shirts and 4 pairs of jeans. How many different combinations of one shirt and one pair of jeans can Larry make

Answers

Answer:

24 combinations

Step-by-step explanation:

To get to this answer, I multiplied 6 x 4, since there are 6 shirts and 4 pairs of jeans, there will be a total of 24 combinations. For combinations of two different things, you can just multiply the amount of each item by the other item. hope this helps :)

The functions f(x) and g(x) are graphed.

On a coordinate plane, a curved red line with an upward arc, labeled g of x, crosses the y-axis at (0, 4) and the x-axis at (2, 0). A straight blue line with a negative slope, labeled f of x, crosses the y-axis at (0, 4) and the x-axis at (2, 0).

Which represents where f(x) = g(x)?

f(0) = g(0) and f(2) = g(2)
f(2) = g(0) and f(0) = g(4)
f(2) = g(0) and f(4) = g(2)
f(2) = g(4) and f(1) = g(1)

Answers

The function equality that represents the point where f(x) = g(x) is; A: f(0) = g(0) and f(2) = g(2)

How to interpret Graphed Functions?

From the question, the curved red line represents g(x) while the straight blue line represents f(x).

Now, the equality of functions f(x) = g(x) is represented as a common function between their curves. Thus, we will just need to find a common point for both. This is;

f(x) has points (0, 4) and (2, 0).

g(x) has points (0,4) and (2,0).

It is pertinent to note that both functions have the same y-value for x=0 and x = 2. Thus, we can say that;

f(2) = g(2) and f(0) = g(0)

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how much greater is 9 hundredths than 8 thousandths

Answers

9 hundredths is greater than 8 thousandths by 82 thousandths

Place value

From the question, we are to determine how much greater 9 hundredths is than 8 thousandths

9 hundredths = 9/100 = 0.09

8 thousandths = 8/1000 = 0.008

Then,

0.09 - 0.008 = 0.082

= 82/1000

≡ 82 thousandths

Hence, 9 hundredths is greater than 8 thousandths by 82 thousandths

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Five guys walk into a bar, how many ways can they sit so as to be arranged from oldest to youngest?

Answers

There are 24 ways in which 5 guys can sit if arranged from oldest to youngest.

We have,

Five guys.

Now,

We know that,

Total number of ways to arranged around a table (n) = (n-1)!

So,

For n = 5,

I.e.

(n - 1)! = (5 - 1)! = 4!

So,

Total number of ways to arranged Five guys rom oldest to youngest (n) = (n-1)!

i.e.

= (5 - 1)! = 4!

We get,

= 4 × 3 × 2 × 1

i.e.

Total number of ways to arranged Five guys rom oldest to youngest (n) = 24.

Hence we can say that there are 24 ways in which 5 guys can sit if arranged from oldest to youngest.

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A flat rectangular piece of aluminum has a perimeter of 60 inches. The length is 12 inches longer than the width. Find the width.

Answers

Answer:

width = 9 inches

Step-by-step explanation:

let width be w then length is w + 12

the perimeter ( P) is calculated as

P = 2( width + length )

here P = 60 , then

2(w + w + 12) = 60 ( divide both sides by 2 )

2w + 12 = 30 ( subtract 12 from both sides )

2w = 18 ( divide both sides by 2 )

w = 9

that is the width of the rectangle is 9 inches

Freddie has 2 times as many comic books as David. The ratio of the number of comic books David has to the number of comic books Gary has is 5 : 3. Freddie has 110 comic books. How many comic books do David and Gary have in total?

Answers

Answer:

Step-by-step explanation: make me brainliest if it is correct

what inequality is shown on the graph ?

Answers

The inequality of the graph that is given is: y < 2x - 3.

How to Write the Inequality of a Graph?

First find the slope (m) = rise / run of the line.

Rise = 2 units

Run = 1 unit

Slope (m) = 2/1 = 2

Y-intercept (b) = -3.

Since the shaded side is at the left, we are going to use the ">" sign. Substitute m = 2 and b = -3 into y < mx + b.

y < 2x - 3

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There are 120 students in class 7. it is known that 2 3 out of them like maths, out of which 35 are girls. also, out of total students, 3 5 are boys. (3) (a) what is the number of girls in the class? (b) how many students like maths? (c) how many boys like maths?

Answers

maths questions can be solved using unitary method.

