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The simplest form of expression [tex]125\frac{2}{3}[/tex] is [tex]25[/tex].

Definition of exponent- Exponent means a quantity, describing the power to which base number is raised. In other words , Exponent refers to the number of times a number is used in a multiplication. The 4 types of exponents are: positive, negative, zero, and rational. Positive exponents tell you how many times to multiply a base by itself. Exponent is defined as the method of expressing large numbers in terms of powers.

Given :- expression: [tex]125^{\frac{2}{3} }[/tex]

Explanation:-In order to solve this expression , we use properties of exponents.

Hence , using the property - [tex]x^{m} x^{n}[/tex][tex]=x^{mn}[/tex]

and rewriting 125 as cube of 5

⇒[tex]125=5^{3}[/tex]

⇒[tex]5^({3)\frac{2}{3} }[/tex]

⇒[tex]5^{2}[/tex]

⇒[tex]25[/tex]

So, finally the simplest form of the given expression is 25.

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The solution for the system of equations is (-19,55).

As given the equations in the question

y = –3x – 2

Simplify the above

y + 3x = -2

5x + 2y = 15

Multiply y + 3x = -2 by 2 and subtracted from 5x + 2y = 15.

2y - 2y + 5x -6x = 15 + 4

-x = 19

x = -19

Putting the value of x in the equation y + 3x = -2.

y + 3 × - 19 = -2

y - 57 = -2

y = -2 + 57

y = 55

Therefore the solution for a system of equations is (-19,55).

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The fact that four identical unit objects arranged in a square formation would produce a square with length 2, rather than length -2, provides the logical basis for the fact that the "principal" square root of 4 is 2 rather than -2.

What is the square root?

The factor that can be multiplied by itself to obtain a given number is known as the square root. The square root is represented by The opposite of squaring a number is finding its square root.

According to the given question:

The value of root 4 exactly corresponds to 2. However, we can say that there are always two roots for any given number, and that the roots can be either positive or negative. Consequently, the value of root 4 is 2, or +2 and -2 (positive 2 and negative 2).

Hence ,the square root of 4 only 2 and not - 2.

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Step 1: Given the functions f(x) and g(x) , compute the limit limx→∞f(x)g(x) lim x → ∞ f ( x ) g ( x ) .

Step 2: If the limit in Step 1 is a finite constant a≠0 a ≠ 0 , then the growth rate of f(x) is a times the growth rate of g(x) at sufficiently large x .

Now, According to the question:

Steps on How to Compare the Rates of Change of Two Functions Using Limits

Step 1: Given the functions f(x) and g(x), compute the limit

limx→∞f(x)/g(x).

Step 2: If the limit in Step 1 is a finite constant a≠0, then the growth rate of f(x) is a times the growth rate of g(x) at sufficiently large x. If the limit in Step 1 is ∞, then the growth rate of f(x) is greater than the growth rate of g(x). If the limit in Step 1 is 0, then the growth rate of g(x) is greater than the growth rate of f(x).

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Yes, x +3 is a factor of polynomial X³ +3x² -4x -12

Polynomial:

In mathematics, a polynomial is an expression which includes indeterminates (also called variables) and coefficients, that entails handiest the operations of addition, subtraction, multiplication, and effective-integer powers of variables. An instance of a polynomial of a single indeterminate x is x² − 4x + 7. An example with 3 indeterminates is

                  x³ + 2xyz² − yz + 1.

Polynomials appear in lots of areas of arithmetic and technological know-how. For example, they're used to form polynomial equations, which encode a wide range of issues, from elementary word troubles to complicated clinical issues; they may be used to outline polynomial functions, which seem in settings ranging from basic chemistry and physics to economics and social science; they're utilized in calculus and numerical analysis to approximate other features. In advanced mathematics, polynomials are used to construct polynomial rings and algebraic varieties, which can be primary concepts in algebra and algebraic geometry.

