Let W Span {V1. every vector in W Vp Show that if x is orthogonal to each Vj, for ] < j < P,then xis orthogonal to'

The height of the triangle that has an area of 7 ⁷/₈ square cm will be 3 cm.

The polygonal shape of a triangle has a number of sides and three independent variables. Angles in the triangle add up to 180°.

Assume 'h' is the height of the triangle and 'b' be the base of the triangle.

Then the area of the triangle is given as,

A = (1/2) × h·b

The area of the triangle is 7 ⁷/₈ cm² and the base length of the triangle is 5 ¹/₄ cm.

Convert the mixed fraction number into a fraction number. Then we have

7 ⁷/₈ = 63/8

5 ¹/₄ = 21/4

The height of the triangle is given as,

63/8 = (1/2) x h x (21/4)

Simplify the equation, then we have

63/8 = (1/2) x h x (21/4)

63/8 = (21/8)h

21h = 63

h = 3 cm

The height of the triangle that has an area of 7 ⁷/₈ square cm will be 3 cm.

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Parallel lines always have the same slope and different y-intercepts.

Slope-intercept form: [tex]y=mx+b[/tex]

We're given:

We can see that in [tex]y=3x-5[/tex] that the slope of the line is 3. Parallel lines have the same slope, so therefore:

m = 3

We're also given that the y-intercept is (0,7).

b = 7

Plug m and b into slope-intercept form:

[tex]y=3x+7[/tex]

[tex]y=3x+7[/tex]

The null hypothesis in the given question is 'the mean annual consumption of high fructose corn syrup by a person in the United States is 48.8 pounds' and the alternative hypothesis in the given question is 'the population standard deviation is 3.6 pounds.'

Population mean = 48.8 pounds

Sample size = 120

Sample mean = 49.5 pounds

Population standard deviation = 3.6

α = 0.05

The standardized statistic z is:

z = (Sample mean - Population mean) x (√n / standard deviation)

z = (49.5 - 48.8) x (√120/3.6)

z = 2.13

The standardized statistic P is:

P = 2 - 2 x Φ(z)

P = 2 - 2 x 0.983

P = 0.034

Since, the result we get is statistically significant. We reject the null hypothesis.

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Answer:

120 children

Step-by-step explanation:

c=children

a=adults

c+a=237

1.75c+2a=444

(c+a=237)*2 ==> make a in the equation equal to 2a

       2c+2a=474

- (1.75c+2a=444)  ==> eliminate a from the equation

2c-1.75c + 2a-2a = 474-444

2.00c-1.75c = 30

0.25c = 30

4 * 0.25c = 4 * 30   ==> remove decimals by multiplying each side by 4

c = 120 children

The depth of the river is 3.192 feet.

A percentage is a value or ratio that may be stated as a fraction of 100. If we need to calculate a percentage of a number, we should divide it's entirety and then multiply it by 100.

The percentage therefore refers to a component per hundred. Per 100 is what the word percent means. It is represented by %.

Since last year the depth of the river was 4.2 feet deep and year it dropped 24%. The depth will be:

= 4.2 - (24% × 4.2)

= 4.2 - 1.008

= 3.192 feet

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On solving the provided question, P = 0.037 reject H o,P-value < 0.05 ( level of significance )

If there is no statistical significance between two variables, the null hypothesis is that there isn't any. The researcher or experimenter generally aims to refute or undermine the theory. An alternate explanation is that the two variables have a statistically significant association.

based on P value -

P = 0.037 reject H o

P-value < 0.05 ( level of signifance )

we are 95% y confidence that the average age of divorced man is how different than it was in 1989.

we would have concluded that the average age is not different wher, it really was different.

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The number of cubical boxes that can contain in the rectangular box is 36.

Volume is defined as the space occupied by an object.

The volume of a cuboid is (length×width×height).

Given, Length of the rectangle box is 16 cm and the width of the box is

12 cm.

We know the volume of a cube is (side)³.

Therefore, given that the volume of a cubical 64 cm³ and we know (4)³ is 64.

Hence the height of the rectangular box is 3 layers so (4×3) = 12 cm

So, the volume of the rectangular box is (16×12×12) cm³ = 2304 cm³.

Therefore, The maximum number of boxes that can be contained is

= (2304/64)

= 36.

