Solve the inequality. Write the solution set in interval notation.
6x + 1
4
≤ x + 1

The domains of each function and each composite function are as follows:

domain of f: (-∞, ∞)
domain of g: (-∞, ∞)
domain of f ∘ g: (-∞, ∞)
domain of g ∘ f: (-∞, ∞)

To find the composite functions f ∘ g and g ∘ f, we need to substitute the function g(x) into f(x) for f ∘ g, and the function f(x) into g(x) for g ∘ f.

(a) f ∘ g = f(g(x)) = f(x3 + 1) = 3(x3 + 1) − 8 = 3x3 + 3 − 8 = 3x3 − 5

(b) g ∘ f = g(f(x)) = g(3x − 8) = (3x − 8)3 + 1 = 27x3 − 72x2 + 64x − 511

The domain of a function is the set of all values of x for which the function is defined.

The domain of f is all real numbers, since there are no restrictions on the values of x. So the domain of f is (-∞, ∞).

The domain of g is also all real numbers, since there are no restrictions on the values of x. So the domain of g is (-∞, ∞).

The domain of f ∘ g is the same as the domain of g, since the values of x are first substituted into g(x) before being substituted into f(x). So the domain of f ∘ g is (-∞, ∞).

The domain of g ∘ f is the same as the domain of f, since the values of x are first substituted into f(x) before being substituted into g(x). So the domain of g ∘ f is (-∞, ∞).

Therefore, the domains of each function and each composite function are as follows:

domain of f: (-∞, ∞)
domain of g: (-∞, ∞)
domain of f ∘ g: (-∞, ∞)
domain of g ∘ f: (-∞, ∞)

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Answer:

Step-by-step explanation:

First we can determine that this is a special 45-45-90 triangle, since the two legs are congruent, meaning the two unknown angles are congruent as well:

180 = 90 + 2(theta)

theta = 45

Knowing this, we can remember that the hypotenuse of this special triangle is equal to the leg*sqrt(2).

This means that 1 = x(sqrt(2)).

[tex]x=\frac{1}{\sqrt{2} }[/tex]

Hope this helps!

Answer:

70

Step-by-step explanation:

When we want to find 10% of a number, we move it's current decimal to the left one time.

700.

     ←

70.

10% of 700 is 70.

Step-by-step explanation:

Step 1: Simplify the identity we need to find

[tex] \sin(2x) = 2 \sin(x) \cos(x) [/tex]

So we need to find sin and cos

Here, we are given tan(x)

A simple way to find sin (x) and cos(x) when given tan(x) is to use the definition of the trig functions of acute angles.

[tex] \tan( \alpha ) = \frac{y}{x} [/tex]

[tex] \cos( \alpha ) = \frac{x}{r} [/tex]

[tex] \sin( \alpha ) = \frac{y}{r} [/tex]

where r is

[tex]r = \sqrt{ {x}^{2} + {y}^{2} } [/tex]

Here, y is 4 and x is 3.

So

[tex]r = \sqrt{ {3}^{2} + {4}^{2} } = 5[/tex]

Since both cosine and sine are in quadrant 3, they are both negative.

Know we can plug in the knowns, since we know x,y, and r.

[tex] \cos( \alpha ) = - \frac{3}{5} [/tex]

[tex] \sin( \alpha ) = - \frac{4}{5} [/tex]

Now plug in the knowns for sin 2a

[tex] \sin(2 \alpha ) = 2( - \frac{3}{5} )( - \frac{4}{5} ) = \frac{24}{25} [/tex]

The linear equation that shows a proportional relationship is y = 5x/6, the correct option is C.

A linear equation is an equation that has the variable of the highest power of 1. The standard form of a linear equation is of the form Ax + B = 0. A proportional relationship is one where two variables are related in such a way that when one variable changes, the other changes by a constant factor. In other words, the ratio of the two variables remains constant.

