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The single logarithms found by logarithm properties are listed below:

Case 31

㏒₃ 277809109776

Case 32

㏒₅ 134456

Case 33

5 · ㏒₇ (a / b²)

Case 34

㏑ (a⁶ / b)⁶

Case 35

㏒₇ (c⁴ · [tex]\sqrt[3]{a}[/tex])

Case 36

㏒₈ [w · (√u · v)]

In this problem we find six cases of logarithmic expressions that must be simplified into a single logarithm by logarithm properties. The complete procedure for each case is shown below:

Case 31

8 · ㏒₃ 11 + 4 · ㏒₃ 6

㏒₃ 11⁸ + ㏒₃ 6⁴

㏒₃ (11⁸ · 6⁴)

㏒₃ 277809109776

Case 32

5 · ㏒₅ 7 + 3 · ㏒₅ 2

㏒₅ 7⁵ + ㏒₅ 2³

㏒₅ (7⁵ · 2³)

㏒₅ 134456

Case 33

5 · ㏒₇ a - 10 · ㏒₇ b

㏒₇ a⁵ - ㏒₇ b¹⁰

㏒₇ (a⁵ / b¹⁰)

㏒₇ (a / b²)⁵

5 · ㏒₇ (a / b²)

Case 34

36 · ㏑ a - 6 · ㏑ b

㏑ a³⁶ - ㏑ b⁶

㏑ (a³⁶ / b⁶)

㏑ (a⁶ / b)⁶

Case 35

4 · ㏒₇ c + (1 / 3) · ㏒₇ a

㏒₇ c⁴ + ㏒₇ [tex]\sqrt[3]{a}[/tex]

㏒₇ (c⁴ · [tex]\sqrt[3]{a}[/tex])

Case 36

㏒₈ w + (1 / 2) · ㏒₈ u + (1 / 2) · ㏒₈ v

㏒₈ w + ㏒₈ √u + ㏒₈ √v

㏒₈ [w · (√u · v)]

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Answer:5 35/36

Step-by-step explanation:

63 13/18 − 57 3/4 =?

Solving the whole number parts

63−57=6

Solving the fraction parts

13/18−3/4 is equal to 26/36 − 27/36

Combining the whole and fraction parts

6−1/36 = 5 35/36

The values -3 , -6 , 3 , 8 are in the solution set of the compound inequality 4(x + 3) ≤ 0 and x + 1 > 3

Given, a compound inequality

4(x + 3) ≤ 0 and x + 1 > 3

Now, on solving the first inequality, we get

4(x + 3) ≤ 0

4x + 12 ≤ 0

On subtracting both the sides by 12, we get

4x + 12 - 12 ≤ 0 - 12

4x  ≤ -12

On dividing both the sides by 4, we get

4x/4  ≤ -12/4

x ≤ -3

So, the solution set of the 4(x + 3) ≤ 0 inequality be, x ≤ -3 i.e. all the real numbers less than -3.

Now, on solving the second inequality, we get

x + 1 > 3

On subtracting both the sides by 1, we get

x + 1 - 1 > 3 - 1

x > 2

So, the solution set of the x + 1 > 3 inequality be, x > 2 i.e. all the real number greater than 2.

Now, the values in the solution set of the compound inequality

4(x + 3) ≤ 0 and x + 1 > 3 be,

x ≤ -3 and  x > 2.

As, we can see -3 , -6 , 3 , 8 are satisfying the condition.

Hence, -3 , -6 , 3 , 8 are the values in the solution set of the compound inequality 4(x + 3) ≤ 0 and x + 1 > 3.

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The amount of grapes Carmen purchase than Hunter is 3/4 lbs

From the question, we have the following parameters that can be used in our computation:

Carmen = 11/12 lbs

Hunter = 2/12 lbs

The difference between these parameters is the amount of grapes Carmen purchase than Hunter

This is represented as

Difference = Carmen - Hunter

Substitute the known values in the above equation, so, we have the following representation

Difference = 11/12 - 2/12

Evaluate

Difference = 9/12 = 3/4

Hence, the amount of grapes is 3/4 lbs

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Answer: $194.25

Step-by-step explanation:

Given data

Principal=  $7400

Rate=  10.5%

time= 1/4 years

The simple interest formula is given as

SI= PRT/100

substitute

SI= 7400*10.5* 0.25/100

SI= 19425/100

SI= $194.25

Hence Mark will pay  $194.25

Equation of line will be y = x/3 - 2

A line's equation can be written in many forms in which one is slope-intercept form.

