how to find the GCF and re write a sum using GCF and the distributive property?

Answer:

Step-by-step explanation:

You want to know the consecutive integers that bound the values of 30^(1/5) and 45^(1/3).

The fifth root of 30 will lie between the two integers whose 5th powers lie on either side of 30:

  1^5 = 1 < 30

  2^5 = 32 > 30

So, the fifth root of 30 lies between 1 and 2:

  [tex]\boxed{1 < \sqrt[5]{30} < 2}[/tex]

The cube root of 45 will lie between the two integers whose cubes lie on either side of 45:

  3^3 = 27 < 45

  4^3 = 64 > 45

So, the cube root of 45 lies between 3 and 4:

  [tex]\boxed{3 < \sqrt[3]{45} < 4}[/tex]

__

Additional comment

A calculator confirms this:

  [tex]\sqrt[5]{30}\approx 1.974\\\sqrt[3]{45}\approx3.557[/tex]

The method used to find the prime factors of a number is called prime factorization. One common method to find the prime factorization of a number is called trial division.

Here is how the trial division method works:

Divide the number by the smallest prime number (2). If the division is exact, then 2 is a prime factor of the number.

If 2 is a factor, divide the number by 2 again, and again, until the division is no longer exact. Then divide the resulting quotient by the next smallest prime number (3), and continue dividing by the smallest prime numbers (5, 7, 11, etc.) until the quotient is prime.

Repeat this process until the quotient is 1. The prime factors of the original number will be the prime numbers that were used in the divisions.

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Yes, Zero is a real answer because zero is a rational number, and any rational number is also a real number. In particular, the set of real numbers is the union of the set of irrational numbers with the set of rational numbers.

Zero is an even number, but it is not a prime as well as a composite number. Also, zero is neither a positive nor a negative number, but it completes the gap between them. Zero factorial is 1 (0! = 1).

Zero is rational, real, and an integer, but it is not a natural number. Any number divided by zero is undefined, as is zero divided by zero. Of course, zero did not always exist.

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Answer: Question 1 - 1/2
Question 2 - 1/4

Step-by-step explanation:

Hello! The answer to the first question is 1/2 because the quarter can have either two possible outcomes.  The quarter could land on heads or tails, and the chance of landing on heads is 1/2 and the chance of landing on tails is 1/2.

For the second question, the answer is 1/4 because the probability of a nickel landing on heads is 1/2 and the probability of a penny landing on heads is also 1/2.  To find the probability of both landing on heads.  You have to multiply both probabilities together.  1/2 x 1/2 = 1/4, which is your answer for the second question.

Hope this helps!

The ratio of the two legs of the triangle 30-90-60 will be 1 : √3.

The branch of mathematics that sets up a relationship between the sides and the angles of the right-angle triangle is termed trigonometry.

The trigonometric functions, also known as a circular, angle, or goniometric functions in mathematics, are real functions that link the angle of a right-angled triangle to the ratios of its two side lengths.

The triangle is 30-90-60. The ratio of the two legs will be calculated as,

tan(30) = AB / BC

1 / √3 = AB / BC

Hence, the ratio of the two legs will be 1 : √3.

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Jane travels 8 miles much farther than her trainer. The solution is obtained using arithmetic operations.

What are arithmetic operations?

By combining operands with one arithmetic operator, an arithmetic operation is given. The built-in functions add, subtract, divide, and multiply additionally allow for the specification of arithmetic operations.

3. Using pythagoras theorem, we get the equation as D² = 12² + 16².

⇒D² = 144 + 256

⇒D² = 400

⇒D = √400

⇒D = 20

4. Jane has traveled 28 miles in total whereas her trainer has traveled 20 miles.

So, the difference is 28-20= 8 miles

Hence, Jane travels 8 miles much farther than her trainer.

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The additive inverse of 19/-7 is 19/7.

The additive inverse of a number is the negative of that number. If a number is added to its additive inverse, the sum of both numbers becomes zero. In other words, the additive inverse is the number that is added to a given number to make the sum zero. That is what you add to a number to get zero is called its additive inverse.

That is, Number + Additive inverse = 0

OR

The negative of a number.

Now we are given the number

19/-7

So, the additive inverse of 19/-7:

⇒19/-7 + Additive inverse = 0

⇒Additive inverse = 19/7

Thus the additive inverse of is 19/-7 is 19/7.

