Simplify (6 +i)(8 – 3i)

51 – 10i
48 – 32i
45 – 10i
48 – 10i – 3i^2 ...?

The numbers, their sum is 7, and their product is 15 are,

(7 + √11 i)/2 and (7 - √11 i)/2.

Use the concept of complex numbers defined as:

A real number and an imaginary number are effectively combined to create a complex number. The complex number is written as a+ib, where a and ib are real and imaginary numbers, respectively.

Let the numbers x and y such that,

x+y = 7  .....(i)

xy = 15  ......(ii)

Since we know that,

(x - y)² = x² + y² - 2xy

Add 2xy on both sides,

(x - y)² + 2xy = (x² + y² + 2xy) - 2xy

(x - y)² + 2xy = (x + y)² -2xy

put the values from equations (i) and (ii),

(x - y)² + 30 = 49 - 30

Simplify it,

(x - y)²  = -11

Take square root on both sides,

x - y = √11 i   .....(iii)                 ( i = √-1)

Add equations (i) and (iii),

2x = 7 + √11 i

x = (7 + √11 i)/2

Substitute this value of x in equation (iii),

y =  (7 - √11 i)/2

Hence,

The required numbers are (7 + √11 i)/2 and (7 - √11 i)/2.

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The complete question is:

Find the complex or real numbers such that,

Their sum is 7, and their product is 15.

The decimal 0.87 can be written as the fraction 87/100, which cannot be simplified further.

The decimal 0.87 can be written as a fraction by placing the decimal over 100 and then simplifying the fraction if possible. In this case, 0.87 can be written as 87/100. However, this fraction can be simplified further. Both 87 and 100 can be divided by 1, making the simplified fraction 87/100.

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Before a theorem is proven, it is known as a proposition. This term indicates that the statement is proposed to be true and is pending proof.

A theorem called before it is proven is known as a proposition. A proposition is a statement that is proposed to be true and is awaiting proof or disproof. In contrast, a postulate is a statement that is assumed to be true without proof for the purposes of reasoning in mathematics or science, whereas a contradiction refers to a statement that is always false. A tautology is a statement that is true by necessity or by virtue of its logical form.

Final answer:

A theorem before it is proven is known as a proposition. Propositions are statements considered in mathematics and philosophy for investigation and proof. Other terms like postulate, contradiction, and tautology have different distinct meanings in logic.

Explanation:

Before a theorem is proven, it is known as a proposition. A proposition is a statement that proposes an idea, which can be either true or false, but has not yet been proven to be true. In mathematics and philosophy, this term is used to describe concepts that are considered and investigated for proof. Once a proposition is proven conclusively, it becomes a theorem. Other options like postulate, contradiction, and tautology have different meanings within mathematical logic and reasoning.

Final answer:

1/4 yard is longer than 1/4 foot because 1/4 yard is equivalent to 3/4 foot, and 3/4 is greater than 1/4.

Explanation:

The question is asking to compare the lengths of 1/4 foot and 1/4 yard. To make this comparison, it is crucial to understand the relationship between feet and yards. From the reference information, we know that 1 yard is equal to 3 feet. Thus, to find how long 1/4 yard is in feet, we use the conversion ratio:

1/4 yd = (1/4) × 3 ft = 3/4 ft.

Now, we compare 1/4 foot with 3/4 foot. Since 3/4 is greater than 1/4, it is clear that 1/4 yard is longer than 1/4 foot.

Answer: B) The typical exponential function, y = ax, has asymptote the x-axis and y-intercept x = 0.

Step-by-step explanation:

For the funcion [tex]y = a^x[/tex], the asymptote is the value that the function would never assum, but gets really close. In this case, there is no value for y = 0, because[tex]y = a^x\\0 = a^x \\ln(0) = x.lna\\ln(0)[/tex]is not defined! This way, asymptote is x-axis (y=0).

y-intercept ⇒ x=0, so

[tex]y=a^x\\y=a^0=1[/tex]

This, way, the point (0,1)

alternative b

The typical exponential function y = ax has a horizontal asymptote y = 0 and a y-intercept at the point (0,1), which makes Option B the correct answer.

In this context, the function will approach an asymptote, which is a line that the graph gets infinitely close to without touching. Specifically, for an exponential function, the horizontal asymptote is y = 0 regardless of the value of a, provided that a is nonzero. As the value of x gets very large or very negative, the value of y will get very close to 0 but will never actually reach it.

Additionally, the y-intercept of the function is the point at which it crosses the y-axis. This occurs when x is 0. Substituting 0 for x in the function, we get y = a0, which simplifies to y = 1 because any nonzero number to the power of 0 is 1. Therefore, the y-intercept of the function is (0,1).

So, for the question asking about the asymptote and y-intercept of the typical exponential function y = ax, the correct answer is Option B: asymptote y = 0; y-intercept (0,1).

By substituting the value of one number in terms of the other from the difference equation into the sum of squares equation, we find the two numbers to be 10 and -3.

Let's solve the problem step-by-step. Given that two numbers differ by 13 and the sum of their squares is 125, we can set up the following equations where x and y are the two numbers:

Substitute the value of y from equation (1) into equation (2), giving us:

x^2 + (x - 13)^2 = 125

Solving this, we find that x = 10 and hence y = -3, which means the two numbers are 10 and -3.