Equation in slope-intercept form is written as y = mx + c where

m = slope

c = y intercept

Slope shows the steepness of the line. It can be given by change in y axis by change in x axis.

y intercept is the point where the line cuts the y axis that is x = 0 at this point.

Points given to us are (6 , 0) and (9 , 1)

slope = [tex]\frac{y_{2} - y_{1}}{x_{2} - x_{1}}[/tex]

m = (1 - 0) / (9-6)

m = 1/3

For c put values of either point in the equation as they both are on the line so they both will satisfy the equation.

y = mx + c

0 = (1/3)*6 + c

0 = 2 + c

c = -2

now put values of m and c in the equation and get a general equation.

Therefore of equation of line will be y = x/3 - 2

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Answer: 31.5

Step-by-step explanation:

Answer:

31.5

Step-by-step explanation:

[tex]45 \times .7 = 31.5[/tex]

These triangles can be proven congruent by using S-A-S rule.

What is congruent triangles?

Two triangles are same to be congruent if all 3 corresponding sides ar equal and every one the 3 corresponding angles ar equal in measure. These triangles will be slides, rotated, flipped and turned to be looked identical. If repositioned, they coincide with one another. The symbol of congruence is’ ≅’.

Main body:

In ΔABC and ΔXYZ

AC = XZ   ---(1)

∠C = ∠ Z    ---(2)

BC = YZ    ---(3)

using 1, 2 ,3  we can prove that  ΔABC and ΔXYZ are congruent.

Hence ΔABC and ΔXYZ  are congruent.

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The x-intercepts 0 and 30 indicate it takes 30 seconds to return to the surface of the water

The vertex (15, -5) indicates the scuba diver descends for 15 seconds to a depth of 5 meters under the surface before ascending

The equation of the function is given as

d(x) = 1/3|x - 15| - 5

Set d(x) to 0, to determine the x-intercept

So, we have

1/3|x - 15| - 5 = 0

This gives

1/3|x - 15| = 5

So, we have

|x - 15| = 15

Remove the absolute brackets

x - 15 = 15 and x - 15 = -15

Evaluate the equations

x = 30 and x = 0

This means that the diver is at the surface at 0 and 30 seconds

An absolute value function represented as f(x) = a|x - h|  + k has a vertex of (h, k)

This means that

(h, k) = (15, -5)

It means that the diver dived for 15 seconds to get to 5 meters below the water surface

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to get the slope of any straight line, we simply need two points off of it, wait a minute!!!! holy smokes!! the graph already has two points for us, yeahhh, let's use those then

[tex](\stackrel{x_1}{5}~,~\stackrel{y_1}{0})\qquad (\stackrel{x_2}{0}~,~\stackrel{y_2}{-7}) ~\hfill \stackrel{slope}{m}\implies \cfrac{\stackrel{rise} {\stackrel{y_2}{-7}-\stackrel{y1}{0}}}{\underset{run} {\underset{x_2}{0}-\underset{x_1}{5}}} \implies \cfrac{ -7 }{ -5 } \implies {\Large \begin{array}{llll} \cfrac{7 }{ 5 } \end{array}}[/tex]

Answer:

x = 2

Step-by-step explanation:

3x + 5= 2x + 7

Subtract 2x from both sides

3x + 5 −2x = 7

Combine 3x and −2x to get x

x + 5 = 7

Subtract 5 from both sides

x = 7 − 5

Subtract 5 from 7 to get 2

x = 2

Answer:

38 people

Step-by-step explanation:

In this case, we can see that, being two people, the cost is 14 and when it is double, that is, 4 people, the value is not 28 but 20.

Which means that there is an additional value that is being charged for the vehicle, that is:

let "c" be the value of the vehicle and "x" be the value per person

c + 2 * x = 14

c + 4 * x = 20 => c = 20 - 4 * x

replacing:

20 - 4 * x + 2 * x = 14

-2 * x = 14-20

-2 * x = -6

x = 6/2

x = 3

for C:

c = 20 - 4 * 3

c = 8

Which means that the formula for the value of the input would then be:

e = 8 + 3 * n

where "n" is the number of people.

