10-2 skills practice simplifying radical expressions

The perimeter of the triangle is given by the equation P = 4√3 feet

A triangle is a plane figure or polygon with three sides and three angles.

A Triangle has three vertices and the sum of the interior angles add up to 180°

Let the Triangle be ΔABC , such that

∠A + ∠B + ∠C = 180°

The area of the triangle = ( 1/2 ) x Length x Base

For a right angle triangle

From the Pythagoras Theorem , The hypotenuse² = base² + height²

if a² + b² = c² , it is a right triangle

if a² + b² < c² , it is an obtuse triangle

if a² + b² > c² , it is an acute triangle

Given data ,

Let the perimeter of the triangle be P

Let the triangle be represented as LMX

Now , An equilateral triangle is a triangle in which all three sides have the same length.

∠L = ∠M = ∠X = 60° and LM = MX = LX

The height of the triangle is LW = 2 feet

Now , from the trigonometric relations ,

sin 60° = opposite side / hypotenuse

Substituting the values in the equation , we get

√3 / 2 = 2 / LX

Multiply by LX on both sides of the equation , we get

LX ( √3/2 ) = 2

Multiply by ( 2/√3 )on both sides of the equation , we get

LX = 4/√3

Now , the perimeter of the triangle = LX + LM + MX

Perimeter of the triangle P = 3 ( 4/√3 )

Perimeter of the triangle P = 4√3 feet

Hence , the perimeter is 4√3 feet

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The area of the Temasa's doll house is 31.5 square feet.

What is the scale factor?

The scale factor is a ratio that compares the measurements of an object on a drawing or map to the actual measurements of that same object in real life. It is usually expressed as a ratio in the format 1:n (or 1/n) where n is the scale factor.

We can start by using the scale factor of 1:10 to find the actual dimensions of the doll house.

Since the scale factor is 1:10, this means that for every 1 inch on the drawing, the actual measurement is 10 inches. So, to find the actual length and width of the doll house, we can multiply the measurements on the drawing by 10.

7 1/5 inches * 10 = 71 inches + 2 inches

4 4/5 inches * 10 = 44 inches + 8 inches

So, the actual length of the doll house is 71 inches + 2 inches = 73 inches, and the actual width is 44 inches + 8 inches = 52 inches.

To find the area of the doll house, we can multiply the length and width:

73 inches * 52 inches = 3788 square inches

To convert this to square feet, we can divide by 12^2, since 1 foot = 12 inches:

3788 square inches / (12 inches)^2 = 31.5 square feet (rounded to the nearest square foot)

Hence, the area of the Temasa's doll house is 31.5 square feet.

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The factors of a polynomial x² +6x + 8 ( binomial, power of variable x is 2) are equals to the ( x + 2) ( x + 2).

In Mathematics, Factorisation of an algebraic expression is a process of writing the given expression as a product of its factors. These factors can be any type like numbers, variables, or an algebraic expression. To make the factor of a number means to break it up or decompose into numbers such that the multiplication of these can be get the original number. For example, In case of x² – 9 is (x – 3) (x + 3). In other words

(x- 3) and (x + 3) are the factors of x² – 9. We have, a polynomial say, p(x) = x² +6x + 8 and we have to factorize it. Now, x² +6x + 8 ----(*)

Split the middel term of above expression such that sum/difference of split terms is equal to 6x and multiplication of these give 8x².

=> x² + 4x + 2x + 8

group the pairs, => (x² + 2x ) + (4x + 8)

factor each group of pairs,

=> x( x + 2) + 2( x + 2)

Factor out the common term (x + 2)

=> (x + 2) ( x + 2)

Hence, required factors are ( x + 2) (x + 2).

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After solving, the remainder when [tex]x^4 + x^3 - 2x^2 + x + 1[/tex]  is divided by x − 1 is 2.

In the given question, we have to find the remainder when [tex]x^4 + x^3 - 2x^2 + x + 1[/tex]  is divided by x − 1.

The given polynomial is [tex]x^4 + x^3 - 2x^2 + x + 1[/tex].

We have to find the remainder of the polynomial when we divide this polynomial x - 1.

To find the polynomial we equate x - 1 equal to zero to find the value of x.

x - 1 = 0

x = 1

Now putting x = 1 in the given equation [tex]x^4 + x^3 - 2x^2 + x + 1[/tex].