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The length of the Arc of the circle is 17.00 units

We should know that the arc of the circle is part of the circumference of the circle cut off by the radii of the circle.

The length of the arc is given by

L=[tex]\frac{A}{360} *2*\pi *r[/tex]   where A= angle made by the radii

[tex]\pi[/tex]=3.14, radius =10, L-length of arc = x

Applying the values in the formula we have

x=(x/3*2*3.14*10)÷360

Simplifying the expression

62.8x=1080x

Taking the ratio of both sides we have

62.8x:1080x

= 17.00 units

Therefore the length of the arc is 17.00 units

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As a result, the likelihood of a flush (five cards of the same suit) is.198% while the likelihood of getting three cards of the same rank and two cards from any other rank is 0.07612%.

Probability is the study of numerical representations of the likelihood that an event will happen or that a statement is true. An event's probability is a number between 0 and 1, where 0 generally denotes the event's impossibility and 1 typically denotes its certainty.

Here,

a) It is irrelevant what suit the opening card is. Any of the suits may be drawn first, provided that the subsequent four cards are dealt from the same deck and are of the same suit.

After the initial card is drawn, there are only 12 of a given suit left in the deck, which has 13 of each suit in total. The third card, which has 11, the fourth card, which has 10, and the fifth and final card, which has nine, are then the cards that are still available. Each time, one fewer card will be present in the deck as a whole.

(ℎ) = (2 ) ∙ (3 ) ∙ (4ℎ )\s∙ (5ℎ )

(ℎ) = [tex]\frac{12}{51} *\frac{11}{50} *\frac{10}{49} *\frac{9}{48}[/tex]

(ℎ) =11880 / 5997600

(ℎ)= 33/16660 = .00198 .198%

c) The likelihood of three identical cards and two cards from one other rank.

Considering that the cards have 13 ranks

So, P =  [tex]\frac{13}{52} *\frac{12}{51} *\frac{11}{50} *\frac{10}{49}[/tex]

P = 267696/351655200

P=0 .0007612 or 0.07612 %

As a result, the likelihood of a flush (five cards of the same suit) is.198% while the likelihood of getting three cards of the same rank and two cards from any other rank is 0.07612%.

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We calculated sample mean 14 and standard deviation 1.32

A sample mean refers to the average of all values included in the data list.

When we have a list of data set, we can calculate the mean value of the data set by adding all the values and divided by the amount of data.

We are given, 8 packages of ground chuck

 number of samples, N = 8

fat contents (%) for 8 packages, X= 12, 15, 16, 13, 15, 13, 15, 13

Sum of X= ∑X= 112

sample mean= sum of all data values/ number of data items in the sample

   X= ∑X/N

  sample mean X = 112/8 =14

we calculate   X² for 8 sample = 144, 225, 256, 169, 225, 169, 225, 169

∑X² = 1582

standard deviation is calculated by using the formula bellow

√[∑X²/N-(∑X/N)²] = √[1582/8-(112/8)²]  

standard deviation = 1.32

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The profit earned per day is $6674.695 at point (449.93,500.02).

Let the number of pair of super-fit jeans = x

Let the number of pair of super-hug jeans = y

We know that x,y≥0.

Total time provided for cutting per day = 18,999 minutes

Total time provided for sewing and finishing per day = 40,999 minutes

Therefore, the inequalities are:

20x+20y ≤ 18,999

30x+55y ≤ 40,999

The shaded region is the feasible region and the point of intersection is (449.93,500.02).

The profit on Super-fit jeans = $6.50

The profit on super-hug jeans = $7.50

To maximise the profit in a day, we maximize z = (6.50 x x) + (7.50 x y)

The maximum profit is obtained at (449.93,500.02).

Therefore, z = (6.50 x 449.93) + (7.50 x 500.02)

z = 2924.545 + 3750.15

z = 6674.695

The profit earned per day is $6674.695.

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Answer:

so (-5.2,-2) and (4.2,-2) are 9.4 units apart  away from each other

Step-by-step explanation:

so (4.2,-2) and (-5.2,-2) add together and get 9.4

A formal power series that encodes the coefficients of a sequence of numbers is a generating function.

Here, the coefficients of the sequence {Sₙ} are encoded by the generating function S(x). where Sₙ is the number of lattice paths as described above.