Now, we are told that for e = 122

122 = 8 * + 3 * n

3 * n = 122 -8

n = 114/3

n = 38

which means that for that bus there are a total of 38 people

Answer: 3: 4^21/5^6

Step-by-step explanation:

Remember when you are raising you need to multiple the powers.

The maximum height of 32 feet is reached after 1.25 second of flight by the rocket

The function is given as

h(t) = -16t² + 40t + 7

Differentiate the function and set it to 0

-32t + 40 = 0

Add 32t to both sides

32t = 40

Divide both sides by 32

t = 1.25

Hence, the time is 1.25 seconds

In part (a) of this solution, we have

t = 1.25

Recall that

h(t) = -16t² + 40t + 7

So, we have

h(1.25) = -16(1.25)² + 40(1.25) + 7

This gives

h(1.25) = 32

Hence, the maximum height is 32 feet

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The compound figure formed by a square and four circular sections has an approximate perimeter of 308.496 centimeters.

In this question we find a compound figure formed by a square with a side length of 30 centimeters and four circular sections with central angle of 270° and a radius of 10 centimeters. The perimeter of the compound figure (p), in centimeters, is the sum of the perimeter of the square and the perimeter of the four circular sections:

p = p' + 4 · p''

Where:

p = 4 · (30 cm) + 4 · (3π / 2) · (10 cm)

p = 120 cm + 60π cm

p ≈ 308.496 cm

The perimeter is approximately equal to 308.496 centimeters.

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The smaller number from the given situation is 16.

Given that, the difference between two whole numbers is 6.

In mathematics, an equation is a formula that expresses the equality of two expressions, by connecting them with the equals sign =.

Let the larger number be x and the smaller number be y.

The difference between two whole numbers is 6.

So, x-y=6

x=6+y --------(I)

Twice the smaller number is 10 more than the larger number.

x+10=2y

x=2y-10 --------(II)

Substitute equation (I) in equation (II), we get

6+y=2y-10

⇒ y=16

Hence, the smaller number from the given situation is 16.

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Answer:

  (b)  35.5w + 136.45 = 455.95; w = 9

Step-by-step explanation:

You want an equation to determine how many weeks, w, you must babysit to earn $455.95 at the rate of $35.50 each week, having already saved $136.45.

If you earn $35.50 each week for w weeks, your total earnings will be ...

  35.50w

When that amount is added to the amount you already have, $136.45, the sum is ...

  35.5w +136.45

You want that total to be $455.95, which gives you the equation ...

  35.5w +136.45 = 455.95

To solve this 2-step equation, subtract the left-side constant:

  35.5w = 319.50

The second step is to divide by the coefficient of the variable:

  w = 319.50/35.5

  w = 9

By using the definitions of combinations and permutations:

a) The first permutation is not defined.

b) N =  (³C₂) / (⁴C₂) = 1/2.

A permutation:

ⁿPk is written as:

ⁿPk= n!/(n - k)!  

(the denominator only appears if nk)

And a combination is written as:

ⁿCk = n!/(n - k)ka)

We have the product of two permutations, but there is a problem:3P4 The first number is the size of the set, and the second is the number of elements we take from that set.

In that permutation we have a set of 3 elements and we need to select 4, so that permutation is not defined.

b) (³C₂) = 3!/(3 - 2)!*2! = (3*2*1)/(2*1) = 3  

(⁴C₂) = 4!/(4 - 2)!*2! = (4*3*2*1)/[(2*1)*(2*1)]

= 3*2 = 6

Then the quotient can be written as:

(³C₂) (⁴C₂)  = 3/6 = 1/2.

Note that A quotient is seen as the outcome of dividing two numbers by each other.

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The equivalent expression of 6⁵ is 6² × 6³.

The equivalent expression of 5⁻² × 5⁵ × 5  is 5⁴.

Two algebraic expressions are said to be equivalent if their values obtained by substituting the values of the variables are same.

Equivalent expressions are expressions that work the same even though they look different.