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A Perfect Number N is defined as any positive integer where the sum of its divisors minus the number itself equals the number.

A perfect number is a positive integer that is equal to the sum of its proper divisors, which are positive integers that divide the number without leaving a remainder. For example, 6 is a perfect number because its proper divisors are 1, 2, and 3, and 1 + 2 + 3 = 6. The first four perfect numbers are 6, 28, 496, and 8128.

Perfect numbers have been studied by mathematicians for centuries, and continue to be an area of active research. They have connections to other areas of mathematics, such as number theory and algebraic number theory. Perfect numbers also have applications in computer science and cryptography.

It is also important to note that in number theory, a natural number is called abundant if it is less than the sum of its divisors and deficient if it is greater than the sum of its divisors. Perfect numbers are exactly equal to the sum of their divisors.

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Answer:

Tewanda reads 162 words per minute

Step-by-step explanation:

To find the unit rate of how fast Tewanda reads:

1458/9 = 162

Tewanda reads 162 words per minute

Together, fixed and variable costs make up the overall cost.When n is 4, the price is $33.00.

Together, fixed and variable costs make up the overall cost.The total cost, for instance, would be $35 if a company had a fixed cost per unit of $30 and a variable cost per unit of $5 that increased as output increased.The formula for average total cost is created by dividing the sum of all production costs by the sum of all payments in money.

The total-cost formula is a simple and user-friendly process that works by dividing the total production cost by the quantity of goods produced.

Y River

c = 33

Z River

Where n is the number of hours, c is equal to 5n + 13.

Graph

Include nothing before to n = 0

The cost will be the same when both sides of the equation are equal.

Where the two lines meet is the answer.

Both of them come together at a single place.

The point is this (4,33)

When n is 4, the price is $33.00.

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The values of a, b and c are given as follows:

a = 5, b = 3, c = 3.

The exponential expression for this problem is defined as follows:

[tex]\left(\frac{4}{3}\right)^{\frac{1}{2}} + \left(\frac{1}{3}\right)^{-\frac{1}{2}}[/tex]

The negative exponent means that the numerator and the denominator are inverted, with the positive exponent, then:

[tex]\left(\frac{4}{3}\right)^{\frac{1}{2}} + \left(3)^{\frac{1}{2}}[/tex]

The fractional exponent of 1/2 can be transformed into a square root, hence:

[tex]\frac{\sqrt{4}}{\sqrt{3}} + \frac{\sqrt{3}}{1}[/tex]

[tex]\frac{2}{\sqrt{3}} + \sqrt{3}[/tex]

[tex]\frac{2 + (\sqrt{3})^2}{\sqrt{3}}[/tex]

Then:

[tex]\frac{5}{\sqrt{3}}[/tex]

We rationalize the expression, thus:

[tex]\frac{5}{\sqrt{3}} \times \frac{\sqrt{3}}{\sqrt{3}} = \frac{5\sqrt{3}}{3}[/tex]

Meaning that the values are given as follows:

a = 5, b = 3, c = 3.

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The  60.42 percentage of 8th graders believe they spend less than an hour per day on social media.

A percentage is a ratio expressed as a fraction of 100.

A random sample of students was asked in a survey if they estimated they spent more or less than an hour per day on social media.

To calculate the percentage of 8th graders who say they spend more than an hour a day on social media, divide the number of students who said "less" by the total number of 8th grade students surveyed, then multiply by 100.

29 students out of 48 students responded "less,".

To find the percentage,

(obtained value / total score ) x 100%

So here,

percentage = (29/48) x 100

accounting for 60.42% of the total.

As a result, 60.42% of 8th graders believe they spend less than an hour a day on social media.

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Answer:

h= 13.6 ft

Step-by-step explanation:

Let,

a= 8 ft

b= 11 ft

The ladder against the top of the house is creating a right triangle

So, we can find the length of the ladder using

[tex]h^{2} = {a}^{2} + {b}^{2} \\ h^{2} = {8}^{2} + {11}^{2} \\ h^{2} = 64 + 121 \\ h = \sqrt{185} \\ = 13.6ft[/tex]

Therefore, the length of the ladder (h) = 13.6 ft ( rounded to the nearest tenth)

Answer:

31.5

Step-by-step explanation:

At the end of the week, the store has 212 tuffed animals.

