The number which is repeated as a factor in an exponential expression exponent 2. the fixed amount multiplied by a term to get to the next term in a geometric sequence radical expression 3. the (superscript) number in an exponential expression which tells how many times a factor is repeated index 4. a sequence in which there is a common ratio between the terms geometric sequence 5. the small number preceding a radical symbol that indicates the desired root of the radicand rationalize 6. expressions having the same root index and the same radicand like radicals 7. an expression that contains a radical; √7 and √21y are examples of radical expressions term 8. the expression under a radical symbol common ratio 9. eliminate the radical from the denominator of a fraction radicand 10. a set of numbers that follows a pattern, with a specific first number sequence 11. an individual quantity or number in a sequence base

Answer:

the first function, y = (x + 1)^2, does pass through the points (1, 4), (2, 9), and (3, 16).

Step-by-step explanation:

Note that 4, 9 and 16 are squares of 2, 3 and 4.  So it's likely that the parent function is of the form f(x) = x^2.  Note that f(1) = 1, f(2) = 4, f(3) = 9 and f(4) = 16.

Let's now determine whether these three points satisfy the first given equation, y = (x + 1)^2:  Does 4 = (1 + 1)^2?  Does 4 = 2^2?  YES.

Does 9 = (2 + 1)^2?  Does 9 = 3^2?  YES

Does 16 = (3 + 1)^2?  YES

So the first function, y = (x + 1)^2, does pass through the points (1, 4), (2, 9), and (3, 16).

Final answer:

The detailed answers cover questions related to functions, sequences, equations of lines, slopes, and graphing lines.

Explanation:

Question 1:

The function that passes through the points (1, 4), (2, 9), and (3, 16) is y = (x + 1)^2.

Question 2:

The equation representing the total number of laps swum y in terms of hours x is y(x) = 10x + 1.

Question 3:

The sequence {4, 2, 0, -2, -4, -6, …} is an arithmetic sequence with a common difference of -2.

Question 5:

The slope of the line containing (3, 5) and (2, 7) is m = -2.

Question 6:

For a line passing through (-3, 2) with a slope of 4, the equation of the line is y = 4x + 14.

The approximate probability that the person will win at least $15 is 0.255.

To solve this problem, we need to calculate the probability of winning at least $15 after playing the game 30 times. Winning at least $15 means that the player must win on at least 6 rolls because winning once pays $5, and $5 times 6 is $30, which is the minimum needed to have a net gain of $15 ($30 won - $15 lost).

First, let's calculate the probability of winning in a single roll. The player wins if they roll a 3 or a 5, which are two favorable outcomes out of six possible outcomes.

[tex]\[ P(\text{win on a single roll}) = \frac{2}{6} = \frac{1}{3} \][/tex]

The probability of losing in a single roll is the complement of the probability of winning:

[tex][ P(\text{lose on a single roll}) = 1 - P(\text{win on a single roll}) = 1 - \frac{1}{3} = \frac{2}{3} \][/tex]

Now, we need to find the probability of winning at least 6 times in 30 rolls. This can be modeled using the binomial probability formula:

[tex]\[ P(X = k) = \binom{n}{k} p^k (1-p)^{n-k} \][/tex]

where [tex]\( n \)[/tex]  is the number of trials (30 in this case), [tex]\( k \)[/tex] is the number of successes (at least 6), [tex]\( p \)[/tex] is the probability of success on a single trial [tex](\( \frac{1}{3} \)), and \( \binom{n}{k} \)[/tex]  is the binomial coefficient.

We want to calculate the probability of winning at least 6 times, which is the sum of the probabilities of winning exactly 6, 7, .. up to 30 times:

[tex]\[ P(X \geq 6) = \sum_{k=6}^{30} \binom{30}{k} \left(\frac{1}{3}\right)^k \left(\frac{2}{3}\right)^{30-k} \][/tex]

[tex]\[ P(X \geq 6) \approx 0.255 \][/tex]

Answer:

3) [tex]a_n=-2a_{n-1}[/tex], [tex]a_1=5[/tex]

Step-by-step explanation:

Given that the explicit rule for a sequence is [tex]a_n=5(-2)^{n-1}[/tex].