Using the Quadratic formula

x^4 = (x^2)^2

hence

(x^2)^2 + 4x +(-1) = 0

ax^2=1(x^2)^2

bx = 4(x)

c=(-1)

The altitude of the equilateral triangle is [tex]\( \frac{9\sqrt{3}}{2} \)[/tex] centimeters.

In an equilateral triangle, all three sides are equal in length. Let's denote the length of each side of the equilateral triangle as s centimeters.

The perimeter of an equilateral triangle is given as the sum of the lengths of all three sides:

Perimeter = 3s

Given that the perimeter is 27 centimeters, we can set up the equation:

27 = 3s

Now, we can solve for the length of each side s:

[tex]\[ s = \frac{27}{3} = 9 \][/tex]

So, each side of the equilateral triangle is 9 centimeters long.

Now, to find the altitude of the equilateral triangle, we can use the formula for the altitude of an equilateral triangle, which is:

[tex]\[ \text{Altitude} = \frac{\sqrt{3}}{2} \times \text{Length of a side} \][/tex]

Substituting the length of a side s = 9 into the formula, we get:

[tex]\[ \text{Altitude} = \frac{\sqrt{3}}{2} \times 9 \\\[ \text{Altitude} = \frac{9\sqrt{3}}{2} \][/tex]

So, the altitude of the equilateral triangle is [tex]\( \frac{9\sqrt{3}}{2} \)[/tex] centimeters.

The smaller angle is 40 degrees and the larger angle is 50 degrees.

Let's represent the smaller angle by x degrees. According to the problem, the larger angle is 10 degrees more than the smaller angle, so it can be represented by x + 10 degrees. The sum of these two angles is 90 degrees, so we can set up the equation:

x + (x + 10) = 90

Simplifying the equation, we get:

2x + 10 = 90

Subtracting 10 from both sides, we have:

2x = 80

Finally, dividing both sides by 2, we find:

x = 40

So, the smaller angle is 40 degrees and the larger angle is 40 + 10 = 50 degrees.

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Final answer:

To find the total distance a tortoise walked in the desert, use the formula distance = speed x time. In this case, the tortoise walked 40.96 meters by walking for 4 minutes at a speed of 10.24 meters per minute.

Explanation:

A tortoise is moving laboriously-khwanya-khwanya-khwanya! To calculate the total distance the tortoise walked, we use the formula: distance = speed x time. Given the tortoise walks for 4 minutes at a speed of 10.24 meters per minute, the calculation is 10.24 m/min x 4 min = 40.96 meters. Therefore, the tortoise walked a distance of 40.96 meters.

The center of the circle that can be circumscribed about a triangle with vertices A(2, 6), B(2, 0), and C(10, 0) is calculated by finding the intersection of the perpendicular bisectors of the sides of the triangle. In this case, the center of the circle is at (6,3).

For the triangle with vertices A(2, 6), B(2, 0), and C(10, 0), the circle that you can circumscribe about it is known as the circumcircle. The center of this circle is the circumcenter which is obtained by the intersection of the perpendicular bisectors of the sides of the triangle.

Since points A and B lie along the same vertical line, the perpendicular bisector of line AB is a horizontal line halfway between them at y=3. The bisector of line BC is a vertical line halfway between points B and C at x=6.

The intersection of these two bisectors (x=6, y=3) is the circumcenter. So, the center of the circumcircle is at (6,3).

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If four out of 12 people chose milk, then 8 did not choose milk, resulting in a fractional part of 2/3 representing those who chose other drinks.

If four of 12 people chose milk, we can calculate the fractional part of the group that chose other drinks. We subtract the number of people who chose milk from the total number of people surveyed:

12 (total people) - 4 (people who chose milk) = 8 (people who chose other drinks).

The fractional part of the group that chose other drinks is therefore the number of people who chose other drinks divided by the total number of people:

8 (people who chose other drinks) / 12 (total people) = 2/3.

Two-thirds of the group chose drinks other than milk.

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The amount of money that is presently in my investment account is $11,820.

The formula that can be used to determine the amount in my account presently is:

P = FV / (1 + r)^n

25,000 / (1.0425)^18 = $11,820

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Final answer:

To find the current amount needed to reach $25,000 in 18 years at a 4.25% interest rate, the formula P = A / (1 + r)^n is used. Plugging in the values gives P ≈ $11,820, indicating that approximately $11,820 is currently in the account. Thus, the correct answer is (b) $11,820.

Explanation:

The question asks how much money needs to be present in an account now to reach a future value of $25,000 in 18 years at an annual compound interest rate of 4.25%. To find the present value, we use the formula for compound interest: P = A / (1 + r)^n, where P is the present value, A is the amount of money accumulated after n years, inclusive of interest, r is the annual interest rate (decimal), and n is the number of years the money is invested for.

In this case, A = $25,000, r = 0.0425 (4.25%), and n = 18 years. Plugging these values into the formula gives: P = $25,000 / (1 + 0.0425)^18. Calculating this gives us P ≈ $11,820, which means the amount of money currently in the account to reach $25,000 in 18 years at a 4.25% annual interest rate compounded annually is approximately $11,820. Therefore, the correct answer is (b) $11,820.