We are given that;

Linear equations to chose from

Now,

when x increases by a certain amount, y increases by a corresponding amount that is exactly 5/6 times as much. In other words, the ratio of y to x is always 5/6, which means that the two variables are proportional to each other.

Therefore, the linear equation will be y = 5x/6.

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The implied domain of the function y = (-6x^2 - 4x)/8 is found to be (-∞, ∞).

The first step in determining the implied domain of the function is to isolate the dependent variable, in this case y, on one side of the equation.

We can do this by adding 5y to both sides of the equation and subtracting 4x from both sides of the equation:

-6x^2 - 4x = 8y

Next, we can divide both sides of the equation by 8 to get y by itself:

(-6x^2 - 4x)/8 = y

Now we have the function in terms of y:

y = (-6x^2 - 4x)/8

The implied domain of the function is the set of all x-values for which the function is defined.

In this case, there are no restrictions on the values of x that can be used in the function, so the implied domain is all real numbers.

Therefore, the implied domain of the function y = (-6x^2 - 4x)/8 is (-∞, ∞).

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The value of h(-9)= 26 using the function h(x)=−2x+8

And, the value of x for which h(x) = -10 is x = 9.

To evaluate h(-9), we simply plug in -9 for x in the function h(x) = -2x + 8:

h(-9) = -2(-9) + 8

h(-9) = 18 + 8

h(-9) = 26

So the value of h(-9) is 26.

To find the value(s) of x for which h(x) = -10, we can set the function equal to -10 and solve for x:

-2x + 8 = -10

-2x = -18

x = 9

So the value of x for which h(x) = -10 is x = 9.

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The 100th term of the arithmetic sequence: 7, -1, -9 is -785.

To find the 100th term of an arithmetic sequence, we can use the formula: an = a1 + (n - 1)d where an is the nth term, a1 is the first term, n is the number of terms, and d is the common difference.

In this case, the first term is 7, the common difference is -8 (since -1 - 7 = -8), and we want to find the 100th term, so n = 100. Plugging these values into the formula, we get:

a100 = 7 + (100 - 1)(-8)

a100 = 7 + (99)(-8)

a100 = 7 - 792

a100 = -785

Therefore, the 100th term of the arithmetic sequence is -785.

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The solution to the inequality is x ≤ 6.8. This means that Elise can afford to rent the paddle boat for a maximum of 6.8 hours if she has $15 to spend.        

Why does inequality matter?

Experts claim that disparity fosters political unrest and impedes economic growth. Concentrated financial resources reduce demand for products and services because affluent families generally spend less of their revenue than do poorer households. If low-income people have fewer choices, the company might suffer.

To solve the inequality 1.25x + 6.50 ≤ 15, we can start by isolating x on one side of the inequality:

1.25x + 6.50 ≤ 15

Subtract 6.50 from both sides:

1.25x ≤ 8.50

Divide both sides by 1.25:

x ≤ 6.8

Therefore, the solution to the inequality is x ≤ 6.8. This means that Elise can afford to rent the paddle boat for a maximum of 6.8 hours if she has $15 to spend.

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The given matrices are checked and their status is mentioned in the following list below :a. Reduced echelon form [100 010 000 -10-100] b. Not in echelon form [ -8000 -4200 -8110 -9101 -8130] c. Reduce echelon form [11 10 -4 -10] d. Echelon form [ 100 001 000 -502] Echelon form and reduced echelon form of matrices

The Matrix A is in the reduced echelon form if and only if the following conditions are satisfied:All rows having 0s are at the bottom of the matrix.There must be no 0 rows in the matrix.In each row of the matrix containing the leading nonzero element, all entries above and below the leading entry are 0.

The leading coefficient of a nonzero row is always strictly to the right of the leading coefficient of the row above it.In each column containing a leading coefficient, all other entries are zero.

For a matrix to be in echelon form, it must fulfill the following conditions : All rows containing all zero elements are located at the bottom of the matrix. All leading coefficients in any nonzero row are always strictly to the right of the leading coefficient in the row above it.