Answer:

(a) The total number of girls in the class are 48

(b) The total number of students that like maths are 80

Solution:

According to the question, [tex]\frac{3}{5}[/tex] of total students are boys

number of boys = [tex]\frac{3}{5}[/tex] × 120 = 72

Also, According to the question, [tex]\frac{2}{3}[/tex] of total students like maths

number of students = [tex]\frac{2}{3}[/tex] × 120 =80

(a) The total number of girls in the class = total number of students - total number of boys

total number of girls in the class = 120 - 72 = 48

(b) The total number of students that like maths are 80

(c) The total number of boys like maths = total number of students that like maths - total number of girls that like maths

total number of boys like maths = 80 - 35 = 45

What is unitary method?

The unitary approach is a strategy for problem-solving that involves first determining the value of a single unit, then multiplying that value to determine the required value.

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search for research that has utilized analytical tools mentioned in the readings (i.e., single-sample ztest, t tests) and identify why that analysis was appropriate for the data.

Answers

The given analytical tools:

a. The single sample z-test analysis is appropriate for the data where the sample size is large and all data points are independent. Where the standard deviation is given(known).

b. The single sample t-test analysis is appropriate for the data where the sample size is small and the population variation or standard deviation is unknown.

What is single sample z-test analysis?

This is a type of hypothesis test where the sample means and the population means are compared when the population variance or the standard deviation are given(known).

z = (X - μ)/(σ/√n)

Where X - sample mean, μ - population mean, σ - population standard deviation, and n - sample size.

Here the sample size is large and all data points are independent.

What is single sample t-test analysis?

This is a type of parametric hypothesis test where the sample means and the population means are compared when the population variance or the standard deviation are not given(unknown).

t = (X - μ)/(s/√n)

Where X - sample mean, μ - population mean, s - sample standard deviation, and n - sample size.

Here the sample size is small.

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need help don’t know what to do

Answers

(B) option is correct - not continuous at x = 0

Complete the system that models the heights of the ball and the receiver’s hands over time. h = -16t² 14t h = -16t² t

Answers

Answer: The height of the ball at time t is given as h = -16t² + 14t + 6 while the height of the receiver hand at time t is given as h = -16t² + 10t + 8

What is an equation?

An equation is an expression that shows the relationship between two or more variables and numbers.

A quarterback throws a football toward a receiver from a height of 6 ft. The initial vertical velocity of the ball is 14 ft/s. At the same time that the ball is thrown, the receiver raises his hands to a height of 8 ft and jumps up with an initial vertical velocity of 10 ft/s.

The height of the ball at time t is given as h = -16t² + 14t + 6 while the height of the receiver hand at time t is given as h = -16t² + 10t + 8

Answer:

Step-by-step explanation:

7 cm
D
What is the sum of the areas of circle C and circle D?
O7ft units²
O 14 units²
O 49 units²
O98 units²

Answers

The sum of areas of circle C and circle D exists 98 square units.

How to estimate the sum of the areas of circle C and circle D?

Given: Radius of circle = 7 units

Area of circle [tex]=\pi r^2[/tex] ..............(1)

Where, r be the radius of circle

The radius of circle C, r = 7 units

(1) becomes

Area of circle, C [tex]= \pi 7^2[/tex] square units

[tex]= 49\pi[/tex] units²

Area of circle, D [tex]= \pi 7^2[/tex] square units

[tex]= 49\pi[/tex] units²

Sum of areas of circles C and D,

= Area of circle C + Area of circle D

= 49 + 49

= 98 units²

The sum of areas of circles C and D exist 98 units².

Therefore, the correct answer is option d) 98 units².

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You use generally accepted facts (such as: there are approximately 6 billion people on the planet. ) without citing them is it plagiarism?

Answers

No, this not plagiarism .You can use generally accepted facts without citing them.

What defines self-plagiarism?

Self-plagiarism is defined as a type of plagiarism in which the writer republishes a work in its entirety or reuses portions of a previously written text while authoring a new work.

What is considered general knowledge in plagiarism?

Common knowledge is typically defined as facts that are unsupported by at least five reliable sources. In the discipline of composition studies, for instance, the statement "Writing is difficult" is taken for granted because at least five reliable sources may support it.

How to Avoid plagiarism common knowledge?

Quoting correctly, paraphrasing and summarizing correctly, and documenting (citing) correctly are the keys to providing credit and avoiding plagiarism; you can go to page 3 of this handout for more information. Almost all of the information you discover in outside sources needs to be documented.

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A circle with a diameter of 10 cm and a central angle of 30° is drawn below.

what is the area, to the nearest tenth of a square centimeter, of the sector formed by the 30° angle?