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

(x³ + 3x²)− 4x− 12

Factor out the greatest common factor (GCF) from each group.

x²(x+3)−4(x+3)

Factor the polynomial by factoring out the greatest common factor,

x+3.(x+3)(x2−4)

Rewrite

4 as 2².

(x+3)(x²−2²)

Since both terms are perfect squares, factor using the difference of squares formula,

a²−b²=(a+ b)(a−b)

where

a = x  and b = 2.

(x+3)[(x+2)(x-2)]

Remove unnecessary parentheses.

(x+3)(x+2)(x−2)

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It is impossible to create an equation for a circle given exactly two points.

Let's consider the two basic equations of circle, i.e, the standard equation and general equation.

Now, Standard equation of circle with center (h, k) and radius r is represented as

(x−h)² +(y−k)² =r²

Note here we need values of three unknowns, h, k, r to get equation of a certain circle.

The other general equation of circle is of the form

x² + y² + 2gx + 2fy + c = 0

Again, values of three unknowns (g, f and c) are necessary for obtaining equation of a certain circle. So in both cases, values of three unknowns are required to have an equation of a distinct circle. In order to calculate, the values of these three unknowns, we must first require three conditions that would be produced by three equations. Solving those equations, allows us to construct the equation of circle. So if only two points are given, then these gives two equations with three unknowns. It would be impossible to get three distinct values from two equations. So, there is no possible distinct circle for two points.

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Congruent shapes or figures can be said to be congruent if they can be related to each other by a transformation, such as a translation, rotation, reflection or scaling.

For example, two triangles are congruent if they can be related to each other by a rotation of 180 degrees.

To establish a student experience that yields significant and equitable improvements in outcomes and educational value, an institution's structures, culture, and business model must be realigned. This process is known as transformation.

Any action or set of actions that takes one or more inputs, changes and adds value to them, and then produces outputs for consumers or clients is referred to as a transformation process.

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Congruent figures are produced through rigid transformations. Congruent figures may always be connected to stiff transformations even though you could conceive of them as forms that "look precisely the same."

Separation between each pair of points of a geometric object is maintained during exact modification of the object. In other words, exact transformations preserve the size and shape of objects. For example, a triangle rigid transformation preserves the triangle's angles and lengths. Imagine the distortion of a rigid shape similar to that of cardboard. The shape does not change even if it is slid left and right, rotated, and mirrored.

It was the opposite when I made a mold with putty. After that, you can twist and stretch things. However, these conversions are not exact. Stretching the shape changes the distance between each pair of points.

We may produce congruent shapes by applying rotations, reflections, and translations—three types of transformations. In reality, all congruent shape pairings may be matched to one another by combining one or more of these three transformations.

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There are some postulates, such as Side-Side-Side, Side-Angle-Side, Angle-Side-Angle, and Angle-Angle-Side, that can be applied to demonstrate the congruence of triangles.

The Angle-Side-Angle (ASA) postulate asserts that two triangles are congruent if two angles and the included side of one triangle are congruent with two angles and the included side of another triangle.

The basic components, primarily the three angles as well as the three sides of a triangle, must be equal to the corresponding angles and sides of the other triangle for two triangles to be congruent.

To demonstrate that the triangle ABC is congruent, we must show that two of its angles and one of its sides are congruent.

If the lengths of the matching sides and angles of two triangles match, the triangles are said to be congruent. In maths, congruence relates to the similarity of two figures in terms of their size and shape. The mirror image of one shape is the same as the other if two shapes are congruent.

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It is a straightforward application of the pigeonhole principle to demonstrate that there must be at least one duplication in nums. Here, each possible number in nums is a "pigeon," and each specific place where a pigeon can occur is a "pigeonhole."

How to determine whether there is at least one duplicate number in Nums:

Simple Method: The goal is to use nested loops and determine whether or not each element appears in the array more than once for each element.

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The actual distance between Eagle Lake and swallow lake is 9/1400 km.