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We can use the notation definition to show that Π does not perform too differently between two environments vj and v0 by comparing the performance of Π in both environments and calculating the difference between them.

In order to show that Π does not perform too differently between two environments vj and v0, we can use the notation definition to compare the performance of Π in both environments.

First, we can use the notation Π(vj) to represent the performance of Π in environment vj and Π(v0) to represent the performance of Π in environment v0.

Next, we can compare the two performances by calculating the difference between them using the notation |Π(vj) - Π(v0)|.

If the difference is small, then we can say that Π does not perform too differently between the two environments. However, if the difference is large, then we can say that Π performs differently in the two environments.

In conclusion, we can use the notation definition to show that Π does not perform too differently between two environments vj and v0 by comparing the performance of Π in both environments and calculating the difference between them.

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The slope of the curve at a particular point on the hyperbolic spiral, we need to take the derivative of the equation with respect to θ and evaluate it at the given value of θ. Slope of the curve found to be = 2pi. The correct answer is option C

The equation of the hyperbolic spiral is given by: r = 1/θ To express this equation in terms of x and y, we use the polar-to-rectangular coordinate transformation: x = r cos(θ) y = r sin(θ)

Substituting the equation for r, we get: x = (cos(θ))/θ y = (sin(θ))/θ Taking the derivative of y with respect to x using the chain rule, we get:

[tex](dy/dx) = (dy/dθ)/(dx/dθ) (dy/dx) = [(cos(θ)/θ^2) + (sin(θ)/θ)] / [(-sin(θ)/θ^2) + (cos(θ)/θ)] At t = π, θ = π/2.[/tex]

Therefore, the slope of the hyperbolic spiral at t = π is: (dy/dx) = 2/π The correct option to this value is C. 2pi

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Answer:

5.30

Step-by-step explanation:

One hundredth is 0.01. 5.31-0.01=5.30.

sorry for answering late!

The formula for the pressure of a gas is given by the Ideal Gas Law, which states that P = nRT/V, where P is the pressure, n is the amount of gas in moles, R is the ideal gas constant, T is the temperature in kelvin, and V is the volume in cubic centimeters (cc).

Given the initial conditions, we can find the value of the ideal gas constant R:
1170 kpa = (6 moles)(R)(260 K) / (720 cc)
R = (1170 kpa)(720 cc) / (6 moles)(260 K)
R = 8.27 kpa·cc/mol·K
Now, we can use this value of R to find the pressure under the new conditions:
P = (3 moles)(8.27 kpa·cc/mol·K)(280 K) / (180 cc)
P = 3898.6 kpa·cc / 180 cc
P = 2166 kpa

Therefore, the pressure of the gas when the number of moles is 3, the temperature is 280 K, and the volume is 180 cc is 2166 kpa.

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In the following question, among the conditions given, the circumference of a cylinder is "20π cm, or approximately 62.83 cm" with a 6 cm radius and a 4 cm height.

The formula for the circumference of a cylinder is:

Circumference = 2πr + 2πh

where r is the radius of the circular base and h is the height of the cylinder.

In this case, the radius is 6 cm and the height is 4 cm, so we can plug those values into the formula:

Circumference = 2π(6) + 2π(4)

Circumference = 12π + 8π

Circumference = 20π

So the circumference of the cylinder is 20π cm, or approximately 62.83 cm (if we use a calculator to approximate the value of π to two decimal places).

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The expression can be simplified using the rule of exponents. The value of the expression 0.5a²b³ * (-2b)⁶ is 32a²b¹⁵.

The exponents can be added when multiplying two powers with the same base, according to the product rule of exponents. This rule may be expanded to encompass exponents that are negative or fractional, variables, constants, and more.

The expression can be simplified using the rule of exponents.

0.5a²b³ * (-2b)⁶ = 0.5a²b³ * 64b⁶

Using the power rule of the exponents we have:

= 32a²b⁹ * b⁶

Simplifying the above value we have:

= 32a²b¹⁵

Hence, the value of the expression 0.5a²b³ * (-2b)⁶ is 32a²b¹⁵.

