The root of this given equation is 2, -3.
What is a quadratic equation?
Any equation that can be written in standard form as where x is an unknown value, a, b, and c are known quantities, and a 0 is a quadratic equation.
Here, we have
Given: x³ - x² - 8x + 12 = 0
We have to solve this equation and find the roots.
x³ - x² - 8x + 12 = 0
= (x-2)(x²+ x - 6) = 0
= (x-2)(x² + 3x - 2x - 6)
= (x-2)(x(x+3)-2(x+3))
= (x-2)(x+3)(x-2) = 0
x = 2, -3
Hence, the root of this given equation is 2, -3.
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After solving [tex]x^3 - x^2 - 8x + 12 = 0[/tex] we get x = 1, x = √8, and x = -√8.
The equation[tex]x^3 - x^2 - 8x + 12 = 0.[/tex]
Identify the given equation.
The given equation is [tex]x^3 - x^2 - 8x + 12 = 0.[/tex]
Look for possible factorization.
In this case, we can factor the equation by grouping:
[tex]x^3 - x^2 - 8x + 12 = (x^3 - x^2) + (-8x + 12)[/tex]
Factor out the common terms in each group.
[tex]x^3 - x^2 - 8x + 12 = x^2(x - 1) - 8(x - 1)[/tex]
Factor out the common binomial.
[tex]x^3 - x^2 - 8x + 12 = (x^2 - 8)(x - 1)[/tex]
Solve the factors for x.
Set each factor to zero and solve for x:
[tex]x^2 - 8 = 0[/tex] and x - 1 = 0
[tex]x^2 = 8[/tex], x = 1
x = ±√8
The solutions to the equation [tex]x^3 - x^2 - 8x + 12 = 0[/tex] are x = 1, x = √8, and x = -√8.
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The director of a basketball camp estimates that the number N of students who will enroll in a basketball camp can be modeled by
N = −0.3t + 250,
where t is the tuition for the camp in dollars.
(a)
What is the N-intercept?
(t, N) =
(b)
Write a sentence that explains the answer to part (a) in the context of the problem.
If the tuition was $
, then
students would enroll in the basketball camp.
(c)
What is the t-intercept? (Round your answers to two decimal places when necessary.)
(t, N) =
(d)
Write a sentence that explains the answer to part (c) in the context of the problem. (Round your answers to two decimal places when necessary.)
If the tuition was approximately $
, then
students would enroll in the basketball camp.
Answer:
Step-by-step explanation:
(a) To find the N-intercept, we set t to 0 in the equation N = -0.3t + 250:
N = -0.3(0) + 250
N = 250
Therefore, the N-intercept is (0, 250).
(b) The N-intercept is the point on the graph where the tuition is $0 and represents the number of students who would enroll in the basketball camp if it were free. In this case, if the tuition was $0, 250 students would enroll in the basketball camp.
(c) To find the t-intercept, we set N to 0 in the equation N = -0.3t + 250 and solve for t:
0 = -0.3t + 250
0.3t = 250
t = 250/0.3
t ≈ 833.33
Therefore, the t-intercept is approximately (833.33, 0).
(d) The t-intercept is the point on the graph where the number of students who enroll in the basketball camp is zero and represents the tuition that would need to be charged for no students to enroll. In this case, if the tuition was approximately $833.33, no students would enroll in the basketball camp.
How many coefficients are in the expression 5x³ 2x²2 +6x-4?
There are three coefficients present in the expression: 5 for the x³ ; 2 for the x² and 6 is for x.
Explain about the coefficients?The integer or constant quantity put before a variable is known as a coefficient.
The multiplicative numbers directly preceding a variable, such as x or y, are called coefficients. A quantity in a solution is not regarded as a coefficient if it is not linked to a variable. It is referred to be a constant instead. Decimals, fractions, whole numbers, as well as positive or negative, real or imaginary coefficients are all possible.The coefficient in the expression 5x is 5. The above coefficient is a real, positive whole number.The coefficient for the expression -3y is -3. The above coefficient is a genuine, entire, negative number.Thus, there are three coefficients present in the expression: 5 for the x³ ; 2 for the x² and 6 is for x.
