The diagram shows a right triangle and the lengths of two of its sides in inches. Which measurement is closest to the value of d in inches?

Answer: Distributive Property!

Step-by-step explanation:

The answer is the Distributive Property. The Distributive Property states that for any numbers a, b, and c, a(b + c) = ab + ac. In this case, the Distributive Property is being applied to the equations, so 2(3x - y) = 6x - 2y. This allows us to add the two equations together.

7/25 = x/100 17•100= 1700÷25=

60 answer is 68%

Answer:

y ≈ 27

Step-by-step explanation:

using the tangent ratio in the right triangle

tan48° = [tex]\frac{opposite}{adjacent}[/tex] = [tex]\frac{30}{y}[/tex] ( multiply both sides by y )

y × tan48° = 30 ( divide both sides by tan48° )

y = [tex]\frac{30}{tan48}[/tex] ≈ 27 ( to the nearest whole number )

In answering the question above, the solution is If you know the lengths Pythagorean theorem of the other two sides of the right triangle, you may use these formulae to determine the length of the missing side.

The fundamental Euclidean geometry relationship between the three sides of a right triangle is the Pythagorean Theorem, sometimes referred to as the Pythagorean Theorem. This rule states that the areas of squares with the other two sides added together equal the area of the square with the hypotenuse side. According to the Pythagorean Theorem, the square that spans the hypotenuse (the side that is opposite the right angle) of a right triangle equals the sum of the squares that span its sides. It may also be expressed using the standard algebraic notation, a2 + b2 = c2.

[tex]a^2 + b^2 = c^2[/tex]

where a and b are the lengths of the other two sides, and c is the length of the hypotenuse.

c = √[tex](a^2 + b^2)[/tex]

You may rewrite the formula as follows to get the length of one of the other sides:

a = √[tex](c^2 - b^2)[/tex]

b = √[tex](c^2 - a^2)[/tex]

If you know the lengths of the other two sides of the right triangle, you may use these formulae to determine the length of the missing side.

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Ben's ball lands approximately 2.5 seconds after Andrew's ball.

Since the value of parameter a is -5 for both balls, the height of each ball follows the equation:

h(t) = -5t² + vt + h0

where;

Let's assume that Andrew's ball is hit with an initial velocity of v1, and Ben's ball is hit with an initial velocity of v₂. We also know that Ben's ball reaches a maximum height 50% greater than Andrew's, which means that:

h_max₂ = 1.5h_max₁

At the maximum height, the velocity of the ball is zero, so we can find the time it takes for each ball to reach the maximum height by setting v = 0 in the equation for h(t):

t_max1 = v₁ / (2 x 5)

t_max2 = v₂ / (2 x 5)

Since Ben's ball reaches a maximum height that is 50% greater than Andrew's, we can write:

h_max2 = 1.5h_max1

-5(t_max2)² + v₂t_max2 + h0 = 1.5(-5(t_max1)²   +   v1 * t_max1 + h0)

Simplifying this equation, we get:

-5(t_max2)² + v₂t_max2 = -7.5(t_max1)² + 1.5v₁t_max1

We also know that Andrew's ball lands after 4 seconds, which means that h(4) = 0:

h(4) = -5(4)² + v1 * 4 = 0

-80 + 4v1 = 0

v1 = 80/4

v1 = 20 m/s

Solving these equations for t_max2 and v2, we get:

t_max1 = v1 / (2 x 5)

t_max1 = 20 / (2 x 5)  = 2 s

t_max2 = 1.5 * t_max1 = 3 s

v2 = 1.5 * v1 = 30 m/s

To find the time it takes for Ben's ball to land, we need to find the time t2 when h(t2) = 0.

We can use the equation for h(t) with v = v2, h0 = 0, and solve for t:

-5t² + v₂t = 0

-5t² + 30 = 0

5t² = 30

t² = 30/5

t² = 6

t = √6

t = 2.5 s

Therefore, Ben's ball lands approximately 2.5 seconds after Andrew's ball.

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The IQR is a measure of the spread or variability of the data. It represents the range of the middle 50% of the data values and is less sensitive to outliers than the range.

