Anyone can help please?

[tex] \Large{\boxed{\sf \dfrac{1 + 2i}{3 - 4i} + \dfrac{2 - i}{5i } = - \dfrac{2}{5}}} [/tex]

[tex] \\ [/tex]

Given sum:

[tex] \sf \: \dfrac{1+2i}{3-4i} + \dfrac{2-i}{5i}[/tex]

[tex] \\ [/tex]

We can simplify the sum only if the denominators of the two fractions are the same. Since they are different, we have to multiply the numerator and the denominator of each fraction by the denominator of the other one.

[tex] \sf \: \dfrac{1+2i}{3-4i} + \dfrac{2-i}{5i} = \dfrac{5i(1 + 2i)}{5i(3 - 4i)} + \dfrac{(3 - 4i)(2 - i)}{(3 - 4i)5i} \\ \\ \\ \sf \: = \dfrac{5i + 10 {i}^{2} }{15i - 20 {i}^{2} } + \dfrac{6 - 3i - 8i + 4 {i}^{2} }{15i - 20 {i}^{2} } [/tex]

[tex] \\ [/tex]

Replace i² with -1:

[tex] \sf \dfrac{5i + 10 {i}^{2} }{15i - 20 {i}^{2} } + \dfrac{6 - 3i - 8i + 4 {i}^{2} }{15i - 20 {i}^{2} } \: \\ \\ \\ \\ \sf \: = \dfrac{5i + 10( - 1)}{15i - 20( - 1)} + \dfrac{ 6 - 11i + 4( - 1)}{15i - 20( - 1)} \\ \\ \\ \\ \sf = \dfrac{5i - 10}{20 + 15i} + \dfrac{2 - 11i}{20 + 15i} [/tex]

[tex] \\ [/tex]

Simplify the expression:

[tex] \sf = \dfrac{5i - 10}{20 + 15i} + \dfrac{2 - 11i}{20 + 15i} \\ \\ \\ \\ \sf \: = \dfrac{5i - 10 + 2 - 11i}{ 20 + 15i} = \sf \dfrac{ - 8 - 6i}{20 + 15i}[/tex]

[tex] \\ [/tex]

To write our solution in the x + iy form, also known as the algebraic form, we have to understand what the conjugate of a complex number is.

[tex] \textsf{Let z be our complex number, and} \: \overline{\sf z} \: \textsf{its conjugate.} [/tex]

[tex] \\ [/tex]

The conjugate of z, [tex] \overline{ \sf z}, [/tex] is the complex number formed of the same real part as z but of the opposite imaginary part.

Since x is the real part of z, and y is its imaginary part, this can be expressed as:

[tex] \sf If \: z = x + iy \:, then \: \overline{ \sf z} = x - iy [/tex]

[tex] \\ [/tex]

Now, we have to multiple both the denominator and the numerator of our fraction by the conjugate of its denominator:

[tex]\sf \dfrac{ - 8 - 6i}{20 + 15i} = \dfrac{( - 8 - 6i)( \overbrace{20 - 15i}^{ \overline{z}}) }{ (20 + 15i)( \underbrace{20 - 15i}_{ \overline{z}}) } \\ \\ \\ \sf = \dfrac{ - 160 + 120i - 120i + 90 {i}^{2} }{400 - 300i + 300i - 225 {i}^{2} } \\ \\ \\ \sf = \dfrac{ - 160 + 90 {i}^{2} }{400 - 225 {i}^{2} }[/tex]

[tex] \\ [/tex]

One more time, substitute -1 for i²:

[tex] \sf \: \dfrac{ - 160+ 90 {i}^{2} }{400 - 225 {i}^{2} } \: = \dfrac{ - 160 + 90( - 1)}{400 - 225( - 1)} \\ \\ \\ \sf = \boxed{\sf - \dfrac{ 250}{625}} [/tex]

[tex] \\ [/tex]

Finally, let's simplify our result:

[tex] \sf - \dfrac{250}{625} = - \dfrac{2 \times 125}{5 \times 125} = \boxed{ \boxed{ \sf - \dfrac{2}{5}}}[/tex]

