What to do if the bases and exponents are different?

One of the important theorems in the field of geometry that deals with the properties of triangles are called the Mid- Point Theorem.

The theory of midpoint theorem is used in coordinate geometry, stating that the midpoint of the line segment is an average of the endpoints. The ‘x’ and the ‘y’ coordinates must be known for solving an equation using this theorem. The Mid- Point Theorem is also useful in the fields of calculus and algebra.

Mid-Point Theorem Statement

The midpoint theorem states that “The line segment in a triangle joining the midpoint of any two sides of the triangle is said to be parallel to its third side and is also half of the length of the third side.”

Mid-Point Theorem Proof

If a line segment adjoins the mid-point of any two sides of a triangle, then the line segment is said to be parallel to the remaining third side and its measure will be half of the third side.

Consider the triangle ABC, as shown in the above figure,

Let E and D be the midpoints of the sides AC and AB. Then the line DE is said to be parallel to the side BC, whereas the side DE is half of the side BC; i.e., DE || BC

DE = (1/2 *  BC).

Mid-Point Theorem Formula

In Coordinate Geometry, the midpoint theorem refers to the midpoint of the line segment. It defines the coordinate points of the midpoint of the line segment and can be found by taking the average of the coordinates of the given endpoints. The midpoint formula is used to determine the midpoint between the two given points.

If P1(x1, y1) and P2(x2, y2) are the coordinates of two given endpoints, then the midpoint formula is given as:

Midpoint = [(x1 + x2)/2, (y1 + y2)/2]

The converse of Mid-Point Theorem

The converse of the midpoint theorem states that ” if a line is drawn through the midpoint of one side of a triangle, and parallel to the other side, it bisects the third side”

Formula to Find the Midpoint

The Midpoint Formula is given as,

(x, y) = [(x1 + x2)/2, (y1 + y2)/2]

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If the slope of the line is 1/10 then it means for every ten units of horizontal distance crosses there is unit change of vertical distance.

We know that the slope of a line is nothing but the change in y coordinate with respect to the change in x coordinate.

It describes the steepness of a line.

Also,  if a line makes an angle θ with the positive direction of the x-axis, then the slope of the line will be tangent of angle θ

m = tan θ

Here we have been given a slope 1/10

Let m = 1/10

From definition of slope of line,

m = tan θ

1/10 = tan θ

θ = arctan(1/10)

θ = arctan(0.1)

θ = 5.71°

This means, when line makes an angle of  5.71° with the positive x-axis then the slope of line is 1/10

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Answer:

Step-by-step explanation:

The absolute value of real number is 0 and the absolute value of complex number is z=a+bi.

So, the complete value of a complex number represents the distance between the starting and the point in the complex plane. On the other hand, the absolute value of a real number means only how far a number is from zero without all in all any direction.

A number system is a way of showing numbers by writing, which is a mathematical way of act for the numbers of a given set, by using the numbers or symbols in a mathematical means.

Both all the negative and positive integers, decimal and fractional numbers without imaginary numbers are called real numbers. Real numbers are act for by the “R” symbol.

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The symbol for an ordered pair is a set of round or parentheses ( ) and it is written as (x,y), where x and y are the coordinates.

The length of DM is 38 cm.

We know that,

Triangles DEF and OEN are similar by the AAA Similarity Postulate.

According to the Angle-Angle-Angle (AAA) criteria, "if in two triangles, corresponding angles are equal, then their corresponding sides are in the same ratio (or proportion), and hence the two triangles are identical."

If two triangles are similar, then the ratio of its corresponding sides is proportional,

Hence,

DE/OE = DF/ON

Substituting the values given,

44/22 = DF /38

DF = 2*38 = 74 cm

Since, DF = 2DM

76 = 2 DM

DM = 38 cm

Therefore, The length of DM is 38 cm.

(Refer to the image attached for the diagram of the triangle)

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So B requires 8 more days to complete the work.

Lets assume total unit of work to be= 30units

A can finish the work in= 15 days

B can finish the work in= 10 days

A work efficiency  can be calculated by formula

Efficiency=  Total work/ Number of days

A efficiency= 30/15

30 divided by 15

A= 2

Similarly for B we can calculate its work efficiency

B efficiency= 30/10

30 divided by 10.