P(x) = [tex](1)^4 + (1)^3 - 2(1)^2 + 1 + 1[/tex]

P(x) = 1 + 1 - 2×1 + 1 + 1

P(x) = 2 - 2 + 2

P(x) = 2

Hence, the remainder when [tex]x^4 + x^3 - 2x^2 + x + 1[/tex]  is divided by x − 1 is 2.

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The complete question is:

What is the remainder when [tex]x^4 + x^3 - 2x^2 + x + 1[/tex]  is divided by x − 1?

Answer:

[tex]x\sqrt{x}[/tex]

Step-by-step explanation:

Use Exponent Power rule:

[tex]b^{1/n} =\sqrt[n]{b}[/tex]

Given [tex]\sqrt{x^3}[/tex] also known as [tex]\sqrt[2]{x^3}[/tex], rewrite with exponent power rule as [tex]x^{3/2}[/tex].

Factor [tex]x[/tex] from [tex]x^{3/2}[/tex] = [tex]x(x^{1/2})[/tex]

Rewrite as [tex]x\sqrt{x}[/tex]

The scale factor is equal to 2.

A scale factor is defined as the ratio between the scale of a given original object and a new object, which is its representation but of a different size (bigger or smaller).

as given, K =(5,0) (10,0)

Solution:

Initial Coordinates=(5,0)=(xi,yi)→xi=5, yi=0

Final Coordinates=(10,0)=(xf,yf)→xf=10, yf=0

Scale factor=xf/xi=yf/yi

We divide the final coordinates by the initial coordinates. In this case the ordinates (yf and yi) are zero and we can't divide by zero, but the abscissas (xf and xi) are different of zero, then we can use them:

Scale factor=xf/xi=10/5

Scale factor=2

Therefore, the scale factor is equal to 2.

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Using Composition of functions, the value of g(h(24)) = 1158.

A function is composed in mathematics when two functions, f and g, are used to create a new function, h, such that h(x) = g(f(x)). The function of g is being applied to the function of x, in this case. Therefore, a function is essentially applied to the output of another function.

The new function h is created by combining the two functions f and g, where h(x) = f(g(x)) for all x in g's domain where g(x) is in f's domain. Function composition is denoted by the notation h = f • g, also written as h(x) = (f • g)(x), which can be interpreted as "f of g of x."

Given,

g(x) = 2x + 4, h(x) = x² +1

Using composition of functions,

⇒ h(24) = 24² + 1

⇒ g(h(24)) = 2 (24² + 1) + 4

⇒ g(h(24)) = 2 × 577 + 4

⇒ g(h(24)) = 1158

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Answer:

follow me first

Step-by-step explanation:

i will answer that in 60 sec

Answer:

B

Step-by-step explanation:

1. Before anything, figure out the inequality symbol. Why? In the solutions, all of the inequalities are different so you narrow it down by figuring out the inequality sign.

2. "At most 50 pounds" means we can't go over 50 pounds, but the package can still equal 50, that's just the most it can be. This means we will have a line underneath because it includes the maximum number, 50, in the solution.

3. "At most" also means the package (x) cannot be over 50. The sign opening must face 50 because that shows x has to be less than. But it can equal 50, so the sign is [tex]x\leq 50[/tex]. That would be B.

4. About the numbers, the box weighs 2 pounds. This means it is x+2 because weight is not negative and it is ADDING weight to what we don't know. To isolate x, just do 50-2, and you get 48. That again proves b is the answer.

Answer: The median is 19

Step-by-step explanation:

To find the median you have to arrange the numbers from lowest to highest.
It would go like this: 13, 15, 16, 22, 24, 28

Pick the number that is in the middle. In this case, it would be 16 and 22. The median of these numbers is the mean of these 2.

To find the mean you add 16 and 22 together to get 38 and divided it by 2 to get 19.

I hope this was able to help you :D

Step-by-step explanation:

fog(x)=f(x-3)

=(x-3)²+2

=x²-6x+9+2

=x²-6x+11

The number of earth days that it will take the planet to orbit the star is 500 earth days.

The rate of distance moved in a specific direction to the time taken can be referred to as the speed.

i.e speed = distance/ time taken

In the given question, the total distance covered is the length of the orbit.