[tex] \\ \\ [/tex]

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Answer:

a)1/216

b)1/36

Step-by-step explanation:

p(2r)and p(3b) and p(4g)

1/6×1/6×1/6=1/216

p(4r) and p(even b) and p(no less than 3)

1/6×3/6×2/6=1/36

Answer:

see explanation

Step-by-step explanation:

(a)

given y is inversely proportional to x² then the equation relating them is

y = [tex]\frac{k}{x^2}[/tex]

to find k use any ordered pair from the table

using (1, 4 ) and substituting into equation

4 = [tex]\frac{k}{1^2}[/tex] = [tex]\frac{k}{1}[/tex] , thus

k = 4

y = [tex]\frac{4}{x^2}[/tex] ← equation of proportion

(b)

when y = 25 , then

25 = [tex]\frac{4}{x^2}[/tex] ( multiply both sides by x² )

25x² = 4 ( divide both sides by 25 )

x² = [tex]\frac{4}{25}[/tex] ( take square root of both sides )

x = ± [tex]\sqrt{\frac{4}{25} }[/tex] = ± [tex]\frac{2}{5}[/tex]

then

positive value of x when y = 25 is x = [tex]\frac{2}{5}[/tex]

The number 'n' is equal to 14

In the question, we have been given that the number n is equal to 7 more than half of the number 'n'.

So in these linear equation-solving types of problems, first of all, we convert the written sentence into an equation.

So the equation for the following problem statement is -

n = 7 + n/2  

{since we have been given the number equals 7 more than half of the number }

Solving the equation we have,

n = 14.

We can easily see that 14 is the number which is 7 more than half of itself.

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Answer:

Step-by-step explanation:

To find the linearization of the function f(x) = sqrt(x) at x = a, we can use the linearization formula:

f(x) ≈ f(a) + f'(a) (x - a)

To use this formula, we first need to find the value of f(a) and f'(a). At x = a = 4, the value of f(x) is f(4) = sqrt(4) = 2. The derivative of f(x) is f'(x) = 1/(2 * sqrt(x)), so the value of f'(a) is f'(4) = 1/(2 * sqrt(4)) = 1/4.

Substituting these values into the linearization formula, we get:

f(x) ≈ 2 + 1/4 (x - 4)

This is the linearization of f(x) = sqrt(x) at x = 4. It is an approximation of the function f(x) that is valid for values of x that are close to 4.

The answer, based on the information provided, fractional exponents will be 3.

A product in which the same integer is used repeatedly as a factor is represented by a number elevated to a power. The exponent provides the power, and the integer is referred to as the base. The exponent indicates the number of factors, while the base is the repeating factor (the multiplied number).

That number will be the outcome! One of the simplest exponent laws is this one. You'll see that this rule doesn't merely apply to numerical data. We might, for example, elevate a statistic to the initial power.

[tex]=3^{\frac{1}{4} }\times3^{\frac{1}{4} }\times3^{\frac{1}{4} }\times3^{\frac{1}{4} }\\=3^{\frac{1}{4}+{\frac{1}{4}+{\frac{1}{4}+{\frac{1}{4}\\\\=3^{1} \\=3[/tex]

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The distribution of the sample mean for all possible simple random samples of size 15 from the population of net worth in the US will be approximately normal, with a mean of μ and a standard deviation of σ/√15.

Courtney's estimate of the net worth of all people in the US is strongly skewed to the right. This means that the majority of the population has a net worth that is lower than the average net worth. When Courtney selects an SRS of size n = 15 from this population and calculates the sample mean, the shape of the distribution of the sample mean for all possible simple random samples of size 15 from this population will be approximately normal.

The Central Limit Theorem states that if a population has a mean of μ and a standard deviation of σ, then the sampling distribution of the sample mean will be approximately normal, no matter what the shape of the population distribution is, as long as the sample size is large enough. Since Courtney's sample size of 15 is large enough, the sampling distribution of the sample mean will be approximately normal.

