Solve the equation of exponential decay.
A company's value decreased by 11.2% from 2009 to 2010. Assume this continues. If the company had a value of
$9,220,000 in 2009, write an equation for the value of the company years after 2009

Answer:

January

Step-by-step explanation:

A counterexample is something that proves the statement false.

January is a month that starts with J that is not a summer month.

That proves the statement false

[tex]\huge{\boxed{\text{January}}}[/tex]

A counterexample is an example that proves the statement wrong.

In this case, we are trying to prove that not all months that start with J are summer months. This means we need to find a month that starts with J that is also not a summer month.

[tex]\boxed{January}[/tex] is the only month that fits this criteria.  It begins with the letter J and is a winter month, which is not summer.

Answer:

Each part is solved and working is shown  

Step-by-step explanation:

1)

[tex]-6\displaystyle\frac{4}{9}-3\displaystyle\frac{2}{9}-8\displaystyle\frac{2}{9}\\\\= -\displaystyle\frac{58}{9}-\displaystyle\frac{29}{9}-\displaystyle\frac{74}{9}\\\\= -\displaystyle\frac{161}{9}\\=-17\displaystyle\frac{8}{9}[/tex]

2)

[tex]-12.48-(-2.99) -5.62\\=-12.48 + 2.99 -5.62 \\=-15.11[/tex]

3)

[tex]-19\displaystyle\frac{2}{9}-4\displaystyle\frac{1}{9}+3\displaystyle\frac{4}{9}\\\\= -\displaystyle\frac{173}{9}-\displaystyle\frac{37}{9}+\displaystyle\frac{31}{9}\\\\= -\displaystyle\frac{179}{9}\\=-19\displaystyle\frac{8}{9}[/tex]

4)

[tex]-353.92 - (-283.56) - 131.29\\= -353.92 + 283.56 - 131.29\\= -201.65[/tex]

5)

[tex]83\displaystyle\frac{1}{5}-108\displaystyle\frac{2}{5} + 99\displaystyle\frac{1}{5}\\\\= \displaystyle\frac{416}{5}-\displaystyle\frac{542}{5}+\displaystyle\frac{496}{5}\\\\= -\displaystyle\frac{370}{5}\\= 74[/tex]

Answer:

Step-by-step explanation:

Answer:

  n = 11

Step-by-step explanation:

100 mod 11 = 1, which is the remainder from division by 11 for each of the terms of the sum. 11 terms of the sum are needed in order to make the remainders add up to a number divisible by 11.

Answer:

2.

Step-by-step explanation:

is a geometric progression with common ratio of 1/4.

there, multiple 150 by 1/4 in 3 places. (4th, 5th and 6th day)

150 ÷ (4*4*4) = 150/64 = 2.3 ≈ 2

Using the formula for the n-th term of a geometric sequence, the number of mosquitoes killed on the sixth day is found to be approximately 2 when rounded to the nearest whole number.

The sequence representing the number of mosquitoes killed after spraying insecticide appears to decrease by a factor of 4 each day.

This pattern can be described as a geometric sequence. To find the number of mosquitoes killed on the sixth day, we will use the formula for the n-th term of a geometric sequence, which is an = a₁ × rⁿ⁻¹, where a1 is the first term, r is the common ratio, and n is the term number.

Day 1 (first term, a1): 2400 mosquitoes

Day 2: 2400 / 4 = 600 mosquitoes

Day 3: 600 / 4 = 150 mosquitoes

From this pattern, we identify the common ratio r as 1/4. To find the number of mosquitoes killed on the sixth day:

Let n = 6 for the sixth day.

Substitute a1 = 2400 and r = 1/4 into the formula: a6 = 2400 × (1/4)⁶⁻¹ = 2400 × (1/4)⁵.

Calculate the value: a6 = 2400 × (1/1024) ≈ 2.34, which rounds to 2 mosquitoes when rounded to the nearest whole number.

Therefore, approximately 2 mosquitoes were killed on the sixth day after the spraying.

Answer:

x=-3 or x=3/2

Step-by-step explanation:

We are given the following equation:

2x^2+5x-9=2x

We are asked to find the roots.  That means just solve it for x.

2x^2+5x-9=2x

Subtract 2x on both sides:

2x^2+3x-9=0

Let's see if we can put this in factored form.

Compare

2x^2+3x-9=0

and

ax^2+bx+c=0.

a=2, b=3 , c=-9

We have to find two numbers that multiply to be ac and add up to be b.

ac=-18

b=3

What are two numbers that multiply to be -18 and add to be 3?

Say -3 and 6.

So we are going to factor 2x^2-3x+6x-9=0

The first two terms have a common factor of x.

The last two terms have a common factor of 3.