Here we will use the fact that the generating function for a sequence satisfies a functional equation that relates the generating function to the sequence itself to prove that 1 + (x – 1)S(x) + xS(x)² = 0,

Here, the functional equation is

S(x) = 1 + xS(x) + xS(x)²

This equation implies that

The term 1 on the right-hand side corresponds to the fact that to reach the point (0,0) (the starting point), there is exactly one way which is to stay at the starting point.

The term xS(x) corresponds to the fact that by taking a step in the positive x-direction and then following a path to (n-1, n-1) we can reach any point (n,n)

The term xS(x)² corresponds to the fact that by taking a step in the positive x-direction and then following a path to (n-2, n-2), and then taking another step in the positive x-direction and following a path to (n-1, n-1) we can reach any point (n,n)

Thus, 1 + (x – 1)S(x) + xS(x)² = 0. Hence proved.

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In the graph B we can see that y intercept is 9 and zero is 3. So the graph b is the correct answer.

In the given question, linear function k has a zero at 3 and a y-intercept of 9.

We have to choose which graph best represents k.

As we know that;

The link between the two unknown variables is represented by a linear equation. The linear equation have one degree.

The most basic linear equation demonstrates the relationship between (x,y), which can be used to create a table, a graph on the coordinate plane, or to perform an assessment at a particular x or y value.

We represent linear function in the form of y=mx+c.

where m is the slope of the line, c is the y intercept of the line, x is independent variable and y is the dependent variable.

As we see the y intercept in the graph crosses the y axis.

Here y intercept is 9.

The zeros of the linear function is that where y value is zero.

As in the graph B we can see that y intercept is 9 and zero is 3. So the graph b is the correct answer.

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Timmy has a collection of 95 coins consisting of dimes and nickels in his bank, The total value of his bank is $6.50. We can infer that he has 60 nickels and 35 dimes.

In algebra, an equation is a declaration of equivalence that includes one or more variables or unknowable quantities.

Given,

Timmy has a collection of 95 coins consisting of dimes and nickels in his bank, The total value of his bank is $6.50.

Set d and n to the appropriate numbers.

Timmy has 95 coins in total, all of which are dimes and nickels, as far as we know.

Equation,

So d + n = 95

His bank is worth a total of $6.50.

A nickel is worth $0.05, while a dime is worth $0.10.

So

0.10d + 0.05n = $6.50

Take note that we have just produced an equation system that we can solve.

The two equations exist.

D + n = 95

and

0.10d + 0.05n = 6.50.

The replacement approach is probably going to be the simplest way to solve this problem out of all the options available to us.

First, we'll want to change the second equation's terms so that one of the variables is defined.

d+ n = 95

Take d off on both sides.

n = 95 - d

Now, let's define "n."

After defining one of the variables, we can now enter (or replace) its value in the other equation. After replacing it, we can find a solution for the other variable.

0.10d + 0.05n = 6.50

n = 95 - d

0.10d + 0.05(95 - d) = 6.50

We now determine d.

0.10d + 0.05(95 - d) = 6.50

distribution of the 0.05 in step 1

0.05 × 95 = 4.75

and

0.05 × -d = -0.05d

0.10d + 4.75 - 0.05d = 6.50

Step 2: Add like terms together (0.10 - 0.05) to get 0.05

0.05d + 4.75 = 6.50

step 3: Take 4.75 off of each side.

0.05d = 1.75

step 4: Subtract 0.05 from both sides.

0.05d / 0.05 = d and 1.75 / 0.05 = 35

d = 35

Now that we know the value of one variable, we can use it to solve for the second variable by plugging it into one of the equations ( note that the equation that we use does not matter, we will acquire the same answer )

d + n = 95

d = 35

35 + n = 95

Take 35 off both sides.

n = 60

We can infer that he has 60 nickels and 35 dimes.

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The solution to the given composite functions are;

f(g(x)) = √(x² + 1) + 8

g(f(x)) = [√(x) + 8]² + 1

Composite functions are usually formed when one function is used as the input of another function.

We are given the functions;

f(x) = √(x) + 8

g(x) = x² + 1

1) f(g(x)); This simply means we will put the entire g(x) function for x in f(x) to get;

f(g(x)) = √(x² + 1) + 8

2) g(f(x)); This simply means we will put the entire f(x) function for x in g(x) to get; g(f(x)) = [√(x) + 8]² + 1

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According to factor theorem, if f(x) is a polynomial of degree n ≥ 1 and 'a' is any real number, then, (x-a) is a factor of f(x), if f(a)=0.