The steps below can be used to determine the interquartile range (IQR) in a box plot:

Find the data set's median (Q2) value. With 50% of the data above and 50% of the data below, this value splits the data in half.

Find the lower half of the data set's median (Q1). With 25% of the data above and 75% below, this value divides the lower half of the data into two quarters.

Find the top half of the data set's median (Q3). With 75% of the data above and 25% below, this value divides the upper half of the data into two quarters.

The interquartile range (IQR) is calculated as the difference between the third and first quartiles (Q3 and Q1, respectively).

Below is an illustration to show you how to find the IQR:

Let's say we have the following collection of data:

10, 11, 12, 15, 16, 18, 20, 22, 25, 30

We must determine the quartiles in order to generate a box plot:

Median (Q2) = (16 + 18) / 2 = 17

Lower half: 10, 11, 12, 15, 16

Median (Q1) = (11 + 12) / 2 = 11.5

Upper half: 18, 20, 22, 25, 30

Median (Q3) = (22 + 25) / 2 = 23.5

The IQR is: because the box ranges from Q1 to Q3.

IQR = Q3 - Q1 = 23.5 - 11.5 = 12

Hence, 12 is the interquartile range.

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We cannot determine if T is a one-to-one linear operator without more information about the values of N, M, and P2

Yes, T: R3 R3 is a one to one linear operator.

Let x, y, z be elements of R3, then T(x, y, z) = (W1, W2, W3), where W1 = 2Nx - 3y + Mz, W2 = x + 3y - P2 and W3 = -x - Ny + z.

To determine if this is a one to one linear operator, we must show that given two inputs, x, y, z and x', y', z' we have that T(x, y, z) = T(x', y', z') implies x = x', y = y' and z = z'.

We can see that W1 = W1', W2 = W2' and W3 = W3' implies that 2Nx - 3y + Mz = 2Nx' - 3y' + Mz', x + 3y - P2 = x' + 3y' - P2' and -x - Ny + z = -x' - Ny' + z'

Expanding the expressions and simplifying, we have that 2N(x - x') - 3(y - y') + M(z - z') = 0, x - x' + 3(y - y') - P2(z - z') = 0 and -(x - x') - N(y - y') + (z - z') = 0.

Since the expressions must be 0 for all inputs x, y, z and x', y', z' these imply that x = x', y = y' and z = z'. Therefore, T: R3 R3 is a one to one linear operator.
A linear operator T: R3 → R3 is one-to-one if and only if the only solution to the equation T(x, y, z) = (0, 0, 0) is (x, y, z) = (0, 0, 0). In other words, the kernel of T is trivial.

Let's substitute the given values of W1, W2, and W3 into the equation T(x, y, z) = (0, 0, 0) and solve for x, y, and z:

2Nx - 3y + Mz = 0

x + 3y - P2 = 0

-x - Ny + z = 0

From the second equation, we can solve for x in terms of y:

x = P2 - 3y

Substituting this into the first equation gives:

2N(P2 - 3y) - 3y + Mz = 0

Simplifying and rearranging terms:

(6N + 3)y = 2NP2 + Mz

y = (2NP2 + Mz)/(6N + 3)

Substituting this back into the equation for x gives:

x = P2 - 3(2NP2 + Mz)/(6N + 3)

Finally, substituting these values of x and y into the third equation gives:

-(P2 - 3(2NP2 + Mz)/(6N + 3)) - N(2NP2 + Mz)/(6N + 3) + z = 0

Simplifying and rearranging terms:

z(6N + 3 - M - 3N) = P2(6N + 3) - 6NP2

z = (P2(6N + 3) - 6NP2)/(6N + 3 - M - 3N)

Now we have expressions for x, y, and z in terms of the constants N, M, and P2. If these expressions are all equal to zero, then the only solution to the equation T(x, y, z) = (0, 0, 0) is (x, y, z) = (0, 0, 0), and T is a one-to-one linear operator.

However, it is not clear from the given information if these expressions are all equal to zero. Therefore, we cannot determine if T is a one-to-one linear operator without more information about the values of N, M, and P2.

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The area of rectangle S is 84.15 square unit.