So that,

length of the orbit = 2πR

where R is the radius i.e the distance of the star to the planet.

length of the orbit = 2 * 3.142 * 67000000

                              = 421028000 km

speed = distance/ time taken

⇒     time taken = distance/ speed

                          = 421028000 km/ 35,000 km/h

                          = 12029.37 h

But,

1 earth day = 24 hours

x = 12029.37 h

x = 12029.37 h * 1 earth day/ 24 h

  = 501.2238 earth days

The number of earth days it will take the planet to orbit the star is 500 earth days.

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The true statements are,

A. Joel's distance is a function of time.

B. Joel's fastest speed occurs in the first hour of the hike.

E. The graph is decreasing at all the times between 3.5 hours and 7 hours.

In mathematics, the graph of a function f is the set of ordered pairs, where {\displaystyle f(x)=y.} In the common case where x and f(x) are real numbers, these pairs are Cartesian coordinates of points in two-dimensional space and thus form a subset of this plane.

here, we have,

from the given graph we get,

The graph represents his journey, and,

the true statements are,

A. Joel's distance is a function of time.

B. Joel's fastest speed occurs in the first hour of the hike.

E. The graph is decreasing at all the times between 3.5 hours and 7 hours.

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A triangle is a three-sided polygon with three angles, each angle is formed by the intersection of two sides of the triangle and these angles are labeled with the corresponding vertex.

There are three angles in a triangle ABC. A triangle is a three-sided polygon and it has three angles. Each angle is formed by the intersection of two sides of the triangle.

In a triangle, the angle between two sides is measured in degrees and these angles are formed by the intersection of the two lines that represent the sides of the triangle. The three angles in a triangle are labeled with the corresponding vertex, for example, angle A, angle B, and angle C.

In a triangle, the sum of the three angles is always 180 degrees. This can be proved using the fact that a straight line is 180 degrees. If we draw a straight line, and divide it into two angles, which are supplementary angles. Therefore, these angles add up to 180 degrees. In a triangle, we can draw one straight line and divide it into three angles, which are supplementary angles, therefore the sum of the angles in a triangle is always 180 degrees.

Therefore, In short, a triangle is a three-sided polygon with three angles, each angle is formed by the intersection of two sides of the triangle and these angles are labeled with the corresponding vertex. The sum of the three angles in a triangle is always 180 degrees.

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To solve a complex fraction, which is a fraction that contains one or more fractions in its numerator and/or denominator, you can follow these steps:

Factor the numerator and denominator of the complex fraction if possible. This will make it easier to cancel common factors.

Multiply both the numerator and denominator by the reciprocal of the fraction(s) in the denominator. This will eliminate the fraction(s) in the denominator, leaving a simplified expression.

Simplify the resulting expression by combining like terms, if possible.

Check your work by multiplying the numerator and denominator of the original complex fraction by the reciprocal of the fraction(s) in the denominator, and verifying that the simplified expression you obtained is equivalent.

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Answer:

The length of the shorter leg of the triangle can be calculated using the Pythagorean Theorem. The formula is a^2 + b^2 = c^2, where a is the length of the shorter leg, b is the length of the longer leg and c is the length of the hypotenuse. In this case, a^2 + 52^2 = c^2. The hypotenuse can be calculated by using the sine formula. c = (52/sinA). Substituting this value into the Pythagorean Theorem equation, a^2 + 52^2 = (52/sinA)^2. Solving for a, a = sqrt(52^2 - (52/sinA)^2).

The angle 5 is an acute angle.

The correct answer is an option (a)

We know that the most common types of angles (based on their magnitude) are:

Acute Angle,

Obtuse Angle, (an angle which is less that 180° and greater than 90°)

and Right Angle (an angle that measures 90°)

There are some types of angle based on their rotation:    

Straight Angle, (an angle measuring 180 degrees)

Reflex Angle, (an angle which is greater than 180 degrees but less than 360 degrees )

and Full Rotation

As we know any angle measuring greater than 0° and less than 90° are called as the acute angles.

Here we have been given an angle 5 which is gretaer than 0 and less than 90

so, angle 5 is an acute angle.

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What type of angle is 5?

a) Acute Angles.

b) Obtuse Angles.

c) Right Angles.

d) Straight Angles

Answer:

9

Step-by-step explanation:

The pre-image is larger than the image because the scale factor is 1/3.  That means that the scale factor going from the image to the pre-image is 3.  