B= 3

A work for 3 days and leaves

work done by A = 2*3 = 6 unit

2 is A work Efficiency  and we multiply by 3 that is number of days A worked on.

 work done formula = Efficiency* Number of days  

Now we calculate the remaining work

We subtract the total work by the work done by A

Remaining work= 30-6= 24 unit

so the days required by B is = Work Left divided by B efficiency

B days= 24/ 3 = 8

So B requires 8 more days to complete the work

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A can do a piece of work in 15 days and B alone can do it in 10 days. A works at it for 3 days and then leaves. After A leaves, B works alone. How many more days will B take to complete the work?

Bianca will earn interest of $382.5 for a period of 3years.

Simple interest is a method to calculate the amount of interest charged on a sum at a given rate and for a given period of time. In simple interest, the principal amount is always the same.

Simple interest is calculated with the following formula:

S.I. = P × R × T,

where P = Principal, R = Rate of Interest in % per annum, and T = Time

Given,

Principal P = $3000

Rate of interest R = 4.25% = 4.25/100 = 0.0425

Time T = 3 years

Simple interest S.I. = P × R × T

                              = 3000 × 0.0425 × 3

                              = 127.5 × 3

                      S.I.   = $382.5

Hence, Interest Bianca will earn in 3 years = $382.5

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Answer: Yes it can construct a negative base.

Step-by-step explanation:

A negative base (or a negative radix) each position holds multiples of the appropriate power of the system's base.

The value for the length of rafters is found using trigonometric function, and that is length = 43.38 inches.

What is trigonometric function?

Simply put, trigonometric functions also referred to as circular functions are the functions of a triangle's angle. This means that these trig functions provide the relationship between the angles and sides of a triangle.

The roof and the ceiling form two back-to-back right triangles .

It bisects the top angle so the top angle of the right triangle would be 35 degrees.

Half the width of 36 foot forms the lower side of the triangle - 36/2 = 18 inch.

The rafter (not counting the 1 foot overhang) is the hypotenuse, call it h.

We can use the sine trigonometric function to find the hypotenuse.

Sin θ = Opposite/Hypotenuse

Sin 35° = 18/h

0.573576 = 18/h

h = 18/0.573576

h = 31.38

Since, 1 foot = 12 inch rafter is extended then -

h = 31.38 + 12 = 32.38 foot

Therefore, the length of rafters is 43.38 inches.

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Observational data that contains a numerical value or amount is called quantitative observation.

Quantitative observation is an objective data gathering that is primarily concerned with numbers and values and is "related to, of or expressed in terms of a quantity." The outcomes of a quantitative observation are derived using statistical and numerical analytic techniques. Thus, quantitative observation refers to observational data that includes a numerical value or amount.

An objective gathering of facts, quantitative observation refers to "related to, of or depicted in terms of a quantity" and is generally focused on numbers and values. Methods of statistical and numerical analysis are used to derive the results of quantitative observation.

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The panels in a joint probability table include the kind of probabilities, with the exception of the cells on the table's margins, which carry marginal probabilities instead of joint probabilities.

The likelihood that two separate occurrences will occur simultaneously is referred to as a joint probability when there are two or more random variables. A joint probability is the likelihood that event Y will occur at the same time as event X. It is a statistical metric that determines the likelihood of two occurrences occurring simultaneously and at the same moment in time. When two or more random variables are represented as a joint probability distribution in a table, the link between the variables is demonstrated (it uses variables or conditions instead of events). The probabilities included in the cells of the joint probability table are joint probabilities, but the probabilities on the margins are marginal probabilities.

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Answer:

Yes, Noah brings enough money to get into the park

Step-by-step explanation:

First, we have to find how much each student pay by taking

$100 / 8 = $12.50 each student

We know Noah brought $13

$13 > $12.50

So, Noah brings enough money to get into the park.

Strong negative correlation is 0.89. While the size of the number (0.89) suggests a strong link, the negative (-) symbol before the number implies it is negative.

Correlation coefficients demonstrate how strongly two independent variables, x and y, are connected linearly.

A linear correlation coefficient greater than zero indicates a relationship that is favorable.

A value less than zero denotes a bad relationship. To summarize, when x and y have values of 0, there is no link between them.

A statistic that describes how closely two variables' movements are related to one another is the correlation coefficient (r2).

The most widely used correlation coefficient is the Pearson product-moment correlation, which is used to detect if two variables are linearly related.

This correlation value, however, might not always be an accurate predictor of reliance in a non-linear link.

Hence, Strong negative correlation is 0.89. While the size of the number (0.89) suggests a strong link, the negative (-) symbol before the number implies it is negative.

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h and k cannot be determined without knowing a value of g(x). The equation given is g(x) = rootindex 3 startroot x minus h endroot k, which is an equation of a cube root.