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Complete question-
How many coefficients are in the expression 5x³ + 2x²+6x-4?
What is the gradient of this line? Give each of your answers as an integer or as a fraction in its simplest form
a. The y-intercept of the line is 4.
b. The gradient of the line is 2.
What is the Gradient and the Y-intercept of a Line?The gradient of a line is the steepness of the line, which is determined by how much the line rises or falls for each unit of horizontal distance [m = change in y / change in x], while the y-intercept is the point where the line crosses the y-axis.
a. The line cuts the y-axis at 4, therefore, the y-intercept is 4.
b. Gradient of the line = rise/run = 2/1
Gradient = 2.
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if it is 5:00 will the clock show a right angel
Answer:
the clock hands do not form a right angle at 5:00.
Step-by-step explanation:
Help me With this please.
m∠NOP = 90° is the value of angle .
What does a math angle mean?
An angle is made up of two rays (half-lines) that have a shared termination. The rays serve as the angle's sides, occasionally serving as the angle's legs and occasionally serving as its arms, while the latter is referred to as the vertex of the angle.
A figure that is created by two rays or lines that have the same endpoint is known as an angle in plane geometry. The Latin word "angulus," which meaning "corner," is the source of the English term "angle". The vertex, which is the shared terminal of the two rays, is referred to as the side of an angle.
m∠NOP = 90° is the value of angle .
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somebody help me. how can i do it
Answer: The tank starts with 10 gallons of gas
Step-by-step explanation:
so the first thing that you are going to do is divide and multiply what might you be dividing great question, you will divide 10 in to 9 then multiplying 160 into 15.
If you want to know how i got this answer please ask me in the commets below.
You randomly choose a marble from a bag that contains 10 black marbles
8 red marbles, 4 white marbles, and 6 blue marbles. Match each marble
with the correct probability of selection.
Answer: 10/20, 8/20, 4/20, 6/20
Step-by-step explanation:
Black: 10/20
Red: 8/20
White: 4/20
Blue: 6/20
There are 20 marbles in total the amount of colored marbles is the numerator and the denominator is the total.
Have a lovely day!!
There are 8 colored chips. 3 of them are blue, and 2 of them are red. Emily and Josie were randomly choosing a colored chip. What is the probability that Emily will choose a blue chip, keep it, and then Josie chooses a red chip?
Answer:
If the entire mass of the Milky Way was due to gas and stars, how would you expect the rotational speed of a star near the edge of the galaxy to compare to the rotational speed of a star near the center?
Step-by-step explanation:
3/2 (as a fraction) by the power of 2
Answer:
9/4
Step-by-step explanation:
3/2 x 3/2= 9/4
Write the polynomial P(x) of degree 4, in standard form, with integer coefficients, and zeros 5i, 4, and -1/3 as some of its zeros. Show all work.
Answer:
[tex]P(x)=3x^4-11x^3+71x^2-275x-100[/tex]
Step-by-step explanation:
Given information:
Polynomial function with real integer coefficients.Zeros: 5i, 4, and -1/3Degree: 4For any complex number [tex]z=a+bi[/tex], the complex conjugate of the number is defined as [tex]z*=a-bi[/tex].
If f(z) is a polynomial with real coefficients, and z₁ is a root of f(z)=0, then its complex conjugate z₁* is also a root of f(z)=0.
Therefore, if P(x) is a polynomial with real coefficients, and 5i is a root of P(x)=0, then its complex conjugate -5i is also a root of P(x)=0.
So the zeros of the function are: 5i, -5i, 4, and -1/3.
The zeros of a function P(x) are the x-values of the function such that the P(x)=0. According to the Factor Theorem, if P(x) is a polynomial, and P(a)=0, then (x - a) is a factor of P(x).
Therefore, the factors of the function are:
[tex]x=5i \implies (x-5i)[/tex]
[tex]x=-5i \implies (x+5i)[/tex]
[tex]x=4 \implies (x-4)[/tex]
[tex]x=-\dfrac{1}{3} \implies (3x+1)[/tex]
So the polynomial in factored form is:
[tex]P(x)=a(x-5i)(x+5i)(x-4)(3x+1)[/tex]
As we have not been given a specific leading coefficient, let us assume it is one.