Area is the region enclosed by the shape of an object. The space covered by a figure or any two-dimensional geometric shape in the plane is the area of ​​the shape, or

Area is a measure of the surface area of ​​a shape. To find the area of ​​a rectangle or square, you need to multiply the length and width of the rectangle or square.

Solution according to the given information in the question:

Area of rectangle R = 187

Ratio of rectangle R = 17

Ratio of rectangle S = 7.65

Area of rectangle S = (7.65/17)×187

                                 = 84.15 square unit

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Answer:

I CANT SEE THE OTHER NUMBER OML

Step-by-step explanation:

We have proved that a set containing three non-zero vectors in V cannot be linearly independent if {v1, v2} is a spanning set for V.

To prove that a set containing three non-zero vectors in V cannot be linearly independent, we will use the definition of linear independence and the fact that {v1, v2} is a spanning set for V. Here are the steps:

Let {v1, v2, v3} be a set containing three non-zero vectors in V.

Since {v1, v2} is a spanning set for V, any vector in V can be written as a linear combination of v1 and v2. In particular, v3 can be written as v3 = a*v1 + b*v2 for some scalars a and b.

Now, consider the linear combination 0 = c1*v1 + c2*v2 + c3*v3, where c1, c2, and c3 are scalars. Substituting v3 = a*v1 + b*v2, we get 0 = (c1 + a*c3)*v1 + (c2 + b*c3)*v2.

Since {v1, v2} is a spanning set, the only way for this linear combination to be equal to 0 is if both coefficients are 0, i.e., c1 + a*c3 = 0 and c2 + b*c3 = 0.

If c3 = 0, then c1 = 0 and c2 = 0, which means that the set {v1, v2, v3} is linearly independent. However, if c3 is not 0, then we can solve for c1 and c2 in terms of c3, and we get c1 = -a*c3 and c2 = -b*c3.

This means that there are infinitely many solutions to the linear combination 0 = c1*v1 + c2*v2 + c3*v3, and therefore the set {v1, v2, v3} is not linearly independent.

we have proved that a set containing three non-zero vectors in V cannot be linearly independent if {v1, v2} is a spanning set for V.

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The fractions that are equivalent to 2/3 are 4/6 and 20/30.

To determine if two fractions are equivalent, you can cross multiply the numerator and denominator of one fraction with the denominator and numerator of the other fraction, respectively. If the products are equal, then the fractions are equivalent.

For example, to determine if 2/3 and 4/6 are equivalent, you would cross multiply as follows:

2 x 6 = 12
3 x 4 = 12

Since the products are equal, 2/3 and 4/6 are equivalent fractions.

Similarly, you can cross multiply 2/3 and 20/30 to determine if they are equivalent:

2 x 30 = 60
3 x 20 = 60

Since the products are equal, 2/3 and 20/30 are equivalent fractions.

Therefore, the fractions that are equivalent to 2/3 are 4/6 and 20/30.

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Step-by-step explanation:

Let the number of students in each van be "x" and the number of students in each bus be "y". We can then set up a system of equations to represent the given information:

From High School A: 8x + 8y = 240

From High School B: 4x + y = 54

We can use the second equation to solve for y in terms of x:

y = 54 - 4x

We can then substitute this expression for y into the first equation and solve for x:

8x + 8(54 - 4x) = 240

8x + 432 - 32x = 240

-24x = -192

x = 8

Now that we know x = 8, we can use the equation for y to solve for y:

y = 54 - 4x = 54 - 4(8) = 22

Therefore, there were 8 students in each van and 22 students in each bus.

Step-by-step explanation:

To find the zeros of the function f(x), we need to solve the equation f(x) = 0.

f(x) = x^2 + 12x + 32

Setting f(x) equal to zero and factoring, we get:

0 = x^2 + 12x + 32

0 = (x + 4)(x + 8)

Using the zero product property, we set each factor equal to zero and solve for x:

x + 4 = 0 or x + 8 = 0

x = -4 or x = -8

Therefore, the zeros of the function f(x) are -4 and -8.

16) The area of the surface area of the canvass is 240 feet²

17) The approximate height of the silo is 24 meters.

16) The A-Frame tent has two identical triangles as its sides and a rectangle as its base.

The area of each triangle is

(1/2)bh

= (1/2)10(12)

= 60 square feet.