The factor theorem states

[tex]$(x-a)$ is a factor of $f(x) \Leftrightarrow f(a)=0$[/tex]

f(x)=4 x^3-3 x^2-8 x+4

we have (x-2)[tex]\Rightarrow a=2[/tex]

& [tex]f(2)=4 \times 2^3-3 \times 2^2-8 \times 2+4 \\& f(2)=4 \times 8-3 \times 4-16+4 \\& f(2)=32-12-16+4 \\& f(2)=8 \neq 0\end{aligned}[/tex]

[tex]$\therefore x-2$ is not a factor of $4 x^3-3 x^2-8 x+4$[/tex]

however by the remainder theorem when

[tex]$f(x)=4 x^3-3 x^2-8 x+4$[/tex] is divided by [tex]$(x-2)$[/tex] the remainder is 8

the factor theorem being a special case of the remainder theorem

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Answer:

Step-by-step explanation:

Use the factor Theorem to determine whether x =3 is a factor of P (x)=x-2x2-6. Specifically, evaluate P at the proper value, and then determine whether X-3 is a factor. P (UI) - 0 P 0 1 - 3 is a factor of P (*) O +- 3 is not a factor of P (x) This problem has been solved!

Answer:

small car: 2

medium car: 2

luxury car: 2

Small van: 2

Large van: 3

Step-by-step explanation:

just multiply the amount per day by numbers increasing from 1 till you get a number that is larger than the week.  I swear whoever made this car rental will not be in business much longer lol.

For 2 days, vehicle can be given to rent before it would be less expensive to rent.

Algebra is a study of mathematical expressions, in which numbers and quantities are represented in formulas and equations by letters and other universal symbols.

Given that,

Rent for small car,

For 1 day = $100.

For 1 week = $250

The maximum days in which small car could be given to rent,

= 250 / 100 = 2.5 ≅ 2 days.

Similarly, for medium car = 290 / 110 = 2.7 ≅ 2 days.

For luxury car = 325 / 120 = 2.54  ≅ 2 days.

For small van = 350 / 150 = 2.3  ≅ 2 days.

And for large can = 390 / 170 = 2.28  ≅ 2 days.

Each car can be rented for 2 days before it would be less expensive for the week.

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There were 37 kisses at the family reunion. Probability is used to make predictions and inform decision-making.

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

To find the number of kisses at the family reunion, you can set up the ratio 4 hugs : 1 kiss as a proportion and solve for the number of kisses. Here's how you can do this:

First, cross-multiply to find the total number of hugs and kisses:

4 hugs * 1 kiss = 148 hugs

4 kisses = 148 hugs

Then, divide both sides of the equation by 4 to find the number of kisses:

4 kisses / 4 = 148 hugs / 4

1 kiss = 37 hugs

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The correct order of the given statements is- 1, 2, 3, 4, 5, 6, followed by 7.

Here, we are given the following statement-

For all sets A, B and C if A ≤ B and B∩C = Φ

then A∩C = Ф

We can use the proof-by-contradiction method to prove the given statement as shown below-

Suppose that there exist 3 sets, that are- A, B, and C

Such that A ≤ B, B∩C = Φ

and A∩C ≠ Φ

Then, there must exist an element x such that x ∈ A∩C

Now, by the definition of intersection, we have- x ∈ A and x ∈ C

Then x ∈ B because A ≤ B. Additionally, we know that x ∈ C

Hence, by the definition of intersection, x ∈ B∩C

Thus, B∩C ≠ Φ by definition of null set (Φ)

But this contradicts the supposition. Hence, our assumption is false and so A∩C = Ф.

The correct order of the given statements is- 1, 2, 3, 4, 5, 6, followed by 7.

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Using theory about stereotypes and different associated thoughts, we have to:

A stereotype, basically, can be defined as an idea that is already pre-established and accepted, it is applied to different individuals.

¡Hope this helped!