What is distance?

Distance is the measure of the amount of space between two points or objects. It can be measured in a variety of ways, including miles, kilometers, feet, and meters. Distance is an important concept in physics, mathematics, and many other scientific fields. In addition, distance is a key factor when considering the speed of objects or events. Distance can also be used to measure the time it takes for an object or event to travel from one point to another. Distance is a fundamental concept in understanding the physical universe, and it is essential for navigation, navigation systems, and communication.

The scale of the map is given as,

                          14 m = 1 Km

We know that, 1 m = 100 cm

So, 14 × 100 cm = 1 Km

      1 cm = 1/1400 Km

So, 9 cm = 9 × 1/1400 km

               = 9/1400 km.

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x-1 is the quotient when [tex]x^2-2x+1[/tex] is divided by x-1

if we equate x-1=0

=> x=1

after putting the value of x=1 in the equation  [tex]x^2-2x+1[/tex] to find the remainder.

so the value of [tex]x^2-2x+1[/tex] when x=1

=>1 -2*1 +1

=>0

so the value is 0 which means that x-1 completely divides the given equation  [tex]x^2-2x+1[/tex]

since the given equation is quadratic equation and one factor is linear so other factor which is quotient also need to be linear.

let the quotient after division is ax+b

and according to rule of division

[tex]x^2-2x+1[/tex] = (x-1)*(ax+b)

as remainder is 0

so (x-1)*(ax+b) = a[tex]x^2[/tex] +(b-a)x -b

comparing a[tex]x^2[/tex] +(b-a)x -b with  [tex]x^2-2x+1[/tex]

we observe that a=1 ,b=-1

so the quotient of the above divsion is x-1

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Complete question:

what is the quotient when [tex]x^2-2x+1[/tex] is divided by x-1

Answer:

B. 2.2 kilometers

Step-by-step explanation:

⭐What is the change in the cyclist's elevation?

Thus, we need to solve for the value of the height (x) of the hill.

Notice that the hill is in the shape of a right triangle. In order to find an unknown side length for right triangles, we need to use a trigonometric function.

⭐What is a trigonometric function?

First, let's see what kind of side lengths we are given (opposite, hypotenuse, or adjacent).

We are given a side length of 10km, which is the hypotenuse of the triangle, and we are given a side length of x, which is the opposite of θ (13°).

Therefore, we will use the trigonometric function sin(θ) to solve for x.

Substitute the side lengths and angle measures we are given:

[tex]sin(13) = \frac{x}{10}[/tex]

Set the equation equal to x:

[tex]10sin(13) = x[/tex]

Solve for x. I recommend using a scientific calculator (e.g. Desmos Scientific Calculator):

[tex]2.2 = x[/tex]

∴ The cyclist's change in elevation is 2.2 kilometers.

Answer:

m=1/5

Step-by-step explanation:

SImplify

4m-8=-6m-6

then 10m=2

so m=1/5

4(m - 2) = -2(3m + 3)

4m - 8 = -6m -6

4m + 6m = -6 + 8

10m = 2

m = 1/5   or 0.2

4(0.2 - 2) = -2[3(0.2) + 3]

4(-1.8) = -2(0.6 + 3)

- 7.2 = -2(3.6)

- 7.2 = - 7.2

Step-by-step explanation:

The measure of the section HK of the parallelogram will be 13 inches.

A parallelogram is a straightforward quadrilateral with two parallel sides. A parallelogram's opposite or facing sides are of equal length, and its opposite angles are of equal measure.

A diagonal in geometry is a line segment that connects two vertices of a polygon or polyhedron that are not on the same edge. Informally, any sloping line is referred to as a diagonal.