The equation provided is g(x) = rootindex 3 startroot x minus h endroot k. To determine the values of h and k, we need to know the value of g(x). Since no value of g(x) is provided, we cannot determine the values of h and k.

The equation given is g(x) = rootindex 3 startroot x minus h endroot k, which is an equation of a cube root. To find the values of h and k, we need to know the value of g(x). Unfortunately, no value for g(x) is given, so we cannot determine the values of h and k. To calculate these values, we must have a value of g(x) that we can substitute into the equation. Without this value, we cannot find the values of h and k.

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Answer:

X - 53=2

Or, x=2+53

Or, x=55

Estimation for the number of miles the runner has run so far is 7.74 mile

From the question, we have 2 infomation :

1.  Total distance of running is 18 mile

2.  Race completed so far is 43%

We can stated that 18 mile will be 100 %

0% ----------------------------------------------------------------------------------------> 100%

1    2    3    4    5    6    7    8    9    10   11    12    13    14    15    16    17    18

If she has run 43% of the race, it will be between 1 mile and 18 mile. We could simply multiply the total distance with total race completed to estimate total distance she has achieved.

So, the best way to estimate the total distance she has achieved is

Estimation = Total Distance x Total race completed

                  = 18 mile x 43%

                  = 7.74 mile

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Answer: 2x5=10 if it was doubled its 20 but wouldnt this be a measure of size and not of volume we would have to see all measurments for this, put 20 down and see if that works.

Step-by-step explanation: i have f's in all of my classes so dont trust me. lol

If he already had enough crushed cookies, Then he doesn't need any more i.e. zero needed.

When we recognise how crucial zero is to computation and how it enabled mathematicians in earlier eras to solve more challenging problems, we can see how significant zero is. They were then able to research the fundamental characteristics of numbers and advance mathematics.

As the additive identity of integers and real numbers, zero's necessity in mathematics is significant. Algebraic structures also depend heavily on zero. The place value system needs the digit zero as a stand-in.

Fractions also help us understand the significance of zero in math. If it weren't for zero, we wouldn't be taught fractions in school. The absorbing zero also produced a better way to define fractions.

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Answer:

Step-by-step explanation:

In a right triangle, the sine of an angle is equal to the length of the side opposite the angle divided by the length of the hypotenuse (the longest side of a right triangle, also called the hypotenuse). Therefore, to find the length of the side opposite angle θ, we can use the following formula:

opposite = sin(θ) * hypotenuse

In this case, we know that θ = 25 degrees and the adjacent side has a length of 3. Since we are dealing with a right triangle, we can use the Pythagorean theorem to find the hypotenuse:

hypotenuse = sqrt(adjacent^2 + opposite^2) = sqrt(3^2 + opposite^2) = sqrt(9 + opposite^2)

So we can substitute that value into the first equation:

opposite = sin(25) * sqrt(9 + opposite^2)

Solving for opposite, we get:

opposite = sin(25) * sqrt(9 + opposite^2)

opposite = 0.4226 * sqrt(9 + opposite^2)

We can calculate the value of sin(25) which is 0.4226.

opposite = 0.4226 * sqrt(9 + opposite^2)

opposite = 0.4226 * sqrt(opposite^2 + 9)

Square both sides of the equation to get:

opposite^2 = (0.4226 * sqrt(opposite^2 + 9))^2

opposite^2 = 0.1799 * opposite^2 + 1.5874

Subtracting 0.1799 * opposite^2 from both sides of the equation gives:

0 = 1.5874

Solving for the opposite, we get:

opposite = sqrt(1.5874)

opposite = 1.259

So the side opposite to the angle of 25 degrees is 1.259 units long, rounding to at least 3 decimal places.

Answer:

The path of the soccer ball can be modeled by a parabolic equation, which can be represented as y = ax^2 + bx + c.

In this equation, y represents the height of the soccer ball, x represents the horizontal distance traveled, a represents the parabolic shape of the path, b represents the horizontal direction and c represents the initial position.

To find the equation that models the path of the soccer ball, you would need to know the values of a, b, and c. These values can be found using the information given in the problem, which is the horizontal distance traveled (80 feet) and the highest point reached (30 feet).

It can be represented as y = -(1/240)x^2 + (1/2)x + 30

In this equation, -(1/240) is the value of a, (1/2) is the value of b and 30 is the value of c.

Step-by-step explanation:

The linear regression equation models  is y = 1.20x - 14.32

The homework grade for a student with a test grade of 50 is 54.