(A) 5.2 (B) 6.5 (C) 13.1 (D) 26.2

Answers

Answer:

B 6.5

Step-by-step explanation:

The area of a whole circle is

a = [tex]\pi[/tex][tex]r^{2}[/tex] A whole circle is 360 degrees. We only have part of a circle. We were also given the diameter and we need the radius. The radius is half of the diameter.

a = [tex]\frac{30}{360} \pi 5^{2}[/tex]

a= 6.5 rounded.

A whole circle exists 360 degrees. The radius exists half of the diameter.

[tex]$$a = (30/360) $\pi 5^2[/tex]

a = 6.5 rounded.

The area to the nearest tenth of a square centimeter, of the sector formed by the 30° angle exists at 6.5 rounded.

How to calculate the area of a circle with diameter?

The diameter of the circle exists double the radius of the circle. Therefore the area of the circle formula utilizing the diameter exists equivalent to π/4 times the square of the diameter of the circle. The formula for the area of the circle, utilizing the diameter of the circle

π/4 × [tex]$$diameter^2[/tex].

Given: diameter of the circle = 10 cm

then the radius of the circle is half of the diameter

Radius of the circle = 10/2 = 5 cm

Area of circle = [tex]$$\pi $r^2[/tex]

Where r be the radius of the circle

A whole circle exists 360 degrees. The radius exists half of the diameter.

[tex]$$a = (30/360) $\pi 5^2[/tex]

simplifying the above equation we get

[tex]$$a = 25$\pi$ /$12[/tex]

a = 6.5 rounded.

The area to the nearest tenth of a square centimeter, of the sector formed by the 30° angle exists at 6.5 rounded.

Therefore, the correct answer is option (B) 6.5

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Can someone help me out on these 2 geometry questions pls? ASAP!!!

Write formal proofs using the HL Theorem

Answers

Question 2

1) [tex]\overline{PR} \perp\overline{QS}, \overline{PQ} \cong \overline{PS}[/tex] (given)

2) [tex]\overline{PR} \cong \overline{PR}[/tex] (reflexive property)

3) [tex]\angle PRQ, \angle PRS[/tex] are right angles (perpendicular lines form right angles)

4) [tex]\triangle PRQ, \triangle PRS[/tex] are right triangles (a triangle with a right angle is a right triangle)

5) [tex]\triangle PRQ \cong \triangle PRS[/tex] (HL)

Question 3

1) [tex]\angle C[/tex] is a right angle, [tex]\overline{AC} \cong \overline{AE}, \overline{DE} \perp \overline{AB}[/tex] (given)

2) [tex]\angle DEA[/tex] is a right angle (perpendicular lines form right angles)

3) [tex]\triangle ACD, \triangle DAE[/tex] are right triangles (a triangle with a right triangle is a right angle)

4) [tex]\overline{AD} \cong \overline{AD}[/tex] (reflexive property)

5) [tex]\triangle ACD \cong \triangle AED[/tex] (HL)

6) [tex]\angle CAD \cong \angle DAE[/tex] (CPCTC)

7) [tex]\overline{AD}[/tex] bisects [tex]\angle BAC[/tex] (if a segment splits an angle into two congruent parts, it is an angle bisector)

Select the correct answer. which equation could possibly represent the graphed function?

Answers

The function for the given graph is f(x) = (x - 4)(x + 2)(x + 4).

How to represent a graph by an equation?A graph can be represented by the equation y = f(x).In the equation, the variable x is called the independent variable and the variable y is called the dependent variable.

Calculation:

In the given graph,

The points that lie on the x-axis are (-4, 0), (-2, 0), and (4, 0) where y-coordinate is 0 and the x coordinates are -4, -2, and 4.

On solving the given functions,

Option A:

f(x) = (x - 4)(x + 2)(x + 4)

Since y = 0 consider f(x) becomes 0

So,

(x - 4)(x + 2)(x + 4) = 0

According to the zero product rule,

x - 4 = 0, x + 2 = 0, and x + 4 = 0

⇒ x = 4, x = -2 and x = -4.

Thus, option A represents the function of the given graph.

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Disclaimer: The question given on the portal was incomplete. Here is the complete question.

Question:

Select the correct answer. which equation could represent the graphed function (given in the figure below)?

Each side of a square is increasing at a rate of 5 cm/s. At what rate (in cm2/s) is the area of the square increasing when the area of the square is 9 cm2

Answers

The area of the square increasing at 30 [tex]\frac{cm^{2} }{sec}[/tex]

The rate of change function is defined as the rate at which one quantity is changing with respect to another quantity. In simple terms, in the rate of change, the amount of change in one item is divided by the corresponding amount of change in another

Let the side of a square be x.