2x^2-3x+6x-9=0

x(2x-3)+3(2x-3)=0

Now we can factor the (x-3) out of those 2 terms there since they share that common factor:

(x+3)(2x-3)=0

(x+3)(2x-3)=0 implies x+3=0  or 2x-3=0.

So we must solve x+3=0   and 2x-3=0

x+3=0

Subtract 3 on both sides:

x=-3

2x-3=0

Add 3 on both sides:

2x=3

Divide both sides by 2:

x=3/2

The solutions are x=3 or x=-3/2

Answer:

y-7 = 1/2x point slope form

y = 1/2x+7  slope intercept form

Step-by-step explanation:

y=-2x+9

This equation is in the form y= mx +b so the slope is -2

We want a line perpendicular

Take the negative reciprocal

m perpendicular is - (-1/2)

m perpendicular = 1/2

We have a slope of 1/2 and a point.  We can use point slope form

y-y1 = m(x-x1)

y-7 = 1/2(x-0)  point slope form

y-7 = 1/2x

Adding 7 to each side

y-7+7 =1/2x +7

y = 1/2x+7  slope intercept form

A line graph would be the best type of chart to display the growth of Mike's hybrid tomato plant over a three-month period.

The type of chart that would best display the growth of Mike's hybrid tomato plant over a three-month period is a line graph. A line graph is suitable for showing the changes in a variable over time, making it ideal for displaying the growth of the tomato plant.

The garden house takes 4 days, so that means it fills 1/4 of the pool per day ( 1/4 x 4 days = 1)

The larger hose takes 2 days, so this means it fills 1/2 the pool per day ( 1/2 x 2 days = 1)

Using both the garden hose and larger hose means 1/4 + 1/2 = 3/4 of the pool is filled in one day.

Now we need to find X ( the number of days to completely fill the pool.

Multiply the amount per day by the number of days to equal 1 ( 1 pool):

3/4 * x = 1

To solve for x, multiply both sides of the equation by the reciprocal of 3/4, which is 4/3:

x = 4/3 *1

x = 4/3 = 1 and 1/3 days.

It will take 4/3 days or approximately 1.33 days to fill the pool when both hoses are used together.

To solve this problem, we need to understand the rate at which each hose fills the pool and then combine these rates to find the total rate when both hoses are used together.

Let's denote the fill rate of the first garden hose as 1/4 pool per day and the larger hose as 1/2 pool per day.

Using both hoses together, you add their rates to get the combined rate.

So, the combined rate of both hoses is:

1/4 + 1/2  = 3/4 pool per day.

This means three-quarters of the pool is filled in one day with both hoses working together.

To find the time (t) it takes to fill one entire pool, you can set up the equation:

3/4 * t = 1  (quantity is equal to 1 full pool)

t = 4/3 days

Therefore, it will take 4/3 days, or approximately 1.33 days, to fill the pool when both hoses are used together.

I hate rounding.

Let's call the diagonal x.  It's the hypotenuse of the right triangle whose legs are the rectangle sides.

According to the problem we have a length x-4 and a width x-5 and an area

82 = (x-4)(x-5)

82 = x^2 - 9x + 20

0 = x^2 - 9x - 62

That one doesn't seem to factor so we go to the quadratic formula

[tex]x = \frac 1 2(9 \pm \sqrt{9^2-4(62)}) = \frac 1 2(9 \pm \sqrt{329})[/tex]

Only the positive value makes any sense for this problem, so we conclude

[tex]x = \frac 1 2(9 \pm \sqrt{329})[/tex]

That's the exact answer.  Did I mention I hate rounding?  That's about

x = 13.6 meters

Answer: 13.6

----------

It's not clear to me this problem is consistent.  By the Pythagorean Theorem the diagonal satisfies

[tex]x^2 = (x-4)^2 + (x-5)^2[/tex]

which works out to

[tex]x=9 \pm 2\sqrt{10}[/tex]

That's not consistent with the first answer; this problem really has no solution.  Tell your teacher to get better material.

To find the length of the diagonal, we can use the formula for the area of a rectangle and quadratic equation. By substituting the given values and solving for D, we can find the length of the diagonal.

To solve this problem, we can use the formula for the area of a rectangle: length * width = area. Let's represent the length of the rectangle as L, the width as W, and the diagonal as D. According to the problem, L = D - 4 and W = D - 5, and the area is given as 82 m2. We can substitute these values into the formula and solve for D.

L * W = area

(D - 4) * (D - 5) = 82

Expanding and rearranging the equation, we get:

D2 - 9D - 82 = 0

Next, we can solve this quadratic equation either by factoring or by using the quadratic formula. After finding the value of D, we can round it to the nearest tenth to obtain the length of the diagonal.