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Expression 0.5a²b³ times (-2b)⁶ is -32a²b⁹

An algebraic expression is a combination of variables, constants, and mathematical operations. It represents a mathematical relationship between quantities and can be used to model real-world situations or solve problems.

Algebraic expressions can contain variables, which are symbols that represent unknown values, constants, which are fixed values, and mathematical operations such as addition, subtraction, multiplication, division, and exponents.

We can simplify this expression using the rules of exponents:

0.5a²b³ times (-2b)⁶ = 0.5a²b³ times (-64b⁶)

Multiplying the coefficients, we get:

-32a²b⁹

Therefore, the simplified expression is -32a²b⁹.

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Christine swam 0.7 miles more than Jack.

The solution can be simply found out by subtracting the distance swam by Jack by the distance swam by Christine.

Distance swam more by Christine= 4.1 - 3.4

Distance swam more by Christine= 0.7

The mile is a customary unit of measurement in the United States and the United Kingdom that is based on the older English unit of length equal to 5,280 English feet, or 1,760 yards. It is often referred to as the international mile or statute mile to distinguish it from other miles. One mile can be covered in under one minute. Nonetheless, there are varying speed limits on the roads. You should include in their average speed of 25–60 mph when calculating how long it will take you to reach your destination.

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The scale factor from the original pool's dimensions to the new pool's dimensions is approximately 3.682.

The volume of a rectangular prism (such as a swimming pool) is given by the formula V = lwh, where l is the length, w is the width, and h is the height.

Let x be the scale factor from the original pool's dimensions to the new pool's dimensions. Then, the dimensions of the new pool can be expressed as lx, wx, and hx, where l, w, and h are the dimensions of the original pool.

We know that the original pool can hold 60,000 L of water, so we can set up the equation:

60,000 = lwh

Substituting in lx for l, wx for w, and hx for h, we get:

60,000 =[tex](lx)(wx)(hx)[/tex]

We also know that the new pool can hold 3,039,180 L of water, so we can set up another equation:

3,039,180 = [tex](lx)(wx)(hx)[/tex]

The result of dividing the second equation by the first equation is:

3,039,180/60,000 = [tex](lx)(wx)(hx)/(lx)(wx)(hx)[/tex]

Simplifying, we get:

50.653 =[tex]x^3[/tex]

The result of taking the cube root of both sides is:

x ≈ 3.682

Therefore, the scale factor from the original pool's dimensions to the new pool's dimensions is approximately 3.682.

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The number of cubic centimeters of space that 3 ping pongs would have is 339. 29 cm ³

A ping pong ball takes the shape of a sphere and the volume of a sphere is ;

= 4 / 3 x π x radius ³

Given the radius of 3 cm, the volume of one ping pong ball is :

= 4 / 3 x π x 3 ³

= 113. 09734 cm ³

If there are three ping pong balls, the volume would then be :

= 113. 09734  x 3 ping pongs

= 339. 29 cm ³

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Other interactions that are confounded with the blocks are BCD and BCD.

(a) Constructing a one replicate of a 24 factorial design in factors A, B, C, and D in blocks of size 4 by confounding the interactions ABC and ACD with the blocks:

1. Number the blocks from 1 to 4.

2. Assign the factor levels to each block, so that the interactions ABC and ACD are confounded. One possible way to do this is as follows:

3. List the contents of two blocks, one of which contains the treatment combination (1) and the other contains treatment combination a.

(b) Other interactions that are confounded with the blocks are BCD and BCD.

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Answer:

scatter plot with points plotted at 1 comma 2, 1 comma 3, 3 comma 2, 3 comma 3, 4 comma 2, 4 comma 5, 5 comma 6, 6 comma 6, 7 comma 8, 8 comma 7, 9 comma 10, and 10 comma 9, with a line drawn through the points 4 comma 5 and 5 comma 6.