The area of the rectangle is

lw = 12(10)

= 120 square feet.

Therefore, the total area that he plans to canvass is

2(60) + 120

= 240feet²

17) The volume of the silo is given by

V = (1/3)πr²h, where r is the radius of the base and h is the height. Solving for h,
we get

h = 3V/πr².

Substituting V = 4000 cubic meters and r = 8 meters, we get

h = 3(4000)/(π(8)²)
≈ 24 meters approximately.

Therefore, the approximate height of the silo is 24 meters.

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From left to right, function f has zeros at x = 0, x = 3, x = -3, x = 1/2, and x = -3/2.

An expression, rule, or law in mathematics that explains how one independent variable and one dependent variable are connected (the dependent variable).

The factors of the function f(x) can be found by factoring the expression:

f(x) = 2x⁴ - x³ - 18x² + 9x = x(2x³ - x² - 18x + 9)

To find the zeros of f(x), we need to find the values of x that make the expression in the parentheses equal to zero:

2x³ - x² - 18x + 9 = 0

We can use synthetic division or other methods to factor this polynomial and find its zeros. Alternatively, we can use the Rational Zeros Theorem to test possible rational zeros:

Possible rational zeros: ±1, ±3, ±9, ±1/2, ±3/2, ±9/2

Testing x = 1: 2(1)³ - (1)² - 18(1) + 9 = -8, not a zero

Testing x = -1: 2(-1)³ - (-1)² - 18(-1) + 9 = 28, not a zero

Testing x = 3: 2(3)³ - (3)² - 18(3) + 9 = 0, a zero

Testing x = -3: 2(-3)³ - (-3)² - 18(-3) + 9 = 0, a zero

Using polynomial division or factoring by grouping, we can factor the polynomial further:

2x³ - x² - 18x + 9 = (x - 3)(2x² + 5x - 3)

The quadratic factor can be factored using the quadratic formula or other methods:

2x² + 5x - 3 = (2x - 1)(x + 3)

Therefore, the zeros of f(x) are:

x = 0 (from the factor x)

x = 3 (from the factor x - 3)

x = -3 (from the factor x + 3)

x = 1/2 (from the factor 2x - 1)

x = -3/2 (from the factor 2x - 1)

So the completed statement is:

From left to right, function f has zeros at x = 0, x = 3, x = -3, x = 1/2, and x = -3/2.

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The cord length is k* theta radian, where theta radian is the central angle and k is the proportionality constant. This means that the rope length is proportional to the central angle in radians.

The complete cord length for a full central angle of 2pai is 2pai * r if the cord arc is a segment of a circle of radius r, where k=2pai is the proportionality constant.

What is arc?

In mathematics, an arc is a portion of the boundary of a circle or curve. It is sometimes referred to as an open curve.

The measurement around a circle that determines its edge is called the circumference, often known as the perimeter. As a result, the distance between any two locations along an arc's circumference is how it is defined.

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The average amount spent per person on healthcare in 2019 in the United States of America is $1.09×10⁴.

The average can be defined as the sum of all numbers divided by the total number of values. The mean can be defined as the mean of the values ​​of a sample of data. That is, the average is also called the arithmetic mean.

Solution according to the information given in the question:

Given, Population of USA = 328.2 million = 3282 × 10⁵

           Total Healthcare costs = $3.6 trillion = 3.6 × 10¹²

∴ Average amount spent = Total Healthcare costs/Population of USA

                                          = (3.6 × 10¹²)/(3282 × 10⁵)

                                          = (36 × 10¹¹)/ (3282 × 10⁵)

                                          = (36/3282) × (10¹¹/10⁵)

                                          = 0.0109 × 10⁶

                                          = $1.09 × 10⁴

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The average amount spent per person on healthcare in the United States in 2019 was $10,918.98.

What is average in mathematics?

Average in mathematics is a measure of the central or typical value in a set of numbers. It is computed by adding all the values together and dividing the total by the number of values in the set. In statistics, the average is the most commonly used measure of central tendency.