The length of the HK will be calculated as,

HK = √ ( 12² + 5²)

HK = √ ( 144 + 25 )

HK = √ (169 )

HK = 13 inches

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Answer:

-28

Step-by-step explanation:

Yes, 4096 is a perfect cube. The number whose cube is 4096 is 16. To find the cube root of a number, you can use the following formula:

cube root of a number = ∛a

For example, to find the cube root of 4096, you would calculate:

∛4096 = 16

This means that 16 is the number whose cube is 4096. You can verify this by calculating 16 x 16 x 16, which is equal to 4096.

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A possible solution to the system of inequalities is given as follows:

(20,35).

Which means that you work 20 hours dog walking and 35 hours on the sales job at Computers & More, Inc.

The variables of the system of inequalities are given as follows:

The number of hours is a countable amount, hence it cannot assume negative values, and the first two conditions are given as follows:

You can work no more than 60 hours each week at your two jobs, hence:

x + y ≤ 60.

You need to earn at least $450 each week to pay your bills, hence, considering the ratios:

7x + 12y ≥ 450.

Then the solution is obtained from the shaded region of the graph given at the end of the answer.

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We can use rational zero theorem to find rational zeros.

Given,

Rational zero theorem;-

According to the Rational Zeros Theorem, if P(x) is an integer-coefficient polynomial and is its zero (P() = 0), then p is a factor of P(x constant )'s term and q is a factor of P's leading coefficient (x). The Reasonable Zeros Theorem can be used to locate every rational zero in a polynomial.

Purpose of rational zero test;-

It is also known as the rational root test and the rational zero test. It can be used to locate the polynomial function's zeros. One uses it to determine whether a polynomial has rational zeros or roots. A comprehensive list of the polynomial's potential rational roots is also provided.

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Answer:

Step-by-step explanation:

To find the time it takes for the rock to hit the water, we need to find the value of t when h is equal to zero. This is because the rock will hit the water when it reaches a height of zero feet above the water.

Substituting zero for h in the equation h = -16t^2 + 12t + 40, we get:

0 = -16t^2 + 12t + 40

This equation is a quadratic in t, so we can use the quadratic formula to solve for t:

t = (-12 +/- sqrt(12^2 - 4 * (-16) * 40)) / (-32)

Simplifying, we get:

t = (-12 +/- sqrt(144 + 640)) / (-32)

t = (-12 +/- sqrt(784)) / (-32)

t = (-12 +/- 28) / (-32)

t = -40/-32 or t = 16/-32

t = 1/8 or t = -1/2

Since the time cannot be negative (because the rock was thrown upwards), the only solution that makes sense is t = 1/8 seconds.

Therefore, it will take 1/8 seconds for the rock to hit the water.

Answer:

[tex]Option C) 120cm^{3}[/tex]

Step-by-step explanation:

As we know the formula to find the volume is,

[tex]V=LXBXH[/tex]

[tex]V=5cm x 3cmx8cm[/tex]

[tex]V=120cm^{3}[/tex]

Hope it helps you

Answer:

D

Step-by-step explanation:

when x=1, y should = 2

2(1)=2

so y=2x is correct

The nature of the roots of the given quadratic equation 3x²+ 4x - 5 = 0 are real and distinct.

As given in the question,

Given quadratic equation is written as:

3x²+ 4x - 5 = 0

Standard form of quadratic equation is :

ax² + bx + c = 0

And nature of the roots depends on the value of discriminant 'D' .

D = b² - 4ac

here b = 4 , a = 3 , and c = -5

Discriminant 'D' = 4² - 4(3)(-5)

⇒ D = 16 + 60

⇒ D = 76 > 0

When the value of discriminant 'D' > 0 then nature of the roots are real and distinct.

Therefore, the nature of the roots of the given equation is real and distinct.

The above question is incomplete ,the complete question is :

How do you describe the nature of the roots of 3x²+ 4x - 5 = 0?

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The equation of line in slope intercept form of line passing through the points (1,1) and (3,7) is y = 3x - 2  .