Step-by-step explanation:

The scatter plot shows a collection of points on a coordinate plane. To determine which of the given graphs shows a line that fits the data, we need to analyze the points and their distribution.

From the given points, we can observe that there is no clear linear relationship between the x and y variables. However, we can see that there is a cluster of points between (4,2) and (7,8), and the rest of the points are scattered around this cluster. Therefore, any line that fits the data should pass through this cluster of points.

Now, let's analyze each option:

The first option shows a line passing through (6,6) and (7,7). While these two points are part of the cluster, the line does not seem to fit the rest of the data very well.

The second option shows a line passing through (4,5) and (5,6). These points are also part of the cluster, and the line seems to fit the data reasonably well.

The third option shows a line passing through (3,3) and (4,4). These points are near the edge of the cluster, but the line seems to fit the data reasonably well.

The fourth option shows a line passing through (1,2) and (7,8). While these two points are part of the cluster, the line does not seem to fit the rest of the data very well.

Based on the analysis above, the second and third options seem to be the best fits for the data. However, the second option seems to be a slightly better fit than the third option since the line passes through two points that are closer to the center of the cluster. Therefore, the answer is:

scatter plot with points plotted at 1 comma 2, 1 comma 3, 3 comma 2, 3 comma 3, 4 comma 2, 4 comma 5, 5 comma 6, 6 comma 6, 7 comma 8, 8 comma 7, 9 comma 10, and 10 comma 9, with a line drawn through the points 4 comma 5 and 5 comma 6.

Answer: B

scatter plot with points plotted at 1 comma 2, 1 comma 3, 3 comma 2, 3 comma 3, 4 comma 2, 4 comma 5, 5 comma 6, 6 comma 6, 7 comma 8, 8 comma 7, 9 comma 10, and 10 comma 9, with a line drawn through the points 4 comma 5 and 5 comma 6

Step-by-step explanation:

The end behavior of y as x goes to negative infinity in the equation y = -3x^5 + 4x^2 – 7 can be determined by looking at the leading term of the polynomial, which is -3x^5. As x goes to negative infinity, this term will dominate the other terms and determine the end behavior of the function.

Since the degree of the leading term is 5, which is an odd number, and the coefficient is negative, the end behavior of the function will be that y goes to positive infinity as x goes to negative infinity. This can be written in notation as:

lim(x→-∞) y = +∞

In conclusion, the end behavior of y as x goes to negative infinity in the equation y = -3x^5 + 4x^2 – 7 is that y goes to positive infinity.

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Answer:

1. plot the point (6,5) and label it Mr. Snuffles

2. 26 treats were eaten in total (21 not counting Mr.Snuffles)

Step-by-step explanation for plotting the point:

On the bottom line (miles walked) go to the number six, then go up five until it is on the five line for the side line (number of treats) plot the point here and then label the point Mr.Snuffles.

plot the point (6,5)

21 treats were eaten in total

Step-by-step explanation:

On the X axis, go to the number six, then go up five until it is on the five line for the side line (number of treats) plot the point here and then label the point Mr. Snuffles.

The value of the variable 'x' in the isosceles right-angle triangle will be 2√2 units.

The Pythagoras theorem states that the sum of two squares equals the squared of the longest side.

The Pythagoras theorem formula is given as,

H² = P² + B²

The length of the hypotenuse is 4. And the triangle is an isosceles right triangle.

In the isosceles right triangle, the perpendicular and base of the triangle will be the same. Then the value of the variable 'x' is given as,

4² = x² + x²

2x² = 16

x² = 8

x = 2√2

The value of the variable 'x' in the isosceles right-angle triangle will be 2√2 units.

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Answer:6 - 1 = 5

Step-by-step explanation: 12 / 2 = 6. 2 / 2 = 1.

The value of the expression is A = 6xy

Equations are statements in mathematics that have two algebraic expressions on either side of the equals (=) sign.

It displays the similarity of the connections between the phrases on the left and right sides.

Coefficients, variables, operators, constants, terms, expressions, and the equal to sign are examples of the parts of an equation. When creating an equation, the "=" symbol and terms on both sides are necessary.