To determine the number of tuffed animals in the store at the end of the week, we need to subtract the number of tuffed animals sold during the week from the number of tuffed animals in the store at the beginning of the week.

349 tuffed animals (beginning of the week) - 137 tuffed animals (sold during the week) = 212 tuffed animals (at the end of the week).

So at the end of the week, the store has 212 tuffed animals.

The concept used to solve this problem is subtraction. Specifically, we are using the inverse operation of addition. By subtracting the number of tuffed animals sold during the week from the number of tuffed animals in the store at the beginning of the week, we are able to find the number of tuffed animals in the store at the end of the week. The subtraction operation is used to find the difference between two quantities, which in this case is the number of tuffed animals remaining in the store.

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2(3b+6) =78

Expanding the left side of the equation:

6b + 12 = 78

Solving for b:

6b = 66

b = 11

Now we will check the left side and the right side of the equation:

2(3(11) + 6) = 2(33+6) = 2(39) = 78

On the other side of the equation:

-2(6c-3) = -72

Expanding the left side of the equation:

-12c + 6 = -72

Solving for c:

-12c = -78

c = 6.5

Now we will check the left side and the right side of the equation:

-2(6(6.5) - 3) = -2(39 -3) = -2(36) = -72

This equation is true, the left side of the equation equals to the right side of the equation.

The solution for the first equation is b = 11

The solution for the second equation is c = 6.5

The total emails received are 546. The solution is obtained using arithmetic operations.

What are arithmetic operations?

For all the real numbers, there are four basic mathematical operations which are:

1. Addition(‘+’) wherein the sum of the numbers is obtained.

2. Subtraction(‘-’) wherein the difference of the numbers is obtained.

3. Multiplication(‘×’) wherein the product of the numbers is obtained.

4. Division(‘÷’) wherein the quotient of the numbers is obtained.

Let the total emails received be 'x'.

We are given that for every 20 total emails a person received, they could expect to get 11 spam emails.

Now, a person received 300 spam emails in one month

So, the total emails received will be

⇒x = 20*300/11

⇒x = 545.45

⇒x = 546 (approx)

Hence, the total emails received are 546.

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The equation of the quadratic function y = x² + 6x + 3 in vertex form is y =  (x+ 3)² - 6

Quadratic function in math is defined as a polynomial function with one or more variables in which the highest exponent of the variable is two.

Here we have given the quadratic function y = x² + 6x + 3 and we need to find the vertex form of the given quadratic function.

As we all know that the vertex form of the equation is given by

=> f(x) = a(x - h)² + k

Here we have given the expression that can be written as,

=> y = x² + 6x + 3

Now, we have to convert it into vertex form, then we get,

=> x² + 6x + 3

Expand the expression as follows:

=> x² + 6x + 3 + 9 - 9

Now, we have rewritten the expression like the following,

=> y + 9 = (x + 3)² + 3

Here we have to subtract 9 on both sides, then we get

=> y =  (x+ 3)² + 3 - 9

Then the resulting equation is written as,

=> y =  (x+ 3)² - 6

Complete Question:

What is the equation of the quadratic function y = x² + 6x + 3 in vertex form?

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The function for the area is:

A = 0.5H² + 2H

And if the height is 6ft, then the area is 30ft²

We know that the area of a triangle is the product between the base and the height over two.

Here the base is 4ft more than the height H, then:

A = B*H/2

A = (H + 4)*H/2

A = 0.5H² + 2H

If the height is 6ft, then the area is:

A = 0.5*(6ft)² + 2*6ft = 30ft²

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At least play one of the three sports find the number of those 25 numbers play both Volleyball and Rugby.

What is meant by sport?

Since they all play at least one sport, the number of people who don't play football is 30-17 = 13.