The sample mean will have a mean of μ, which is the population mean, and a standard deviation of σ/√n, where n is the sample size. Since Courtney's sample size is 15, the standard deviation of the sample mean will be σ/√15. This means that the sample mean will be normally distributed with a mean of μ and a standard deviation of σ/√15.

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Answer:

Step-by-step explanation: for the first part, 525/7 = 75.

for the second part, 55y = x, which means 55 words per day for a total of (55*7) or 385

for the third part, 1200-525-385=290, not totally sure, but I think this is the answer, the text was a bit blurry and I couldn't understand everything, try not to following this blindly and just see if it makes sense. Hope I could help

Answer:

5x + 100 = 7x + 30

100 - 30 = 7x - 5x

70 = 2x

x = 35

3y + 16 = 5y

16 = 5y - 3y

16 = 2y

y = 8

Step-by-step explanation:

________________________________

=> 5x + 100 = 7x + 30

=> 100 – 30 = 7x – 5x

=> 70 = 2x

=> x = 70/2

=> x = 35

The value of x is 35.

________________________________

=> 16 = 5y – 3y

=> 16 = 2y

=> y = 16/2

=> y = 8

The value of y is 8.

_______________________________

Answer:

Step-by-step explanation:

There is a scale factor of 2 so since the trapezoid is 9 we can divide by 2 so AD is 4.5 and CB is 4.5

YZ is the same as ZW so 8

Using the scale factor we can divide 8 by 2 to get 4

Perimeters can be found from the sums so

ABCD would be 19 and WXYZ would be 38

Answer:

y = (-2/3)x + 6

Step-by-step explanation:

To find the equation of the line that passes through the points (-3, 8) and (6, 2), we can use the slope-intercept form of the equation of a line, which is:

y = mx + b

where m is the slope of the line and b is the y-intercept, which is the point where the line crosses the y-axis.
To find the slope of the line, we can use the following formula:

m = (y2 - y1) / (x2 - x1)

where (x1, y1) and (x2, y2) are two points on the line. Plugging in the coordinates of the two points given in the problem, we get:

m = (2 - 8) / (6 - (-3)) = -6/9 = -2/3

Next, we need to find the y-intercept of the line. This is the point where the line crosses the y-axis, which means that the x-coordinate of this point is 0. Since we know the slope of the line and one of the points that it passes through, we can use the point-slope form of the equation of a line to find the y-intercept:

y - y1 = m(x - x1)

Substituting the values we have calculated above and setting x = 0, we get:

y - 8 = (-2/3)(0 - (-3))

Solving for y, we find that the y-intercept is 8 - (2/3) * 3 = 8 - 2 = 6.

Therefore, the equation of the line that passes through the points (-3, 8) and (6, 2) is:

y = (-2/3)x + 6

Answer:

Step-by-step explanation:

Let, the Speed of the water be 'x'

and the Radius be 'r'

Now, the speed of Water Varies inversely with the Radius

So, x ∝ 1/r

      x = k/r   -----------(i)

Here, k is the proportionality constant

Now,

Given that, x = 2.5 ft per sec

then, 30 ft,

So, from eq. (i)

               2.5 = k/30

          So, k = 75

Again From eq. (i)

                    x = 75/r

          Now, if r = 3ft

           then, x = 75/3

                     x = 25

Hence, Speed of the water (x) = 25 ft per sec.

Answer:30x^2 - 54x - 18

Step-by-step explanation: To find the product of f and g, we can simply multiply the expressions for f and g together. The product of f and g is:

(f * g)(x) = (6x + 2)(-5x - 9)

          = -30x^2 - 54x - 18

So, the product of f and g is -30x^2 - 54x - 18

Answer:

  £46

Step-by-step explanation:

You want to know the cost of luggage in the hold when 2 in the hand and 1 in the hold is £86, and 1 in the hand and 1 in the hold is £66.

Let 'a' and 'b' represent the costs of a piece of hand luggage and one in the hold, respectively. Then the given relations are ...

  2a +b = 86

  a +b = 66

Subtracting the first equation from twice the second gives ...