Given data ,

Let the expression be represented as A

Now , the value of A is

A = 2 ( x ) ( 3 ) ( y )

On simplifying the equation , we get

A = ( 2 ) (3 ) ( xy )

A = 6xy

Therefore , the value of A is 6xy

Hence , the expression is A = 6xy

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The solution set is: [tex]$\left{\frac{-1+\sqrt{5}}{3}, \frac{-1-\sqrt{5}}{3}\right}$[/tex]
Option (D) is correct.

What is a quadratic equation?

A quadratic equation is a type of equation in algebra that can be written in the form of ax^2 + bx + c = 0, where x is the unknown variable, and a, b, and c are constants with a not equal to zero

The equation is:

[tex]$-\frac{3}{2}x^2 = x + 1$[/tex]

We can rewrite this equation as:

[tex]$-\frac{3}{2}x^2 - x - 1 = 0$[/tex]

To solve for x, we can use the quadratic formula:

[tex]$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$[/tex]

Where a = -3/2, b = -1, and c = -1. Substituting these values into the quadratic formula, we get:

[tex]$x = \frac{-(-1) \pm \sqrt{(-1)^2 - 4(-\frac{3}{2})(-1)}}{2(-\frac{3}{2})}$[/tex]

Simplifying this expression, we get:

[tex]$x = \frac{1 \pm \sqrt{5}}{3}$[/tex]

Therefore, the solution set is:

[tex]$\left{\frac{-1+\sqrt{5}}{3}, \frac{-1-\sqrt{5}}{3}\right}$[/tex]

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The solution of the system of equations is ([2+9.48i]/2, 17+9.48i).

A system of linear equations consists of two or more equations made up of two or more variables such that all equations in the system are considered simultaneously. The solution to a system of linear equations in two variables is any ordered pair that satisfies each equation independently.

The given system of equations are y=(x-2)²+35 ----(i) and y=-2x+15 ----(ii).

Here, equation (i) = (ii)

(x-2)²+35= -2x+15

x²-4x+4+35= -2x+15

x²-4x+39= -2x+15

x²-4x+39+2x-15=0

x²-2x+24=0

x²-2x+24=0

By using quadratic formula, we get

x = [-b ± √(b² - 4ac)]/2a

x=[2±√((-2)² - 4×1×24)]/2×1

x=[2±√(-90)]/2

x=[2±9.48i]/2

Here, x=[2+9.48i]/2 and x=[2-9.48i]/2

So, y=2+9.48i+15=17+9.48i

Therefore, the solution of the system of equations is ([2+9.48i]/2, 17+9.48i).

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All of the possible outcomes in which Hana wins the game is {6, 8, 10, 12, 14, 16}.

Probability is a measure of the likelihood or chance of an event occurring. It is expressed as a number between 0 and 1, where 0 indicates that an event is impossible and 1 indicates that it is certain to occur.

To win the game, the sum of the two spinners must be an even number greater than 6. Let's list all the possible outcomes:

So the possible outcomes in which Hana wins the game are: 2 + 4 = 6, 2 + 8 = 10, 4 + 2 = 6, 4 + 4 = 8, 4 + 6 = 10, 4 + 8 = 12, 6 + 2 = 8, 6 + 4 = 10, 6 + 6 = 12, 6 + 8 = 14, 8 + 2 = 10, 8 + 4 = 12, 8 + 6 = 14, and 8 + 8 = 16.

Therefore, all of the possible outcomes in which Hana wins the game is {6, 8, 10, 12, 14, 16}.

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99 as a product of primes is, 99 = 3² * 11

What is  multiplication?

In mathematics, multiplication is a method of finding the product of two or more numbers. It is one of the basic arithmetic operations, that we use in everyday life.

here, we have,

To Write 99 as a product of primes,

we get,

99 = 3 * 3 * 11

99 = 3² * 11

Hence, 99 as a product of primes is, 99 = 3² * 11

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