Therefore, let's find the expression, equivalent to 6⁵,

Using law of indices,

[tex]A^{x} X A^{y} = A^{x +y }[/tex]

Hence,

6² × 6³ = 6²⁺³ = 6⁵

5⁻² × 5⁵ × 5 = 5⁻²⁺⁵⁺¹ = 5⁴

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The total mixture to contain 8% vinegar so: .06F plus .11S equal to .08(320).Now, we can solve the system of equations using substitution.F + S is 310

.06F + .11S = .08(310)

F = 310 - S      [solve 1st equation for F]

.06(310 - S) + .11S = .08(320)     [substitute into 2nd equation]

18.6 -.06S +.11S = 24.8

.05S = 6.2

S = 124

F + 124 = 310  

F = 186

make a 320mL mixture that contains 8% vinegar, use 186mL of the first brand and 124mL of the second brand.

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Answer:

its the third one

oh this is easy

The equation to line is x-y+7=0.

What is a line?

A line is an indefinitely long, straight object that, despite being drawn with a minimum width, is considered to have no particular width in mathematics.

Main Body:

We know that equation of line in two point form =

y-y₁ =( y₂-y₁)/(x₂-x₁)(x-x₁)

points given are (-5,2) and (-4,3).

Therefore

y-2 =( 3-2)/(-4+5) *( x+5)

y-2 = x+5

x-y+7=0

Hence the equation is x-y+7=0.

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Answer:B 3 1/4 + (5/8)

D 3 1/4 + (+5/8)

Step-by-step explanation:

Calculation

an expression is the combination of numbers and operators like +, - * or /

If we have any equal sign then it is an equation .

Given expression is :

to find out equivalent expression  we  remove the parentheisis  by multiplying negative inside the parenthesis

Negative  times negative becomes positive

first fraction is positive . So it remains the same

Multiply negative inside the parenthesis . Negative times negative is positive

so, the equivalent expressions are

The domain of the given relation is: {2, 3, 4].

A relation has ordered pairs of values that are given as x and y coordinates, (x, y) In a relation, the values of the x coordinates are the domain of the relation while the values of the y coordinates are the range of the relation.

Given the relation, {(3,4),(2,3),(2,0),(4,2),(3,6)}, the set of x-values is: {2,3,4}. the set of y-values is: {0, 2, 3, 4, 6}.

{2,3,4} represents the domain while {0, 2, 3, 4, 6}. represents the range.

Therefore, the domain of {(3,4),(2,3),(2,0),(4,2),(3,6)} is given as: {2, 3, 4].

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The time taken for the rocket to reach its maximum height is; 2.25 seconds.

The maximum height reached by the toy rocket by evaluation is; 88 feet.

It follows from the task content that the maximum height reached by the rocket is to be determined.

The given function is; h (t) = -16 t² + 72t + 7.

Since it follows from evaluation that; at maximum height, the rate of change of height to time is equal to 0.

Therefore; at maximum height;

h'(t) = 0

h'(t) = -32t + 72 = 0

-32t = -72

t = -72/-32

t = 2.25 seconds.

Ultimately, the time taken for the rocket to reach its maximum height is; 2.25 seconds.

Also, the maximum height reached by the rocket is at; h (2.25);

h(2.25) = -16(2.25)² + 72 (2.25) + 7

= -81 + 162 + 7

= 88.

The maximum height reached is; 88 feet.

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Answer:

X = 67, Y = 113

Step-by-step explanation:

To find y: 180-62-51=67

To find x: 180-67 = 113

Total angle in a triangle sums to 180 degrees

Total angle in a straight line is also 180 degrees

(C) represents a function with the same rate of change at the function       [tex]y = \frac{7}{4}x+3[/tex].

Given equation,

[tex]y = \frac{7}{4}x+3[/tex]

Slope = m = 7/4

Slope = (x2-x1)/(y2-y1)

Now we will check every option and find slope which we be 7/4.

(A) A(1,0) and B(5,-4)

    Slope = (-4-0)/(5-1)

               = -4/4

               = -1

               ≠ m

(B) A(-1,1) and B(0,4)

     Slope = (4-1)/(0+1)

                = 3/1

                = 3

                ≠ m

(C) A(-4,-6) and (0,1)

    Slope = (1+6)/(0+4)

               = 7/4

               = m

(D) A(0,-2) and B(-4,5)

    Slope = (5+2)/(-4-0)

               = 7/-4

               = -7/4

               ≠ m

Hence, only (C) satisfies the situation.

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