[tex]P(x)=(x-5i)(x+5i)(x-4)(3x+1)[/tex]
To write the polynomial in standard form, expand and simplify.
[tex]\begin{aligned}P(x)&=(x-5i)(x+5i)(x-4)(3x+1)\\&=(x^2+5ix-5ix-25i^2)(3x^2+x-12x-4)\\&=(x^2-25(-1))(3x^2-11x-4)\\&=(x^2+25)(3x^2-11x-4)\\&=3x^4-11x^3-4x^2+75x^2-275x-100\\&=3x^4-11x^3+71x^2-275x-100\end{aligned}[/tex]
PLEASE HELP DUE TONIGHT
Answer: $1.46 less
Step-by-step explanation:
Lets find how much each bottle is first for Store B, to do that divide 29.10 divided by 15 and you should get 1.94 meaning each bottle of juice is $1.94. Next, lets find how much each bottle costs in store A. You can take any number from the total cost and divide it by the bottles and you should get the same thing. Lets do the first one 2.40 divided by 5 equals 0.48. If you want to check you can multiply 0.48 by any number of bottles and you should get the total cost. Now we have to find how much less Store A costs, to do so we subtract store B (1.94) and store A (0.48) and you should get $1.46.
Mrs. Vo is going to put wallpaper around her daughter's rectangular bedroom over the summer. What formula is needed to find the amount of border needed?
Answer: The formula needed to find the amount of border needed is the perimeter formula for a rectangle:
Perimeter = 2(length + width)
Step-by-step explanation:
what is the sum of the first 5 terms of the geometric series: 1+3+9+…+19,686
a. 212
b. 58
c. 136
d. 121
The sum of the first 5 terms of the series is 121.
What is geometric series?An infinite number of terms with a fixed ratio between them are added to form a geometric series.
The series is a geometric series with the first term (a) equal to 1 and the common ratio (r) equal to 3.
To find the sum of the first 5 terms of the series, we can use the formula:
[tex]\rm S = a(1 - r^n) / (1 - r)[/tex]
where S is the sum of the first n terms of the series. Substituting the values we get:
[tex]\rm S = 1(1 - 3^5) / (1 - 3) \\\\= 1(1 - 243) / (-2)\\\\ = -242 / -2 \\\\= 121[/tex]
Therefore, the sum of the first 5 terms of the series is 121.
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help pls show ur work
5. The missing length = D. 91
6. The missing length = C. 5
7. The missing length = B. 121
8. The missing length = D. 30
What are similar triangles?Similar triangles are triangles that have the same shape but possibly different sizes. In other words, they have the same angles but different side lengths.
The corresponding sides of similar triangles are proportional to each other, meaning that if you were to multiply all the sides of one triangle by a certain factor, you would get the corresponding sides of the other triangle.
5. In order to find the missing length:
WU/FU = UV/UG = 130/60 = UV/42
Cross-multiplying, we have:
60UV = 130 x 42
UV = 5 460/60 = 91.
6. To find the missing length, we have:
PQ/DE = PR/DF = PQ/20 = 10/40
Cross-multiplying, we have:
40PQ = 20 x 10
PQ = 200/40 = 5.
7. To find the missing length, we have:
CE/EW = DE/EV = CE/55 = 110/50
Cross-multiplying, we have:
50CE = 110 x 55
CE = 6 050/50 = 121.
8. To find the missing length, we have:
DE/LM = CE/LN = 42/36 = 35/LN
Cross-multiplying, we have:
42LN = 35 x 36 = 1 260
LN = 1 260/42 = 30.
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PLEASE HELP ME ANSWER THIS QUESTION ASAP!!
Answer:
1/4(2^3 + 4^2) = 6
Step-by-step explanation:
I
Solve for v.