The average amount spent per person on healthcare in the United States in 2019 was $10,918.98. This figure is calculated by taking the total healthcare costs of $3.6 trillion and dividing it by the population of 328.2 million people. This figure represents the amount of money each person in the country spent on healthcare in 2019, ranging from medical services and prescriptions to insurance premiums and other costs. It is important to note that this figure does not take into account any out-of-pocket expenses that individuals may have incurred during the year.

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If you have a 3600 m³ cube, then you can fit 3600 kiloliters in the cube.

A kiloliter (kL) is defined as a unit of volume which is used to measure liquids. It is equal to 1,000 liters, and it is used to measure large volumes of water, oil, and other liquids.

We know that 1 cubic meter is equal to 1 kiloliter,

We have to convert 3600 meter cube to Kiloliter,

So , we multiply by 3600,

On multiplying,

We get,

⇒ 3600 meter cube = 3600 × 1 Kiloliter,

⇒ 3600 meter cube = 3600 Kiloliter,

Therefore, 3600 Kiloliter can fill in the cube having the volume as 3600 meter cube.

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Answer:

X is Greater than or equal to 31/10

The equation for the given tangent function is f(x) = 3 tan(x - π/2) - 1.

What is Trigonometric function ?

Trigonometric functions are mathematical functions that relate to angles of a right-angled triangle. The three most common trigonometric functions are sine, cosine, and tangent, which are denoted by sin, cos, and tan, respectively.

The general form of a tangent function is given by:

f(x) = A tan(B(x - C)) + D

where A is the amplitude, B is the frequency (inverse of the period), C is the horizontal shift, and D is the vertical shift.

Given the information, we have:

A = 3

period = π

frequency = 1/period = 1/π

horizontal shift = π/2 to the right

vertical shift = down 1

So, we can plug in the values into the general form and get:

f(x) = 3 tan(1(x - π/2)) - 1

Simplifying:

f(x) = 3 tan(x - π/2) - 1

Therefore, the equation for the given tangent function is f(x) = 3 tan(x - π/2) - 1.

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The solution of system of equation is (0, -3).

System A:
Line 1: y = -3x
Line 2: y = -3x + 1

This system of equations is inconsistent. This means the system has no solution. The two lines are parallel and will never intersect, therefore there is no solution to this system of equations.

System B:
Line 1: y = x - 3
Line 2: 2x + 3y = -9

This system of equations is consistent independent. This means the system has a unique solution. The two lines will intersect at one point, which is the solution to this system of equations. We can solve for x and y by using substitution or elimination method.

Using substitution method:
2x + 3(x - 3) = -9
2x + 3x - 9 = -9
5x = 0
x = 0

Substituting x = 0 into Line 1:
y = 0 - 3
y = -3

Solution: (0, -3)

System C:
Line 1: y = -2x - 3
Line 2: y = x + 3

This system of equations is consistent independent. This means the system has a unique solution. The two lines will intersect at one point, which is the solution to this system of equations. We can solve for x and y by using substitution or elimination method.

Using elimination method:
y + 2x = -3
y - x = 3

Adding the two equations:
3x = 0
x = 0

Substituting x = 0 into Line 1:
y = -2(0) - 3
y = -3

Solution: (0, -3)

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The value of (f· g)(x) from the given functions is  -x^4 - x^2 - x.

Functions are the fundamental part of the calculus in mathematics. The functions are the special types of relations. A function in math is visualized as a rule, which gives a unique output for every input x.

Here,

The given functions are f(x)=x³+x+1 and g(x)=-x.

(f·g)(x)= f(x) × g(x)

= (x³+x+1) × (-x)

=  -x^4 - x^2 - x

so, e get,

(g·f)(x)= g(x) × f(x)

= (-x) × (x³+x+1)

= -x^4 - x^2 - x

Therefore, the value of (f· g)(x) from the given functions is  -x^4 - x^2 - x.

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Its direction is south of east which is 49.56° east of south, the answer is not listed among the answer choices provided.

Magnitude refers to the size or amount of something, often measured numerically or on a scale. In mathematics, it is commonly used to describe the absolute value of a number or the length of a vector in a geometric space. Magnitude can also be used to describe the intensity or strength of physical phenomena, such as the magnitude of an earthquake or the magnitude of a force.

To find the magnitude and direction of MN with tail and head points M(-3, -2) and N(4, 0), we can use the distance formula and trigonometry.