  2(a +b) -(2a +b) = 2(66) -(86)

  b = 46 . . . . . . . simplify

It costs £46 to put an item of luggage in the hold.

Answer:

[tex]w \approx 96.6 \, \textrm{m}[/tex]

Step-by-step explanation:

See the attached image for my labeled diagram.

To solve for [tex]w[/tex], we can use the trigonometric ratio tangent.

[tex]\tan(\theta)=\dfrac{\textrm{opposite}}{\textrm{adjacent}}[/tex]

[tex]\tan(44\textdegree)=\dfrac{w}{100 \, \textrm{m}}[/tex]

↓ multiply both sides by 100m

[tex]100 \, \textrm{m} \cdot \tan(44\textdegree)=w[/tex]

↓ plug [tex]100\, \textrm{m} \cdot \tan(44\textdegree)[/tex] into a calculator

[tex]96.56887 \, \textrm{m} \approx w[/tex]

[tex]w \approx 96.56887 \, \textrm{m}[/tex]

↓ round to 3 significant figures

[tex]w \approx 96.6 \, \textrm{m}[/tex]

Answer:

Step-by-step explanation:

Let total mass be x

(3/4)*x=15

on solving

x=20

now,

mass of 2/5 of piece of metal is-

=> (2/5)*20

=> 8

Ans- 8kg

Answer: the answer is A

Step-by-step explanation:

The first, second, third, and fourth consecutive even integers must be greater than 10, 12, 14, and 16 respectively.

A numerical expression is a mathematical statement written in the form of numbers and unknown variables. We can form numerical expressions from statements.

Let the four consecutive even integers be n, (n + 2), (n + 4) and (n + 6).

Such that twice the sum of the second and third exceeds 3 times the first by 32.

2{(n + 2) + (n + 4)} > 3(n) + 32.

2n + 4 + 2n + 8 > 3n + 32.

4n - 3n > 32 - 12.

n > 10.

So, the consecutive even numbers must all be greater than 10.

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Answer:

Translation: 4 to the right and 8 up.

Scale factor: 3

Step-by-step explanation:

If you move CDE 4 spaces to the right and 8 spaces up C and C' will match up. If you dilate (make bigger) CDE 3 times it will perfectly be C'D'E'.

To expand the expression (x + 10)(3x² + 5x - 2), we can use the distributive property, which states that for any numbers a, b, and c, the product (a + b)(c) can be written as a * c + b * c. Applying this property to the given expression, we get (x + 10)(3x² + 5x - 2) = (x * 3x² + 10 * 3x²) + (x * 5x + 10 * 5x) + (x * -2 + 10 * -2) = 3x³ + 30x² + 5x² + 50x - 2x - 20. Combining like terms, we get 3x³ + 35x² + 51x - 20. Thus, the expanded form of the expression (x + 10)(3x² + 5x - 2) is 3x³ + 35x² + 51x - 20.

Using mathematical operations on the word problem, the amount he will receive as his change is $1.43

A word problem is a few words are presented in a form of problem and  needs to be solved by way of a mathematical calculation.

This problem can be solved using mathematical operation such as addition and subtraction.

The total amount he has with him = $15

The current bill = $8.33

The fruit salad = $2.99

Tip = $2.25

Total amount spent = 8.33 + 2.99 + 2.25

The total amount spent = 13.57

The amount of change = total amount he has with him - total amount spent

The amount of change = 15 - 13.57

The amount of change = 1.43

The change he is going to receive is $1.43

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Answer: 24

Step-by-step explanation:

Answer:

h = 24

Step-by-step explanation:

Answer:

2

Step-by-step explanation:

Answer:

[tex]\frac{10}{30}[/tex]

Step-by-step explanation:

Answer:

10/30

Step-by-step explanation:

Step-by-step explanation:

since it is geometric sequence we will use the formula

[tex]tn = {ar}^{n - 1} [/tex]

a = 6

r = 4

The first term

T1(a) = 6

The second term

[tex]t2 = {a \times r}^{2 - 1} [/tex]