37=-v+182
a business purchases a computer system for $2000if the value of the system is modeled by f(x)=2000X0.85
does the function represent a growth or decay model.
by what percent
The scales of a map is 1 inch to 100 miles. The map is 9 x 12 inches in size. Points A and B are 7 1/2 inches apart. If the distance actually traveled is 11% greater than the distance show on the map, how far would a truck actually travel in going from a to b
Answer: one inch measured the the map represented 100 feet on the ground
Step-by-step explanation:
i need help with the question
The intersection points are: (0, -3) and (10, 2)
The bounded area is -20.83 unit²
How to find the intersection points?(a) The intersection points of the line and curve are the points where the line and curve meet. Thus, the intersection points are:
(0, -3) and (10, 2)
(b) The integral terms of y which gives the area bound are:
a = -3, b = 2 (These are the y values of the intersection points)
Since we have x = y² + 3y and x = 2y + 6, we can say:
y² + 3y = 2y + 6
y² + 3y - 2y - 6 = 0
y² + y - 6 = 0
Thus, h(y) =
Using a = -3 and b = 2 with h(y) = y² + y - 6, we have the bounded area as:
[tex]Area = \int\limits^b_a {y} \, dy[/tex]
[tex]Area = \int\limits^{2}_{-3} {(y^{2} + y - 6}) \, dy[/tex]
Integrating the area:
Area = [y³/3 + y²/2 - 6y + C]²₋₃
Area = [2³/3 + 2²/2 - 6(2)] - [(-3)³/3 + (-3)²/2 - 6(-3)]
Area = [8/3 + 2 - 12] - [-9 + 4.5 + 18]
Area = [-22/3] - [27/2]
Area = -125/6
Area = -20.83 unit²
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Determine the domain of the function F(x) =2-x/12x^2+8x
A delivery truck is transporting boxes of two sizes: large and small. The combined weight of a large box and a small box is 80 pounds. The truck is transporting 60 large boxes and 70 small boxes. If the truck is carrying a total of 5050 pounds in boxes, how much does each type of box weigh?
The weight of small box and large box is 25pounds and 55 pounds respectively.
What is Simultaneous equation?Simultaneous equations are two or more algebraic equations that share variables e.g. x and y . Example of a Simultaneous equation include;
3x+2 = y equation 1
5x +3 = 2y equation 2
Represent the weight of small box by x and the weight of large box by y
x + y = 80 equation 1
70x + 60y = 5050 equation 2
therefore 70( 80-y) + 60y = 5050
5600 -70y +60y = 5050
-10y = 5050-5600
-10y = -550
y = -550/-10
y = 55 pounds
From equation 1
x +55 = 80
x = 80-55
x = 25 pounds
therefore the weight of small box is 25 pounds and the weight of large box is 55 pounds
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Delivery of 5 large boxes, 3 small boxes; total weight 135 kilograms. And 7 large boxes, 9 smaller boxes total weight,279 kilograms. How much does each box weigh?
The weight of each of the boxes are:
Large box = 15.75 kg
Small box = 18.75 kg
How to solve simultaneous Equations?Let the weight of each large box be x
Let the weight of each small box be y.
Thus, if 5 large boxes and 3 small boxes have a total weight of 135 kilograms, then we have the equation:
5x + 3y = 135 -------(1)
Thus, if 7 large boxes and 9 small boxes have a total weight of 279 kilograms, then we have the equation:
7x + 9y = 279 ------(2)
Multiply eq 1 by 3 and eq 2 by 1 to get:
15x + 9y = 405 -----(3)
Subtract eq 2 from eq 3 to get:
8x = 126
x = 126/8
x = 15.75 kg
Thus:
5(15.75) + 3y = 135
3y = 135 - 78.75
3y = 56.25
y = 18.75 kg
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Identify and write the range of possible values for x
The range of possible values for x is approximately -14.7 to 14.7. However, since x represents a length, it cannot be negative. Thus, the range of possible values for x is approximately 0 to 14.7.
What is range?The range refers to the difference between the maximum and minimum values in a dataset.
The range is easy to calculate and provides a quick estimate of how much variation there is in the dataset.