Now we need to find about what is the recursive rule for the sequence and match with the given choices to find the correct choice.

1) [tex]a_n=-2a_{n+1}[/tex], [tex]a_1=5[/tex]

2) [tex]a_n=-5a_{n+1}[/tex], [tex]a_1=2[/tex]

3) [tex]a_n=-2a_{n-1}[/tex], [tex]a_1=5[/tex]

4) [tex]a_n=-5a_{n-1}[/tex], [tex]a_1=2[/tex]

Plug n=1 into given formula to get first term

[tex]a_n=5(-2)^{n-1}[/tex]

[tex]a_1=5(-2)^{1-1}=5(-2)^{0}=5(1)=5[/tex]

base of the exponent part is (-2) so that means we need to multiply -2 to the previous term to get nth term

Hence correct choice is: 3) [tex]a_n=-2a_{n-1}[/tex], [tex]a_1=5[/tex]

Answer:

3) an=−2(an−1)

a1=5

Step-by-step explanation:

Answer:

50.24

Step-by-step explanation:

area of a circle =pi r^2

diameter = 2 * radius

radius=8/2=4

area=pi 4^2

=pi 16

=50.26

Answer:

Step-by-step explanation:

- 5 + (-3) + 3 is the correct option.

Answer:

-5 + (-3) + 3

Step-by-step explanation:

Answer:

  x = 1

Step-by-step explanation:

The base is 3. The value of 3^x will be 3 only when x=1.

_____

The value of any exponential term will be equal to the base when the exponent is 1.

To determine how many plants Ms. Holmes had, divide the total amount of water she had by the amount of water she gave each plant. The expression that can be used is (2 1/2 gallons) divided by (1/4 gallon per plant). Simplifying the expression gives the answer of 10 plants.

To determine how many plants Ms. Holmes had, we can divide the total amount of water she had (2 1/2 gallons) by the amount of water she gave each plant (1/4 gallon per plant).

So, the expression that can be used is (2 ½ gallons) ÷ (1/4 gallon per plant). To divide a fraction by a fraction, we multiply the first fraction by the reciprocal of the second fraction. Therefore, the expression simplifies to (2 ½ gallons) × (4 gallons per plant).

Simplifying further, we get 10 gallons. This means that Ms. Holmes had 10 plants.

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Answer:

4.4, 4.4 : 1, 22 : 5

Step-by-step explanation:

There are many ways to solve this, but I'll do it by finding the ratio between V'T' and VT.

That's easy, we're given the lengths of these two. The ratio would be 8.8/2. 8.8/2 = 4.4, or (8.8*5)/(2*5) = 22/5

put simply, the volume of a rectangular prism is just the product of all its 3 dimensions.

[tex]\bf V=\cfrac{2}{3}\cdot \cfrac{1}{3}\cdot \cfrac{2}{3}\implies V=\cfrac{4}{27}[/tex]

Answer:

Thehe answer is 75

Step-by-step explanation:

Answer with Step-by-step explanation:

The length of a rectangle is 3 meters more than the width.

If the width is b.

⇒ Length=b+3

The perimeter of the rectangle is 26 meters.

Since, Perimeter=2(Length+width)

⇒ 2(b+3+b)=26

dividing both sides by 2

⇒ b+3+b=13

⇒ 2b=13-3

⇒ 2b=10

dividing both sides by 2

⇒ b=5

⇒ Width=5 meter

and Length=5+3=8 meter

You didn't post any option, but the ration roots theorem states that all possible rational roots of a polynomial come in the form

[tex]\pm\dfrac{p}{q}[/tex]

where p divides the constant term and q divides the leading term of the polynomial. So, in your case, p divides 3 (i.e. it is 1 or 3), and q divides 5 (i.e. it is 1 or 5).

So, the possible roots are

[tex]\pm 1,\quad \pm 3,\quad \pm\dfrac{1}{5},\quad \pm\dfrac{3}{5}[/tex]

For the record, this parabola has no real roots.

The possible rational roots of the polynomial function f(x) = 5x² - 3x + 3 are;

±1/5, ±3/5, ±1, ±3

      We are given the polynomial function;

f(x) = 5x² - 3x + 3

       The rational root theorem states that for a polynomial to have any rational roots, then the roots must be of the form;

±(Factors of coefficient of constant term/factors of the coefficient of the highest power)

       Now, the coefficient of the highest power is 5 and the constant term is 3.