First, we can find the coordinates of MN by subtracting the coordinates of M from those of N:

MN = (4 - (-3), 0 - (-2)) = (7, 2)

The magnitude of MN is the distance between M and N, which can be found using the distance formula:

|MN| = √((7 - (-3))² + (2 - (-2))²) = √(10² + 4²) = √116 ≈ 10.77

Next, we can find the direction of MN by using trigonometry. The angle between MN and the positive x-axis (east) is:

θ = arctan(2/7) ≈ 16.26°

The direction of MN is 180° + θ (measured counterclockwise from the positive x-axis) because the tail point M is to the left of the head point N:

Direction of MN = 180° + 16.26° ≈ 196.26°

To convert this direction to a compass direction, we can subtract it from 360° and then divide by 4, rounding to the nearest degree. This gives:

Compass direction of MN = (360° - 196.26°)/4 ≈ 40.44°

Since MN is to the right of the y-axis (north-south), its direction is south of east, which is 90° - 40.44° = 49.56° east of south.

Therefore, its direction is south of east which is 49.56° east of south, the answer is not listed among the answer choices provided.

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Answer: 76

Step-by-step explanation:

58 and 18 are both numbers that don't have any variables, so they are like terms, so you just add them together

58+18=76

If you need to break it down a bit more:

50+10=60

8+8=16

60+16=76

The value of x of the triangle is given by the trigonometric relation x = 30°

Trigonometry is the study of the relationships between the angles and the lengths of the sides of triangles

The six trigonometric functions are sin , cos , tan , cosec , sec and cot

Let the angle be θ , such that

sin θ = opposite / hypotenuse

cos θ = adjacent / hypotenuse

tan θ = opposite / adjacent

tan θ = sin θ / cos θ

cosec θ = 1/sin θ

sec θ = 1/cos θ

cot θ = 1/tan θ

Given data ,

Let the triangle be represented as ΔPQR

Now , the measure of side PQ = 48 cm

The measure of side PQ = 96 cm

The measure of ∠x is calculated by

From the trigonometric relations ,

sin x = opposite side / hypotenuse

On simplifying , we get

sin x = 48 / 96

sin x = 1/2

Taking inverse on both sides , we get

x = sin ⁻¹ ( 1/2 )

x = π/6

x = 30°

Therefore , the value of x is 30°

Hence , the angle of the triangle is 30°

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Similarly, the fact that the cross-products both equal 120 and 200 is further evidence that the equation is a proportion.

A relationship between two expressions in mathematics is known as an equation, and it is written as an equality on both sides of the equal to sign. An example of an equation is 3y = 16.

The false statement is: "This is a proportion because the both sides are equal to 0.6 in decimal form."

While both sides of the equation do indeed equal 0.6 when simplified, this is not the definition of a proportion.

A proportion is an equation that states that two ratios are equal. In this case, the equation is a proportion because the ratio of 12 to 20 is equal to the ratio of 6 to 10. Specifically,

12/20 = 6/10

3/5 = 3/5

which is true, indicating that the two ratios are indeed equal.

Similarly, the fact that the cross-products both equal 120 and 200 is further evidence that the equation is a proportion.

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The relationship that has a zero slope is option a, he first column, x, has the entries, negative 3, negative 1, 1, 3. The second column, y, has the entries, 2, 2, 2, 2.

Positive slopes and negative slopes are absent from horizontal lines. A horizontal line's slope is always zero. This is due to the fact that it has a constant height—if we use the rise over run technique, rise will always be 0 regardless of what run is.

Option A is the connection with a zero slope.

Regardless of the x-values, option A's y-values are all fixed at (2). This implies that for any change in x (run), the change in y (rise) is always zero, resulting in a slope of zero.

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Answer:

2 times larger

Step-by-step explanation:

Mikes basement is 2 time larger then Jacks base meant because jacks basement area is 33 ft and mikes is 66 ft.

33 x 2 = 66

Answer:

[tex]\displaystyle 1-\dfrac{5}{4}x+\dfrac{5}{8}x^2-\dfrac{5}{32}x^3+\dfrac{5}{256}x^4-\dfrac{1}{1024}x^5[/tex]

Step-by-step explanation:

A binomial expansion is the result of multiplying out the brackets of a polynomial with two terms.