F.ds = 4 pi, F.ds = 9 pi

circulation of F around C1, assuming that curl (F) = 2 on the shaded region. F. ds =

c1-9 (52-12-12) π+9л+4л=(207+12)л=219л

Green's Theorem =Green's Theorem states that a line integral around the boundary of the plane region D can be computed as the double integral over the region D. where the path integral is traversed anti-clockwise. Green's Theorem Area

Therefore, the line integral defined by Green's theorem gives the area of the closed curve. Therefore, we can write the area formulas as: A = − ∫ c y d x. A = ∫ c x d y. Green's theorem relates a line integral around a simply closed plane curve C and a double integral over the region enclosed by C. The theorem is useful because it allows us to translate difficult line integrals into more simple double integrals, or difficult double integrals into more simple line integrals.

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Answer:

We can say that he used a total of 5/8 gallons of glue

let x - glue to make purple slime

5/8 = 2/5 + 1/5 + x

5/8 = 3/5 + x             (LCD of 5 and 8 is 40)

25/40 = 24/40 + x

25/40 - 24/40 = 24/40 - 24/40 + x    (subtract 24/40 both sides)

x = 1/40

5/8 = 2/5 + 1/5 + 1/40

5/8 = 3/5 + 1/40

25/40 = 24/40 + 1/40

25/40 = 25/40

Therefore, the glue to make purple slime is 1/40 gallons.

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Step-by-step explanation:

The t statistic used to determine whether the slope is significant is 6.709 when applying regression analysis to sales data (in $1,000s) and advertising data (in $100s).

Given that,

The following information was discovered after applying regression analysis to sales data (in $1,000s) and advertising data (in $100s).

y=12+1.8x

n=17

SSR=225

SSE=75

S₀₁=0.2683

We have to find what is the t statistic used to determine whether the slope is significant.

We know that,

Given by, the t test is used to determine whether the slope is significant.

t= b1/s.e

Where b1= slope of the given regression line

s.e= standard error of the line ( Sb1)= 0.2683

The regression line is given by

Y=12+1.8X

X is the independent variable here, and Y is the dependent variable.

Intercept = 12

Slope = 1.8

The t test is used to determine whether the slope is significant.

t= b1/s.e=1.8/0.2683= 6.7089=6.709

Therefore, The t statistic used to determine whether the slope is significant is 6.709 when applying regression analysis to sales data (in $1,000s) and advertising data (in $100s).

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Answer: 2

Step-by-step explanation:

(a) The probability that X, = 1 will be =1/2

(b) The probability that X, = 1 given X,= 1 will be =2/3

(c) The probability that in the first 4 trials you get (at least) three white balls in a row will be=2/5

(d) The expected value of the number of white balls in 4 trials will be=2

(e) The variance of the number of white balls in 4 trials will be=2

A box starts with 2 balls; [ black and 1 white, x draw on these balls are supposed to be uniformly at random then we will replace it with 2 of the same colour. The Process repeats for Each i=1,2,3..........

Let x_1=1, if we draw a white ball on trial 1

The Property of the x_1 is

The Event where The drawn fall is white =A

Let the number of white balls in 4 trials is denoted by 'x'.

[tex]P(x=0) P\left(A^c A^c A^c A^c\right) . \\=1 / 2 \times 3 / 3 \times 3 / 4 \times 4 / 5=1 / 5 . \\P(x-1)=P\left(A^c A^c A^c A^c\right)+P\left(A^c A^c A^c A^c\right)+P\left(A^c A^c A^c A^c\right)+P\left(A^c A^c A^c A^c\right) \\=1 / 2 \times 1 / 3 \times 1 / 4 \times 3 / 5+1 / 4 \times 1 / 3 \times 2 / 3 \times 3 / 5+1 / 2 \times 2 / 3 \times 1 / 4 \times 3 / 5+1 / 2 \times 2 / 3^2 \times 3 / 4 \times 1 / 5 . \\=\frac{1}{20}+\frac{1}{20}+\frac{1}{20}+\frac{1}{20} . \\=4 \times \frac{1}{20}=1 / 5 . \\[/tex]

[tex]\therefore P(x=2)=1 / 2 \times 2 / 3 \times 1 / 4 \times 2 / 5 \times 4 / 2=6 \times 1 / 3=1 / 5 .[/tex]

[tex]$\begin{aligned} \therefore P(x=2) & =1 / 2 \times 2 / 3 \times 1 / 4 \times 2 / 5 \times 4 / 2=6 \times 1 / 3=1 / 5 . \\ P(x=3) & =1 / 2 \times 2 / 3 \times 3 / 4 \times 1 / 5 \times 4 / 3 . \\ & =1 / 20 \times 4=1 / 5 . \\ P(x=4) & =1 / 2 \times 2 / 3 \times 2 / 4 \times 4 / 5=P(\text { AAAA })=1 / 5 .\end{aligned}$[/tex]