Factors of 5 = 1, 5

Factors of 3 = 1, 3

Thus, the possible rational roots are;

±1/5, ±3/5, ±1, ±3

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f(x) = 2(x+8)

replace f(x) with 12 and then solve for x:

12 = 2(x +8)

Use the distributive Property on the right side:

12 = 2x +16

Subtract 16 from each side:

-4 - 2x

Divide both sides by 2:

x = -4 / 2

x = -2

Answer:

−2

Step-by-step explanation:

f(x) = 2(x+8)

f(x) = 12

12 = 2(x+8)

Divide each side by 2

12/2 = 2/2(x+8)

6 = x+8

Subtract 8 from each side

6-8 =x+8-8

-2 = x

Answer:

A. Lucy Only

Step-by-step explanation:

Got it right on Khan Academy

The possible sample space is: FV, WV, SV, FB, WB, and SB

Probability is simply how likely something is to happen. Whenever we're unsure about the outcome of an event, we can talk about the probabilities of certain outcomes

As, the sample space with all possibilities in which lucy and katy register a sports League.

Now, they have 2 sports and 3 seasons for the Sample space.

Mow let us consider, V is registered for Volleyball and B is registered for basketball .

and, F is registered Fall, W is registered for Winter and S is for Spring.

so, that the sample space is:

FV, WV, SV, FB, WB, and SB

where, FV is the  registered possibility for fall and in volleyball.

WV is the  registered possibility for Winter and in volleyball.

SV is the  registered possibility for Spring and in volleyball.

and so on..

Therefore, the correctly representation for the sample space is the one that has all these possibilities.

Learn more about probability here:

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Answer:

1. no like terms

2. x^ 3 − x^ 2 − x + 1

3.x y + x + y + 1

4.x^ 3 + x^ 2 y + x y^ 2 + y^ 3

Lol

Answer:

[tex]10x^{2}-34[/tex]

Step-by-step explanation:

Because we are told equivalent expressions for y and z we can plug those in to 2(y+z).

[tex]2((3x^{2}+x^{2}-5)+(x^{2}-12))[/tex]

Then simplify by combining like terms of the expressions. Values ending in x^2 can be combined with each other.

[tex]2(5x^{2}-17)[/tex]

Now we can distribute the 2 by multiplying each value in the parentheses by 2.

[tex]10x^{2}-34[/tex]

Final answer:

62.5% of those surveyed prefer hamburgers.

Explanation:

The question is asking to find the percentage of people who prefer hamburgers over hot dogs when given the ratio of 5 to 3.

To calculate this, you would add the two parts of the ratio together to get the total number of parts, which is 5 + 3 = 8.

The number of parts that represent hamburgers is 5.

To find the percentage, you would divide the hamburger part by the total number of parts and then multiply by 100.

This gives you (5/8) × 100, which equals 62.5%.

Therefore, 62.5% of those surveyed prefer hamburgers.

Answer:

Height ( h) = 12 units.

Step-by-step explanation:

Given  : An equilateral triangle with side = 8 √3.

To find : What us the length of the altitude of the equilateral triangle .

Solution : We have given equilateral triangle with side = 8 √3.

By taking the half triangle,

By the Pythagorean theorem :

(Hypotenuse)² = ( adjacent)² + (opposite)².

Plug the values Hypotenuse = 8√3 , adjacent = 4√3 , opposite = h.

(8√3)² = ( 4√3)² + (h)².

192 = 48 +  (h)².

On subtracting by 48 both sides.

192 -48 =  (h)².

144 =  (h)².

On taking square root .

h = 12 .

Therefore, Height ( h) = 12 units.

The guy is right trust me mate

Answer:

23.78

Step-by-step explanation:

CAH

x/25

cos18=x/25

25cos18=x

x=23.78

Answer:

23.78 cm

Step-by-step explanation:

Since, We know that,

In a right angle triangle,

[tex]cos \theta=\frac{B}{H}[/tex]

Where, B represents base adjacent to [tex]\theta[/tex]

H represents the hypotenuse adjacent to [tex]\theta[/tex]

Thus, by the given diagram,

[tex]cos 18^{\circ}=\frac{x}{25}[/tex]

[tex]\implies x = 25\times cos 18^{\circ}=23.7764129074\approx 23.78\text{ cm}[/tex]

Hence, Last option is correct.