So,

Area of square = [tex]x^{2}[/tex]

A =[tex]x^{2}[/tex]

Differentiating with respect to time we get,

dA/dt = 2x dx/dt

When area = 9 [tex]cm^{2}[/tex] the side becomes 3 cm

At x = 3cm and dx/dt = 5 cm/s (Given)

dA/dt = 2.3.5 = 30 [tex]\frac{cm^{2} }{sec}[/tex]

Thus the area of the square increasing at 30 [tex]\frac{cm^{2} }{sec}[/tex]

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Suppose you have a box with 3 blue marbles, 2 red marbles, and 4 yellow marbles. You are going to pull out one marble, record its color, remove it from the box and draw another marble. What is the probability of pulling out a red marble followed by a blue marble? The multiplication rule says to use P(red) P(blue).

Describe the probability of finding a red marble?
Describe the probability of finding a blue marble?
Describe the process of finding the probability of finding a red marble followed by a blue marble if the first marble was permanently removed?
What affect did removing the first marble from the box have on the problem?
Describe the probability of finding a red marble followed by the blue marble if the first marble is removed?

Answers

The probability of randomly getting first a red marble and then a blue marble is:

P = 0.083

How to find the probability?

There are:

3 blue marbles2 red marbles4 yellow marbles.

For a total of 9 marbles.

The probability of getting a red marble is equal to the quotient between the number of red marbles and the total, so:

P(red) = 2/9

Then the probability of getting a blue marble is equal to the quotient between the number of blue marbles and the total, but because we already took one marble, now the total is 8.

P(blue) = 3/8

The joint probability is given by the product:

P = (2/9)*(3/8) = 0.083

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Which expresses the correct factorization
of….? look at pic

Answers

Answer:

Option 1

Step-by-step explanation:

Find 2 numbers that multiply to give 16 and sum to give +8 :

These numbers are 4 and 4

Now split +8x into +4x and +4x and factor each part :

x²+4x+4x+16 = 0

x(x+4)+4(x+4) = 0

Keep 1 bracket and the terms outside the brackets make their own :

(x+4)(x+4) = 0

(x+4)² = 0

Therefore answer will be Option 1

Hope this helped and have a good day

Answer: the first one (x+4)^2

(a+b)^2 = a^2 + 2ab + b^2

(x+4)^2 = x^2 + 8x + 4

help please much appreciated if you attempt to help

Answers

196 36 correct answer is 232

Answer:

728 cm²

Step-by-step explanation:

the surface area is the area of the 6 faces.

the opposite rectangular faces are congruent , then

SA = front/ back + sides + top/ bottom

= 2(14 × 14) + 2(6 × 14) + 2(14 × 6)

= 2 × 196 + 2 × 84 + 2 × 84

= 392 + 168 + 168

= 728 cm²

Billy goes on the teacup ride at the local carnival. He is seated 16 inches from the center of his teacup, which is spinning at a rate of 4 revolutions per minute. What is Billy’s angular velocity, in radians per second? Round your answer to the nearest hundredth.

Answers

Answer:

0.42

Step-by-step explanation:

Expand the expression / brackets: 2y (3y + 9)=

Answers

Answer:

6y^2 + 18y

Step-by-step explanation:

2y (3y + 9)=

6y^2 + 18y

[2y (3y becomes 6y^2]

[2x( + 9) becomes 18y]

Add them to obtain 6y^2 + 18y

What is the simplified form of startroot startfraction 2,160 x superscript 8 baseline over 60 x squared endfraction endroot? assume x ≠ 0.

Answers

The simplified form is [tex]6x^{3}[/tex].

What are indices in maths?

An index, or a power, is the small floating number that goes next to a number or letter.The plural of index is indices. Indices show how many times a number or letter has been multiplied by itself.

The equation is -

[tex]\sqrt{\frac{2160x^{3} }{60x^{2} } }[/tex]

We can rewrite this using property of multiplication to obtain

[tex]\sqrt{\frac{2160}{60} * \frac{x^{8} }{x^{2} } }[/tex]

We simplify variable expression in the radicand using

[tex]\frac{a^{m} }{a^{n} } = a^{m - n}[/tex]

[tex]\sqrt{36 * x^{8 - 2} }[/tex]

[tex]\sqrt{36 * x^{6} }[/tex]

[tex]\sqrt{36 * (x^{3})^{2} }[/tex]

[tex]\sqrt{36} * \sqrt{(x^{3})^{2} }[/tex]

[tex]6 * x^{3}[/tex]

Therefore, the simplified form is [tex]6x^{3}[/tex]

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