An estimate of the line that depicts the actual, but unidentified, linear relationship between the two variables is called a regression line. When the value of the explanatory variable is known, the regression line's equation is used to predict (or estimate) the value of the response variable.

As, linear regression calculator which gives the linear equation in the form :

y = bx + c

y = 1.20x - 14.32

where y = Test grade and x = homework grade

Also, Slope, b = 1.20,  intercept, c = - 14.32

So, Using the model equation we have Test grade, y =

y = 1.20x - 14.32

50 = 1.20x - 14.32

50 + 14.32 = 1.20x

64.32 = 1.20x

x = (64.32 ÷ 1.20)

x = 53.6

Hence, the estimated homework grade for a test score of 50 is 54.

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Answer:

x=25

Step-by-step explanation:

Vertical angles are congruent

37+45=82

180-82=98

Liner pairs=180

82+123=155

180-155=25

x=25

The right triangle in question 3 have the value of m = 10√2 and n = 10 using the trigonometric ratio of sine and cosine of the angle 45°.

The trigonometric ratios involves the relationship of an angle of a right-angled triangle to ratios of two side lengths. Basic trigonometric ratios includes; sine cosine and tangent.

Considering the angle 45° for the right triangle, recall that sin 45° and cos 45° are both equal to √2/2

sin 45° = 10/m {opposite/hypotenuse}

√2/2 = 10/m

m = 20/√2 {cross multiplication}

m = 20/√2 × √2/√2 {rationalize}

m = 20√2/2

m = 10√2

cos 45° = n/m {adjacent/hypotenuse}

√2/2 = n/10√2

n = 10(√2)²/2

n = 10 × 2/2

n = 10

Therefore, the value of m is equal to 10√2 and that of n is equal to 10, using the trigonometric ratio of sine and cosine of the angle 45° for the right triangle.

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The borderline if the linear inequality's symbol is > (greater than) or (greater than or equal to).

In mathematics, an inequality is a connection that does not have an equal sign between two expressions or values. Inequality follows imbalance, therefore. When two numerical values are not equal, an inequality establishes a connection between them. Difference between equality and inequality Use the not equal symbol, which is most often used, when two values are not equal (). No matter how little or huge the values, various inequalities are employed to contrast them. The two sides can be changed until the variables are all that is left in order to solve several basic inequalities. Unevenness is caused by a number of factors, including: The sum or division of negative values on both sides. The left and right are exchanged.

given

inequality for y

5x – 2y ≤ 10

– 2y ≤ – 5x + 10

y ≥ 5/2x - 5

When dividing by a negative number, remember to switch the inequality sign.

In a linear inequality, you will shade below the boundary line if the sign is (less than) or (less than or equal to).  The borderline if the linear inequality's symbol is > (greater than) or (greater than or equal to).

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Answer:

x - 1 is inside a cubic root.

The cubic root of a positive number is positive.

The cubic root of zero is zero.

The cubic root of a negative number is negative.

That means that the cubic root of any real number is also a real number.

The domain is all real numbers.

Basic inequality is a mathematical statement that expresses the relative size of two values. It usually involves the use of an inequality symbol, such as “>” (greater than), “<” (less than), “≥” (greater than or equal to), or “≤” (less than or equal to). An example of a basic inequality is x > 5, which means that x is greater than 5.

The concept of basic inequality is an important one in mathematics, as it helps us to understand relationships between values, and to identify patterns and trends. Inequality is a type of comparison between two or more values, and can be used to determine the size or order of values.

Inequality symbols such as “>” (greater than), “<” (less than), “≥” (greater than or equal to), and “≤” (less than or equal to) are used to indicate the magnitude of the difference between values.

Basic inequality can be used to solve equations, solve for variables, and compare data sets. It can also be used to identify trends in data sets, such as the average, rate of change, and correlations. By understanding basic inequality, we can make predictions and better understand the relationships between values.

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The steps to bisect the line segment are as detailed below.

The steps to bisect a line segment are as follows;

Step 1: Place the compass at one end of line segment.

Step 2: Adjust the compass to slightly longer than half the line segment length

Step 3: Draw arcs above and below the line.

Step 4: Keeping the same compass width, draw arcs from other end of line.

Step 5: Place ruler where the arcs cross, and draw the line segment.

Looking at the given image that was used to bisect a line segment, the steps above are the steps that were taken to bisect it.

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Answer:

(1,9;(-0,4))

Step-by-step explanation:

7(4x-8)+2x=1

28x-56+2x=1

28x+2x=1+56

30x=57|:30

x=1,9

y=4×1,9-8

y=-0,4