Answer:

16 Boxes

Step-by-step explanation:

Total Bottles:125

Amount of Bottles A Box Is Able To Hold:8

125/8=15.625

Round Up To Nearest Whole Number

16 Boxes Are Needed To Hold 125 Bottles

Answer:

D

Step-by-step explanation:

Let the following point represent the first job:  (4, $144).  Let (6, $188) represent the second.  Then we find the equation of the line through these two points.  As we go from (4, $144) to (6, $188), x increases by 2 and y increases by $44.  Thus, the slope of this line is m = rise/run = $44/(2 hrs), or $22/hr.

Using the slope-intercept formula for a  straight line, y = mx + b, we find the equation of this line by finding the y-intercept, b:

Subbing the given info ($22/hr for m, 4 hr for x and $144 for y), we get:

$144 = ($22/hr)(4 hr) + b, or

$144 = $88 + b, which yields b = $56.

Thus, the desired equation is y = mx + b, or y = ($22/hr)x + $56.  This matches the last answer (D)

Answer:

•0.5

Step-by-step explanation:

We'll assume the dilation was done at the origin point, so it was done evenly.

Since the k point went from (10,0) to (5,0), we know it was reduced... so the scale factor has to be below 1. A scale factor means the size wasn't changed and a dilation/scale factor larger than 1 means it was enlarged.

So, we take the new X-value (5) and divide it by the original X-value (10).

Scale factor = 5/10 = 1/2... so 0.5

ANSWER

[tex](4n - 1)\sqrt{3 n} + 3\sqrt{n}[/tex]

EXPLANATION

The given expression is

[tex] \sqrt{48 {n}^{3} } + \sqrt{9n} - \sqrt{3n} [/tex]

We remove the perfect squares under the radical sign.

[tex]\sqrt{16 {n}^{2} \times3 n} + \sqrt{9n} - \sqrt{3n} [/tex]

We can now take square root of the perfect squares and simplify them further.

[tex] \sqrt{16 {n}^{2}} \times \sqrt{3 n} + \sqrt{9} \times \sqrt{n} - \sqrt{3n} [/tex]

This simplifies to:

[tex]4n\sqrt{3 n} + 3\sqrt{n} - \sqrt{3n} [/tex]

This further simplifies to:

[tex](4n - 1)\sqrt{3 n} + 3\sqrt{n} [/tex]

Answer:

Option B is Correct

Step-by-step explanation:

[tex]\sqrt{48n^3}+\sqrt{9n} - \sqrt{3n}[/tex]

We need to solve the above expression.

48 can be written as: 2x2x2x2x3

9 can be written as : 3x3

Putting values

[tex]\sqrt{2*2*2*2*3*n*n*n} +\sqrt{3*3*n}-\sqrt{3n}[/tex]

2*2 = 2^2 and n*n = n^2 and 3*3 = 3^2

we also know √ = 1/2

so, putting these values we get,

[tex]\sqrt{2^2*2^2*3*n^2*n} +\sqrt{3^2*n}-\sqrt{3n}\\(2^2)^{1/2} * (2^2)^{1/2} * (3)^{1/2} * (n^2)^{1/2} * n ^{1/2} + ((3^2)^{1/2} *n^{1/2}) -(\sqrt{3n})\\2*2*n * (3^{1/2} *n ^{1/2}) +(3 +n^{1/2}) -(\sqrt{3n})\\4n (\sqrt{3n})+(3 \sqrt{n}) -(\sqrt{3n})\\Rearraninging\\4n(\sqrt{3n}) - (\sqrt{3n})+(3 \sqrt{n})\\Taking \,\,\sqrt{3n} \,\,common\,\, from\,\, 1st\,\, and\,\, 2nd\,\, term\\\sqrt{3n}(4n-1)+(3 \sqrt{n})\\or \,\,it\,\,can\,\,be\,\,written\,\,as\,\,\\(4n-1)\sqrt{3n}+(3 \sqrt{n})[/tex]

So, Option B is Correct.