We have:
∠ABC + ∠ABD + ∠BDA + ∠BAC = 360°
120° + 115° + ∠BDA + ∠BAC = 360° (substituting given values)
∠BDA + ∠BAC = 125°
Now, we can use the law of cosines to relate the sides and angles in each triangle:
AC² = AB² + BC² - 2ABBCcos(∠ABC)
(4x-10)² = AB² + 24² - 2AB24cos(120°)
(4x-10)² = AB² + 576 + 24AB
AD² = AB² + BD² - 2ABBDcos(∠ABD)
40² = AB² + 24² - 2AB24cos(115°)
AB² - 48AB + 496 = 0
Using the quadratic formula to solve for AB, we get:
AB = (48 ± √(48² - 4×496))/2
AB = 24 ± 2√211
Since AB = 24 + 2x, we have:
24 + 2x = 24 + 2√211 or 24 - 2√211
By Solving for x in each case, we get:
x = √211 + 0.5 ≈ 14.7 or x = -√211 + 0.5 ≈ -14.7
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6. Tim has used 11 squares tiles to cover a surface. What is the height of this surface if its base is 12 cm.
12 cm
b) 13 cm
c) 10 cm
d) 14 cm
e) 11 cm
Height of this surface if its base is 12 cm 42 tiles are required.=14400
What is unitary method?"A method to find a single unit value from a multiple unit value and to find a multiple unit value from a single unit value."
We always count the unit or amount value first and then calculate the more or less amount value.
For this reason, this procedure is called a unified procedure.
Many set values are found by multiplying the set value by the number of sets.
A set value is obtained by dividing many set values by the number of sets.
According to our question-
Area of region = 100 cm x 144 cm = 14400 cm.sq
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COMPLETE QUESTION-
Tim has used 11 squares tiles to cover a surface. What is the height of this surface if its base is 12 cm?
The radius of a spherical globe is 9 inches. What is the volume of the globe?
the volume of the spherical globe with a radius of 9 inches is approximately 3053.63 cubic inches.
What is globe?
A globe is a three-dimensional, spherical representation of the Earth or another celestial body, such as a planet or a moon. It is typically made of a hollow sphere of material, such as paper, plastic, metal, or glass, with a map of the Earth or the celestial body printed or drawn on its surface.
Globes are often used as educational tools to help people visualize and understand the geography, topography, and other features of the Earth or other planets. They can be used in classrooms, museums, or other settings to teach geography, astronomy, or other related subjects.
Globes can also be decorative objects, used to add visual interest to a room or office. They come in a variety of sizes, styles, and designs, ranging from small desktop globes to large, ornate globes mounted on stands.
The formula for the volume of a sphere is V = (4/3)πr³, where V is the volume, π is the mathematical constant pi (approximately 3.14), and r is the radius of the sphere.
Using this formula and substituting the given radius of 9 inches, we can calculate the volume of the globe as follows:
V = (4/3)πr³
V = (4/3)π(9³)
V = (4/3)π(729)
V = 3053.63 cubic inches (rounded to two decimal places)
Therefore, the volume of the spherical globe with a radius of 9 inches is approximately 3053.63 cubic inches.
It's important to note that the volume of a sphere is a measure of the amount of space inside the sphere, so the volume represents the maximum amount of material that can fit inside the sphere. This calculation may be useful in a variety of contexts, such as designing packaging or calculating the capacity of a container.
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in the linear equation y = 3x+8, the y-intercept is
Answer:
(0,8)
Step-by-step explanation:
The slope intercept formula is set up as y=mx+b, where b is the y-intercept value. In the equation y=3x+8, we can see that 8 is the y-intercept. So we plug 8 for y in a coordinate, and 0 for x. The answer is (0,8)
Show the solution how to get 10 with the numbers 3,4,7,8
To get 10, we can create an expression which sums up the
[tex]\left(3\cdot 4+8\right)\ -\left(7\ +\ 3\right)[/tex] = 10
What is sum?In mathematics, the sum refers to the result of adding two or more numbers or quantities together. It is a basic arithmetic operation that yields a total or a whole quantity.
Here, we can make a solution of 20 by
3 × 4 + 8
= 20
We know that 20 - 10 = 10
So another expression must be 10 and it is possible by:
7 + 3
So, we have the expression which has a solution 10
(3 × 4 +8 ) - (7 + 3)
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You have $300,000 saved for retirement. Your account earns 4% interest. How much will you be able to pull out each month, if you want to be able to take withdrawals for 15 years?