3 x 3 = 9

Answer:

the numbers are 13 and 17

Step-by-step explanation:

let the first number be a

let the second number be b

a+b=30.......a=30-b

a-b=4

substitute the first eq into the second one

30-b-b=4

30-2b=4

30-4=2b

26=2b

13=b

a=30-b

a=30-13

a=17

Answer: Yes, $9.

Step-by-step explanation:

From what I understand, since the table isn't showing: \

Dinner bill $25:$4.50 tip, $30:$5.40, etc..

To find if Bill is proportional, we need to find the rate of each tip to Bill  and see if they match.

$25/$4.50 =5.555555555...

$30/$5.40 =5.555555555...

$35/$6.30 =5.555555555...

$40/$7.20 =5.555555555...

The rate is the same for each one, so yes, the amount of the donation proportional to the dinner bill.

To find the donation for a bill of $50, divide by the rate we found above.

[tex]Tip=$50/5.555555555=9.000...[/tex]

No, a 2-digit number cannot start from Zero (0), as it can only have values ranging from 1 to 9.

When a number has two digits, it is called a 2-digit number. 2-digit numbers have two digits and range from 10 to 99. They are unable to begin with zero since it will then be regarded as a single-digit number. Any number between 1 and 9 can be used as the digit in the tens place.

The placement of each digit in a number is known as place value. There is just one place value that matters when discussing 1-digit numbers, and that is the ones place. There are two place values for 2-digit numbers: ones and tens. Two-digit numbers can be expressed verbally, numerically, or in their expanded form. 45 is a 2-digit number, for instance.

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Answer:

19r⁵ - r + 8

Step-by-step explanation:

The perimeter of a two-dimensional shape is the distance all the way around the outside.

Therefore, the perimeter of a triangle is the sum of the lengths of its sides.

[tex]\begin{aligned}\textsf{Perimeter}&=(9r^5-7r+8)+(r^5+4r-15)+(9r^5+2r+15)\\&=9r^5-7r+8+r^5+4r-15+9r^5+2r+15\\&=9r^5+9r^5+r^5+4r+2r-7r+8+15-15\\&=19r^5-r+8\end{aligned}[/tex]

Answer:

Perimeter of triangle = 19r⁵ - r + 8

Step-by-step explanation:

Perimeter of triangle formula,

→ A + B + C

→ Sum of all sides

Let's solve for the perimeter,

→ (9r⁵ - 7r + 8) + (9r⁵ + 2r + 15) + (r⁵ + 4r - 15)

→ (9r⁵ + 9r⁵ + r⁵) + (-7r + 2r + 4r) + (8 + 15 - 15)

→ (19r⁵) + (-r) + (8)

→ 19r⁵ - r + 8

Hence, perimeter is 19r⁵ - r + 8.

Answer:

Step-by-step explanation: 3×6+1,3×6 is 18+1 is 19/6 is your answer

We showed that EFGH is a square by proving that EFGH is a rhombus and all the angles are right angles. This is one of the classic proofs of the Pythagorean theorem.

(a) First, explain why the EFGH must be a rhombus.

In the figure, square ABCD has points E, F, G, and H on each side, so AE = BF = CG = DH, as shown. This means that all four sides of the rectangle EFGH are the same length. This is because they are all congruent with the side lengths of square ABCD. According to the definition of a rhombus, a rhombus is a quadrilateral with all four sides congruent. All four sides of the quadrilateral EFGH are congruent, so it must be a rhombus.

(b) Explain why all interior angles in EFGH must be right angles?

In square ABCD, by definition, all interior angles are right angles (90 degrees). Since the points E, F, G, and H lie on the sides of the square ABCD, the angles EFG and FGH are congruent to the right angle of the square ABCD. Similarly, the angles FGH and HGE are congruent to the right angle of the square ABCD.

Therefore, all four interior angles of quadrilateral EFGH are congruent with right angles of square ABCD, and all four interior angles of quadrilateral EFGH are right angles.

A square is a quadrilateral with all four sides congruent and all four angles congruent at right angles. And since it has already been proved that the quadrilateral EFGH is a rhombus and all its angles are right angles, we know that the quadrilateral EFGH is a square.