Use the binomial formula to expand the given expression.

[tex]\displaystyle \left(1+ax\right)^n=1+\binom{n}{1}(ax)+\binom{n}{2}(ax)^2+\binom{n}{3}(ax)^3+...+(ax)^n[/tex]

where:

[tex]\displaystyle \binom{n}{r}=\dfrac{n!}{r!(n-r)!}=\phantom{l}^nC_r[/tex]

Given expression:

[tex]\left(1-\dfrac{1}{4}x\right)^5[/tex]

Therefore:

Substitute a = -1/4 and n = 5 into the binomial formula:

[tex]\displaystyle =1+\binom{5}{1}\left(-\dfrac{1}{4}x\right)+\binom{5}{2}\left(-\dfrac{1}{4}x\right)^2+\binom{5}{3}\left(-\dfrac{1}{4}x\right)^3+\binom{5}{4}\left(-\dfrac{1}{4}x\right)^4+\left(-\dfrac{1}{4}x\right)^5[/tex]

[tex]\displaystyle =1+5\left(-\dfrac{1}{4}x\right)+10\left(\dfrac{1}{16}x^2\right)+10\left(-\dfrac{1}{64}x^3\right)+5\left(\dfrac{1}{256}x^4\right)+\left(-\dfrac{1}{1024}x^5\right)[/tex]

[tex]\displaystyle =1-\dfrac{5}{4}x+\dfrac{10}{16}x^2-\dfrac{10}{64}x^3+\dfrac{5}{256}x^4-\dfrac{1}{1024}x^5[/tex]

[tex]\displaystyle =1-\dfrac{5}{4}x+\dfrac{5}{8}x^2-\dfrac{5}{32}x^3+\dfrac{5}{256}x^4-\dfrac{1}{1024}x^5[/tex]

Therefore, the expansion of (1 - ¹/₄x)⁵ is:

[tex]\displaystyle \left(1-\dfrac{1}{4}x\right)^5=1-\dfrac{5}{4}x+\dfrac{5}{8}x^2-\dfrac{5}{32}x^3+\dfrac{5}{256}x^4-\dfrac{1}{1024}x^5[/tex]

Please note there was note enough room to add the binomial coefficients calculations to the main calculation, so please find them below:

[tex]\displaystyle \binom{5}{1}=\dfrac{5!}{1!(5-1)!}=\dfrac{5\times \diagup\!\!\!\!4\times\diagup\!\!\!\!3\times\diagup\!\!\!\!2\times\diagup\!\!\!\!1}{1\times\diagup\!\!\!\!4\times\diagup\!\!\!\!3\times\diagup\!\!\!\!2\times\diagup\!\!\!\!1}=\dfrac{5}{1}=5[/tex]

[tex]\displaystyle \binom{5}{2}=\dfrac{5!}{2!(5-2)!}=\dfrac{5\times 4\times\diagup\!\!\!\!3\times\diagup\!\!\!\!2\times\diagup\!\!\!\!1}{2 \times 1\times \diagup\!\!\!\!3\times\diagup\!\!\!\!2\times\diagup\!\!\!\!1}=\dfrac{20}{2}=10[/tex]

[tex]\displaystyle \binom{5}{3}=\dfrac{5!}{3!(5-3)!}=\dfrac{5\times 4\times\diagup\!\!\!\!3\times\diagup\!\!\!\!2\times\diagup\!\!\!\!1}{\diagup\!\!\!\!3\times\diagup\!\!\!\!2\times\diagup\!\!\!\!1\times2 \times 1\times}=\dfrac{20}{2}=10[/tex]

[tex]\displaystyle \binom{5}{4}=\dfrac{5!}{4!(5-4)!}=\dfrac{5\times \diagup\!\!\!\!4\times\diagup\!\!\!\!3\times\diagup\!\!\!\!2\times\diagup\!\!\!\!1}{\diagup\!\!\!\!4\times\diagup\!\!\!\!3\times\diagup\!\!\!\!2\times\diagup\!\!\!\!1 \times 1}=\dfrac{5}{1}=5[/tex]