(a)

[tex]P\left(X_2=1\right) & =P(A A)+P\left(A^C A\right) . \\& =1 / 2 \times 2 / 3 \times 1 / 2 \times 1 / 3 . \\& =2 / 6+1 / 6=3 / 6=1 / 2 . \\P\left(X_4=1\right)= & P(A A A)+P\left(A A A A^{\prime} A\right)+P\left(A^C A A\right)+P\left(A^{\prime} A^{\prime} A\right)+P \\& \left.\quad\left(A^{\prime} A A A\right)+P\left(A^{\prime} A^{\prime} A\right)+P\left(A^A A A\right)\right) \\[/tex]

[tex]= & 1 / 5+1 / 20+1 / 20+1 / 20+1 / 20+1 / 30+1 / 30+1 / 20 . \\= & 1 / 5+4 / 20+3 / 20 .=2 / 5+1 / 10 . \\= & \frac{4+1}{10}=5 / 10=1 / 2 .[/tex]

[tex]$\therefore$[/tex] Probability that x_2=1 is equal with probability that x_4 >1

(b) Probability that x_2=1 given x_4=1

[tex]$$\begin{aligned}& =P\left(x_2=1 / x_{4=1}\right)=\frac{P\left(x_2=1 \& x_4=1\right)}{P\left(x_4=1\right) .} \\& =\frac{P(A A A A)+P\left(A A A^{\prime} A\right)+P(A A A A)+P\left(A^C A A^C A\right)}{1 / 2} \\& =2 \times(1 / 5+1 / 20+1 / 20+1 / 30] \\& =2 \times[1 / 5+1 / 10 \times 1 / 30] \\& =2 \lambda\left[\frac{6+30+1}{30}\right] \\& =2 \times \frac{10}{30} .=2 / \mathrm{3} .\end{aligned}$$[/tex]

(c) Probability that in that the nuts is 3 which is P(x=3)+P(x=4) is [tex]=1 / 5+1 / 5=2 / 5[/tex]

(d) Expected volume of number of white ball is 4 trial is,

[tex]$$\begin{gathered}{[0 \times 1 / 5+1 \times 1 / 5+2 \times 1 / 5+3 \times 1 / 5+3 \times 1 / 5+4 \times 1 / 5]} \\=\frac{1+2+3+4}{5}=10 / 5=2 .\end{gathered}$$[/tex]

(e)

[tex]E\left(x^2\right) & =0^2 \times 1 / 5+1^2 \times 1 / 5+2^2 \times 1 / 5+3^2 \times 1 / 5 .+4^2+1 / 5 . \\& =\frac{1+4+9+16}{5 .} \\& =30 / 51=6 . \\\therefore var/x =E\left(x^2\right)-E[(x)]^2 . \\& =6-2^2 =6-4=2[/tex]

[tex]$\therefore$[/tex] The variance of the number of white balls in 4 trails =2

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Answer:

i got the last two but i am confused on the first two

Step-by-step explanation:

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Answer:

t ≥ 5

Step-by-step explanation:

In the tank now: 19

Fill rate per minute: 6

Number of minutes: t

Amount filled in t minutes: 6t

Total amount in tank after t minutes: 6t + 19

Amount desired: at least 49 liters

At least 49 liters means "greater than or equal to 49"

Inequality:

6t + 19 ≥ 49

Subtract 19 from both sides.

6t ≥ 30

Divide both sides by 6.

t ≥ 5

The operations, along with their inverses are given as follows:

The operations are solved applying the inverse operations.

The first operation is solved as follows:

x + 10 = 25

x = 25 - 10

x = 15.

Hence the addition operation was solved applying the subtraction operation, which is the inverse operation of the addition.

The second operation is solved as follows:

2x = 28

x = 28/2

x = 14.

Hence the multiplication operation was solved applying the division operation, which is the inverse operation of the multiplication.

The third operation is given as follows:

4^x = 64.

log4(4^x) = log4(64)

x = 3. (as 4³ = 64, hence log4(64) = 3).

Hence the exponential operation was solved applying the logarithm operation, which is the inverse operation of the exponential.

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