We showed that EFGH is a square by proving that EFGH is a rhombus and all the angles are right angles. This is one of the classic proofs of the Pythagorean theorem.

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Answer:

B) /AJB & /BJC are a pair of supplementary angles.

Step-by-step explanation:

The product of (X + A) + (X + B) is- [tex]x2 + xb + ax + ab[/tex]

The information provided in the question,

The outcome of multiplying two numbers together is known as the product. 12 is the result of the addition of 3 and 4, 20 is the sum of 4 and 5, and so on. Discovering the Product A multiplication problem has a solution in the product.

We must determine the of (x + a) and (x + b).

The product is the result of multiplying two numbers, as we all know.

So,

[tex]( x+ a)[/tex]   x  [tex]( x + b)[/tex]

[tex]x( x+b) + a( x + b)[/tex]

[tex]x[/tex]²[tex]+ bx + ax + ab[/tex]

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Here you go mate!

7x + 10 = -11

answer : x = -3

-12(x + 5) = -144

Answer : x = 7

-9x - 9 = 27

Answer : x = -4

x/4 + 2 = 10

Answer : x = 32

The events from "Let George Do It!" in the right sequence will be:

From 1946 to 1954, Owen and Pauline Vinson produced the radio drama series Let George Do It in the United States.

In 1954, Olan Soule voiced the role of private eye George Valentine, which was portrayed by Bob Bailey. The scripts by David Victor and Jackson Gillis were directed by Don Clark.

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The amount left in the container by Alice is solved to be 3/8 tsp

To find the amount left in the container the amount used to feed the fish in each case will factored and then used for subsequent feeding. of the fish.

From the problem it can be deduced that the initial amount in the container was 2 tsp

The routine feeding of the fish include:

Alice fed her fish 1/4 tsp, of food before school = 1/4 * 2 = 1/2

The 2 tsp has reduced to 1/2, this is further used as shown below

Alice fed her fish 3/4  tsp, of food after school = 3/4 * 1/2 = 3/8 tsp

We can therefore say that the amount left in the container is 3/8 tsp

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For given functions for zombies and vampires,  zombies will reach 50,000 first.

What is a function?

A function is defined as a relation between a set of inputs having one output each. In simple words, a function is a relationship between inputs where each input is related to exactly one output. Every function has a domain and co-domain or range. A function is generally denoted by f(x) where x is the input. The general representation of a function is y = f(x).

There are several types of functions in math. Some important types are:

Injective function or One to one function: When there is mapping for a range for each domain between two sets.

Surjective functions or Onto function: When there is more than one element mapped from domain to range.

Polynomial function: The function which consists of polynomials.

Inverse Functions: The function which can invert another function.

Now,

Initial numbers, 100 Zombies and 100 Vampires.

In function z(t), the zombies increase in population by 1000 per day,

and in function V(t), the vampire population increases by 25% each day,

Therefore, z(t)=100+1000t where t=no, of days and

v(t)=100+25t where t= no. of days

for z(t)=50000

100+1000t=50000

t=49900/1000=49.9≈50days

for v(t)=50000

100+25t=50000

t=49900/25=1996 days

Hence,

          zombies will reach 50,000 first.

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For given functions for zombies and vampires,  zombies will reach 50,000 first.

What is a function?

A function is defined as a relation between a set of inputs having one output each. In simple words, a function is a relationship between inputs where each input is related to exactly one output. Every function has a domain and co-domain or range. A function is generally denoted by f(x) where x is the input. The general representation of a function is y = f(x).

There are several types of functions in math. Some important types are:

Injective function or One to one function: When there is mapping for a range for each domain between two sets.

Surjective functions or Onto function: When there is more than one element mapped from domain to range.

Polynomial function: The function which consists of polynomials.

Inverse Functions: The function which can invert another function.

Now,

Initial numbers, 100 Zombies and 100 Vampires.

In function z(t), the zombies increase in population by 1000 per day,

and in function V(t), the vampire population increases by 25% each day,

Therefore, z(t)=100+1000t where t=no, of days and

v(t)=100+25t where t= no. of days

for z(t)=50000

100+1000t=50000

t=49900/1000=49.9≈50days

for v(t)=50000

100+25t=50000

t=49900/25=1996 days

Hence,

         zombies will reach 50,000 first.

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