You will be able to pull out approximately $1,983.45 each month from your retirement savings if you want to be able to take withdrawals for 15 years.
What is interest rate?Interest rate is the percentage of the principal amount (a loan or an investment) that is charged or paid over a certain period of time, usually a year. It is used to calculate the amount of interest that accrues on the principal amount.
According to question:To calculate the monthly withdrawal amount that you can make from your retirement savings, we can use the present value of an annuity formula. This formula relates the present value of a series of equal periodic payments to the interest rate, the number of periods, and the payment amount.
Let's assume that you want to make monthly withdrawals for 15 years, which corresponds to 180 months. Also, let's assume that you want to withdraw the same amount each month. We can call this amount "P".
Using the present value of an annuity formula, we can solve for P:
P = PV * (r / (1 - [tex](1 + r)^(-n)[/tex]))
where PV is the present value of your retirement savings,
r is the monthly interest rate (which is 4% / 12 = 0.00333), and
n is the number of months over which you want to make withdrawals (which is 180 in this case).
Substituting the given values, we get:
P = 300,000 * (0.00333 / (1 - (1 + [tex]0.00333)^(-180)[/tex]))
≈ $1,983.45
Therefore, you will be able to pull out approximately $1,983.45 each month from your retirement savings if you want to be able to take withdrawals for 15 years. Note that this amount is an estimate and does not take into account any taxes, fees, or other factors that may affect your retirement income.
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Solve for [tex]y[/tex] as a function of [tex]x[/tex]:
[tex]\frac{dy}{dx}=2xy^3\sin\left(x^2\right)[/tex] and [tex]y(0)=-1[/tex]
The solution to the differential equation dy / dx = 2 · x · y³ · sin x² is equal to - [1 / (2 · y²)] = - cos x² - 3 / 2.
How to solve a separable variable differential equation
In this problem we find a differential equation with separable variables, that is, an ordinary differential equation whose variables can be separated on each side of the equivalence and solved by integrals. First, write the complete expression:
dy / dx = 2 · x · y³ · sin x²
Second, separate the variables:
dy / y³ = 2 · x · sin x² dx
Third, integrate the equation:
∫ dy / y³ = ∫ sin x² · (2 · x) dx
- [1 / (2 · y²)] = - cos x² + C
Fourth, find the value of the constant of integration:
- 1 / 2 = 1 + C
C = - 3 / 2
Fifth, write the solution to the differential equation:
- [1 / (2 · y²)] = - cos x² - 3 / 2
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Two find the point in the solution set for both these functions equation the two functions (f(x)=g(x)) and solve for x. See full process below.
Explanation:
Two find the point in the solution set for both these functions equation the two functions (f(x)=g(x)) and solve for x.
Therefore:
2x−1=−43x+9
We can now solve for x:
2x−1+43x+1=−43x+9+43x+1
2x+43x−1+1=−43x+43x+9+1
2x+43x−0=0+9+1
2x+43x=10
(33×2)x+43x=10
63x+43x=10
103x=10
310×103x=310×10
3
10×103x=310×10
x=3 is the point which lies in the solution set for both functions.
Which answer choice shows the correct classification(s) of the number- 15/3
?
HINT: Simplify the number first. Remember that the fraction bar represents division.
whole, integer, rational
integer, rational
integer
rational
The correct classification(s) of the fractional number 15/3 is: whole, integer, rational.
What is fractional number?
A fractional number, also known as a fraction, is a number that represents a part of a whole. It consists of two parts: a numerator and a denominator, separated by a slash (/). The numerator is the number above the line, and the denominator is the number below the line.
The number 15/3 is a fraction. When we simplify this fraction, we divide 15 by 3 to get:
15/3 = 5
Therefore, the number 15/3 is equivalent to the whole number 5.
An integer is a number that can be expressed without fractions or decimals, and can be positive, negative, or zero. The number 5 is also an integer, because it is a positive whole number.
A rational number is a number that can be expressed as a ratio of two integers, where the denominator is not zero. In this case, 15/3 can be expressed as the ratio of two integers (15 and 3), so it is a rational number.
Therefore, the correct classification(s) of the number 15/3 is:
whole, integer, rational.
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