The Slope Intercept form of equation of line is denoted as y = mx + c , where m is the slope and c is y intercept .

the required line passes through the points (1,1) and (3,7) ,

So , the slope(m) is = (7-1)/(3-1) = 6/2 = 3 .

putting the value in slope intercept form  , we get ;

1 = 3(1) + c ;

1 = 3 + c ;

c = 1 - 3 ;

c = -2 .

the equation of line will become ⇒ y = 3x - 2 .

Therefore , the equation of the line in slope intercept form is y = 3x - 2 .

The given question is incomplete , the complete question is

What is the equation in the slope intercept form of a line passing through the given points (1,1) and (3,7)  ?

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Slope-intercept form is often expressed using the formula y = mx + b, where m denotes the line's slope and b denotes the y-value of the line's y-intercept.

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The return rate is unreasonably low. This is so since the investment is $26,245 and the account balance is $26,292 according to the quarterly statement. It is obvious that the investment has increased. Furthermore, the actual rate of return is 0.18% (($26,292-$26,244)/ $26,292)*100).

What does return on investment mean?

The percentage change in an investment's value is represented by the annual rate of return. For instance, if you estimate a 10% annual rate of return, you're anticipating that your investment's value would rise by 10% annually.

What exactly does compounding mean?

Compounding is the method through which interest is added to both the principle balance already in place and the interest that has already been paid. Thus, compounding can be thought of as interest on interest, with the result that returns on interest are magnified over time, or the so-called "magic of compounding."

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Henrik will exceed Kharif in the number of cards in the 4th month because in the 4th month Henrik has a total of 160 cards and Kharif has only 100 cards left.

The four fundamental operations of arithmetic are addition, subtraction, multiplication, and division of two or even more items.

Included in them is the study of integers, especially the order of operations, which is important for all other areas of mathematics, notably algebra, data management, and geometry.

As per the data given in the question,

Total cards brought by Henrik per month = 40

Total cards sold by Kharif per month = 20

Total collection of Cards of Kharif = 180

Then,

Let's check after 3 months how many cards both have right now.

Henrik = 3 × 40 = 120

Kharif = 180 - (3 × 20) = 120

It means that after 3 months both are having the same number of cards.

Now, let's check for 4th month,

Henrik = 4 × 40 = 160

Kharif = 180 - (4 × 20) = 100

It means in the fourth month, Henrik will have more cards than Kharif.

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The percent error of Alex’s estimate will be 10.5% which is the correct answer would be an option (D).

To find the percent error of Alex's estimate, we need to first calculate the difference between the estimated value and the actual value.

In this case, the difference is 42 mL - 38 mL = 4 mL.

Now we can divide the difference by the actual value and multiply by 100% to find the percent error.

As per the given question, the percent error is :

⇒ (4 mL / 38 mL) x 100%

⇒ 10.5%.

Therefore, the answer is 10.5%.

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Based on the graph provided, the range of the function is the set of all real numbers. Therefore, the correct answer is (B) y is all real numbers.

Generally, To find the range of a function, you need to find the set of all possible output values of the function.

In this case, the range of the function is given by the set of all y-values that appear on the graph of the function.

The set of all real numbers is considered to represent the range of the function according to the graph that has been supplied. As a result, the response that is accurate is (B), which states that y only contains real values.

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when the discriminant is negative, that is, less than 0, then the roots are imaginary.

The discriminant in a quadratic equation is equal to:

b^2 - 4*a*c, where b is the coefficient of the degree 1 variable, a is the coefficient of the degree 2 variable and c is the constant term. There can be 3 possibilities regarding the discriminant of a quadratic equation:

Case 1: When b^2 - 4*a*c = 0

In this case, the roots of the equation are equal and real.

Case 2: When b^2 - 4*a*c > 0

In this case, the roots of the equation are real.

Case 3: When b^2 - 4*a*c < 0

In this case, the roots of the equation are imaginary.

So, when the discriminant is negative, that is, less than 0, then the roots are imaginary.

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