The "no football" category is divided in 3 categories :

only volleyball : 9

only rugby : 12

both = x

So we have :

9 - 3 - x = 13

12 - x  = 13

x = 13 + 12 = 25

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Answer:

see explanation

Step-by-step explanation:

4(x - 1)² - 9(x - 1) =-  2

let t = x - 1 , then rewrite the expression as

4t² - 9t = - 2 ( add 2 to both sides )

4t² - 9t + 2 = 0 ← in standard form

Step 2

consider the factors of the product of the coefficient of the t² term and the constant term which sum to give the coefficient of the t- term

product = 4 ×  2 = 8 and sum = - 9

the factors are - 8 and - 1

use these factors to split the t- term

4t² - 8t - t + 2 = 0 ( factor first/second and third/fourth terms )

4t(t - 2) - 1(t - 2) = 0 ← factor out (t - 2) from each term

(t - 2)(4t - 1) = 0 ← in factored form

equate each factor to zero and solve for t

4t - 1 = 0 ⇒ 4t = 1 ⇒ t = [tex]\frac{1}{4}[/tex]

t - 2 = 0 ⇒ t = 2

Step 3

Now solve for x using

t = x - 1 ( add 1 to both sides ), then

x = t + 1

t = [tex]\frac{1}{4}[/tex] , then x = [tex]\frac{1}{4}[/tex] + 1 = [tex]\frac{5}{4}[/tex]

x = 2 , then x = 2 + 1 = 3

There is no real number that, when multiplied by itself, produces a negative result. Therefore, the square root of a negative number is undefined in the context of real numbers.

A square root is defined as the value that, when multiplied by itself, equals a given number. The square of a real number is always positive, so the square root of a positive number is always positive or zero. However, there is no real number that, when multiplied by itself, produces a negative result. Therefore, the square root of a negative number is undefined in the context of real numbers.

Square root of a negative number is a complex number.

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The best measure of center for the data is the mode only.

a central tendency is a central or typical value for a probability distribution. Colloquially, measures of central tendency are often called averages. The term central tendency dates from the late 1920s. The most common measures of central tendency are the arithmetic mean, the median, and the mode.

In this particular data, it is formed of letter with no definite values, therefore to find the mean value would be and the median would be impossible but the mode can be found.

Firstly we re arrange the letters in alphabetical order e, a, u, i, e, o, o, e, u, a, i, e, o, u, a = a,a,a,e,e,e,e,i,i,o,o,o,u,u,u.

Then the letter that appeared most can be picked out.

Only e appear 4 times. This means that e is the mode of the data.

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Yes, there can be a triangle with three angles equal to 60.

The triangle is called the equilateral triangle where all three angles of a triangle is equal to 60 degree.

As we all knows that the sum of all the angles of a triangle is 180 and there are three angles that present in a triangle.

So, 60+60+60 = 180.

Now, Some properties of equilateral triangle:

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Since a square is either positive or zero, negative numbers don't have real square roots.

No, we are unable to compute the negative number's square root. Positive is always the result of adding two numbers with the same sign.

Theorem: Square Roots of a Negative Number Simplifying an expression involving the square root of a negative number by simplifying the number as if it were a positive number and putting an i in front of it as a coefficient is one method. To put it another way, x=ix x = i x for every x where x is a positive number and where x is a negative number.

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A parallelogram is formed when one pair of opposite sides of a quadrilateral are equal and parallel that is option B and D.

A parallelogram is a simple quadrilateral with two pairs of parallel sides in Euclidean geometry. A parallelogram's opposite or facing sides are of equal length, and its opposite angles are of equal measure. A parallelogram is a type of quadrilateral with opposite pairs of sides that are parallel and equal. The diagram depicts a parallelogram ABCD with AB II CD and AD II BC. In addition, AD = BC and AB = CD. Non-Example: A trapezium is a parallelogram's non-example.

Here,

When one pair of opposite sides of a quadrilateral are equal and parallel, a parallelogram is formed that is option B and D.

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The Tangent Law may be used to calculate the height of the mountain's summit. The ratio of the opposing side to the adjacent side is defined as the tangent. Thus, the mountain's top is 3.44 kilometres high.

Tangent law is a mathematical law that states that the ratio of the tangent of an angle to the tangent of its complementary angle is equal to the ratio of the sine of the angle to the sine of its complementary angle. It can be written mathematically as:

tan(θ) / tan(90-θ) = sin(θ) / sin(90-θ)

The opposing side in this example is the height of the peak, while the adjacent side is the distance between the hotel and the summit (6.5 km).

Opposite/Adjacent = tan is the equation for the tangent law (angle of elevation). We may rewrite this equation to find the height of the mountain's peak: Adjacent = tan(angle of elevation) * Opposite.

We may calculate the height of the mountain's peak by plugging in our values: Opposite = tan(19 degrees) * 6.5 km = 3.44 km.

As a result, the mountain's top is 3.44 kilometres high.

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