[tex]t2 = {6 \times 4}^{1 } = 24[/tex]

The Third Term

[tex]t3 = {a \times r}^{3 - 1} = {a \times r}^{2} [/tex]

[tex]t3 = {6 \times 4}^{2} = 6 \times 16 = 96[/tex]

The fourth term

[tex]t4 = {a \times r}^{4 - 1} = {a \times r}^{3} [/tex]

[tex]t4 = {6 \times 4}^{3} = 6 \times 64 = 384[/tex]

The fifth term

[tex]t5 = {a \times r}^{5 - 1} = {a \times r}^{4} [/tex]

[tex]t5 = {6 \times 4}^{4} = 6 \times 256 = 1,536[/tex]

i hope these helped

Answer:

6, 24, 96, 384, 1536, ...

Step-by-step explanation:

[tex]\boxed{\begin{minipage}{5.5 cm}\underline{Geometric sequence}\\\\$a_n=ar^{n-1}$\\\\where:\\\phantom{ww}$\bullet$ $a$ is the first term. \\\phantom{ww}$\bullet$ $r$ is the common ratio.\\\phantom{ww}$\bullet$ $a_n$ is the $n$th term.\\\phantom{ww}$\bullet$ $n$ is the position of the term.\\\end{minipage}}[/tex]

Given:

Substitute the given values of a and r into the formula to create an equation for the nth term:

[tex]a_n=6(4)^{n-1}[/tex]

To find the first 5 terms of the geometric sequence, substitute n = 1 through 5 into the equation.

[tex]\begin{aligned}\implies a_1&=6(4)^{1-1}\\&=6(4)^{0}\\&=6(1)\\&=6\end{aligned}[/tex]

[tex]\begin{aligned}\implies a_2&=6(4)^{2-1}\\&=6(4)^{1}\\&=6(4)\\&=24\end{aligned}[/tex]

[tex]\begin{aligned}\implies a_3&=6(4)^{3-1}\\&=6(4)^{2}\\&=6(16)\\&=96\end{aligned}[/tex]

[tex]\begin{aligned}\implies a_4&=6(4)^{4-1}\\&=6(4)^{3}\\&=6(64)\\&=384\end{aligned}[/tex]

[tex]\begin{aligned}\implies a_5&=6(4)^{5-1}\\&=6(4)^{4}\\&=6(256)\\&=1536\end{aligned}[/tex]

Therefore, the first 5 terms of the given geometric sequence are:

Answer:

72kg

Step-by-step explanation:

Stephen = 72

Saravanan = x

Shamsul = (x • 1.5) since Shamsul's mass is 1.5 times Saravanan's mass

1. make an equation:

( 72 + x + (x • 1.5) ) ÷ 3 = 64

2. solve it:

( 72 + x + (x • 1.5) ) ÷ 3 = 64

( 72 + x + (x • 1.5) ) = 64 • 3

( 72 + x + (x • 1.5) ) = 192

x + 1.5x = 192 - 72

2.5x = 120

x = 120 ÷ 2.5

x = 48

Thus shamsul's mass is:

48 • 1.5 = 72kg

Answer:

No solution

Step-by-step explanation:

Answer:

Below

Step-by-step explanation:

w = 7 - sqrt (g)    ?? ( syntax is unclear in your post)

sqrt(g) = 7-w

g = (7-w)^2

Answer:

Step-by-step explanation: So First you  would need to multiply 2x3 because if you need to triple two cups you have to multiply so the answer is 6 cups

The different combinations of groups for the group project is (b) 560

From the question, we have the following parameters that can be used in our computation:

Total number of people, n = 16

Numbers to selection, r = 3 people

The number of ways of selection could be drawn is calculated using the following combination formula

Total = ⁿCᵣ

Where

n = 16 and r = 3

Substitute the known values in the above equation

Total = ¹⁶C₃

Apply the combination formula

ⁿCᵣ = n!/(n - r)!r!

So, we have

Total = 16!/(13! * 3!)

Evaluate

Total = 560

Hence, the number of ways is 560

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