Answer:

Option A

53.5

Step-by-step explanation:

Answer:

  [tex]\boxed{18}[/tex]

Step-by-step explanation:

One of the rules of logarithms is ...

  log(a^b) = b·log(a)

So ...

[tex]6\log_5{x^3}=6(3\log_5{x})=\bf{18}\log_5{x}[/tex]

Answer:

8 years

Step-by-step explanation:

Answer:

she will be at the company for 8 years.

Step-by-step explanation:

The number of days of vacation for each year = 3

Maryanne is employed up to a maximum of 24 days of vacation time.

We need to find the number of years she will be in the company before she reaches the maximum amount of vacation of time.

The number of years she will be in the company = The number of maximum days of vacation time divided by the number of days of vacation for each year

= [tex]\frac{24}{3}[/tex]

= 8 years

Therefore, she will be at the company for 8 years.

Answer: the coefficient of 13m is 13.

Answer:

The coefficient of 13 m is 13.

Step-by-step explanation:

Consider the provided expression.

[tex]13m[/tex]

Numbers appear with variables are called coefficients.

Alphabets in an expression are called variables.

Numbers appear without variables are called Constant terms.

Here we need to identity the coefficient.

From the above definition we know the number appear with the variables are coefficients.

Since m is variable, thus the number 13 must be the coefficient.

Hence, the coefficient of 13 m is 13.

Answer:

  B. a three-quarter circle of radius 18 ft, quarter circles of radii 10 ft and 6 ft

Step-by-step explanation:

A diagram can be a useful aid to answering this question. At one corner of the shed, 8 ft of rope is no longer available, so the quarter circle has a radius of 18-8 = 10 ft. (That side of the shed is 12 ft, so the area is only a quarter circle.)

At the other corner of the shed, 12 ft of rope is no longer available, so the quarter circle has a radius of 18-12 = 6 ft. (That side of the shed is 8 ft, so the area is only a quarter circle.)

In the attached diagram, the 3/4 circle with radius 18 ft is shown red, and the quarter circles are shown in green.

Answer:2/3

Step-by-step explanation:

Answer:

1/8

Step-by-step explanation:

one way we can solve this is by making a tree  using H for heads and T for tails

since it can land H or T, we put them down like this. then we can get H or T again because its a coin, so we put that under each letter. finally we do the same thing again to get 8 possibilities. now we can count how many of them was HHH, which is one

         H              T

         /  \            /   \

       H   T         H   T

      /\     /\        /\    /\

    H T H T     H T H T

    ^      this one

or do 2 x 2 x 2 because there are 2 possibilities each three times

Answer:

see explanation

Step-by-step explanation:

To evaluate the expression for the given values.

Substitute the given values for x into g(x)

a

g(0) = 3(0)² - 4(0) + 3 = 0 - 0 + 3 = 3

b

g(- 1) = 3(- 1)² - 4(- 1) + 3 = 3 + 4 + 3 = 10

c

g(2) = 3(2)² - 4(2) + 3 = 12 - 8 + 3 = 7

d

g(- x) = 3(- x)² - 4(- x) + 3 = 3x² + 4x + 3

e

g(1 - t)

= 3(1 - t)² - 4(1 - t) + 3

= 3(1 - 2t + t²) - 4 + 4t + 3

= 3 - 6t + 3t² - 4 + 4t + 3

= 3t² - 2t + 2

Answer:

6/1

Step-by-step explanation:

So we are looking for a ratio of flour which is 4 and brown sugar which is 2/3.

So we divide 4 by 2/3.

4÷2/3 since we are diving by a fraction we keep 4÷2/3 change 4*2/3 flip 4*3/2. So now we are multiplying 4 and 3/2.

So that is 4/1*3/2= 12/2 that simplies to 6/1

Answer:

⁶/₁

Step-by-step explanation:

4 cups of flour, ⅔ cup of brown sugar.  The ratio of flour to brown sugar is:

4 / ⅔

We need to simplify this.  First, write 4 as a fraction:

⁴/₁ / ⅔

To divide by a fraction, multiply by the reciprocal:

⁴/₁ × ³/₂

¹²/₂

⁶/₁