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Answer: 4x^2 -13x + 14

Step-by-step explanation:

(x^2 - 8x + 5)-(-3x^2 + 5x-9)

Subtract -3x^2 from x^2

(4x^2 - 8x + 5)-(5x-9)

Subtract 5x from -8x

(4x^2 -13x + 5)-(-9)

Subtract 9 from 5

4x^2 -13x + 14

Answer:

No, because the reflection of ABC is not congruent to A’B’C’.

Step-by-step explanation:

we know that

The rule of the reflection across the line y=x is equal to

(x,y) -------> (y,x)

so

A(-6,6) -------> A'(6,-6) ----> is ok

B(-3,3) ------> B'(3,-3) ----> is ok

C(-8,2) ------> C'(8,-2) ----> is not ok ( is not a reflection acros the line y=x)

therefore

The triangles are no t congruent, because the reflection of ABC is not congruent to A’B’C’

[tex]25mn^2 +5mn =5mn(5n+1)[/tex]

The complete factored form of the polynomial 25mn^2 + 5mn is mn(25n + 5).

The given polynomial is 25mn^2 + 5mn. To find the complete factored form, we can factor out the GCF (Greatest Common Factor) from each term, which in this case is mn:

So, the complete factored form of the polynomial 25mn^2 + 5mn is mn(25n + 5).

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Answer:

true

Step-by-step explanation:

yes, that is true.  Parallel lines have equal slopes and perpendicular lines have negative reciprocal slopes (or opposite reciprocals, the "opposite" being the sign).

The statement "If two lines are perpendicular, their slopes are negative reciprocals." is: True

The general form for the equation of a line in slope intercept form is:

y = mx + c

where:

m is slope

c is y-intercept

We know that when two lines are parallel, that their slopes are the same. However, when two lines are perpendicular, then their slopes are  negative reciprocals of each other.

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Answer:

-2B

Step-by-step explanation:

log₉ (1/16)

log₉ (16^-1)

log₉ (4^-2)

Using exponent property of logs:

-2 log₉ (4)

Substituting:

-2B

Answer:

-2B

(I guess this is what you are looking for; didn't need A or C).

Step-by-step explanation:

It seems like to wants us to to find [tex]\log_9(\frac{1}{16})[/tex] in terms of [tex]A,B,C[/tex].

First thing I'm going to do is rewrite  [tex]\log_9(\frac{1}{16})[/tex]  using the quotient rule.

The quotient rule says:

[tex]\log_m(\frac{a}{b})=\log_m(a)-\log_m(b)[/tex]

So that means for our expression we have:

[tex]\log_9(\frac{1}{16})=\log_9(1)-\log_9(16)[/tex]

Second thing I'm going to do is say that [tex]\log_9(1)=0 \text{ since } 9^0=1[/tex].

[tex]\log_9(\frac{1}{16})=\log_9(1)-\log_9(16)[/tex]

[tex]\log_9(\frac{1}{16})=-\log_9(16)[/tex]

Now I know 16 is 4 squared so the third thing I'm going to do is replace 16 with 4^2 with aim to use power rule.

[tex]\log_9(\frac{1}{16})=\log_9(1)-\log_9(16)[/tex]

[tex]\log_9(\frac{1}{16})=-\log_9(16)[/tex]

[tex]\log_9(\frac{1}{16})=-\log_9(4^2)[/tex]

The fourth thing I'm going to is apply the power rule. The power rule say [tex]\log_a(b^x)=x\log_a(b)[/tex]. So I'm applying that now:

[tex]\log_9(\frac{1}{16})=\log_9(1)-\log_9(16)[/tex]

[tex]\log_9(\frac{1}{16})=-\log_9(16)[/tex]

[tex]\log_9(\frac{1}{16})=-2\log_9(4)[/tex]

So we are given that [tex]\log_9(4)[/tex] is [tex]B[/tex]. So this is the last thing I'm going to do is apply that substitution:

[tex]\log_9(\frac{1}{16})=\log_9(1)-\log_9(16)[/tex]

[tex]\log_9(\frac{1}{16})=-\log_9(16)[/tex]

[tex]\log_9(\frac{1}{16})=-2\log_9(4)[/tex]

[tex]\log_9(\frac{1}{16})=-2B[/tex]

Answer:

  c.  Look at the first 7 digits in the table. Let digits from 0 to 3 ...

Step-by-step explanation:

The digits in a random number table are intended to be uniformly distributed, so that each digit has a probability of 0.1. By using combinations of digits you can fairly easily define an outcome that has a probability that is a multiple of 0.1.

Here, you want a probability of 40% = 0.4 = 4×0.1. By defining your outcome as any of 4 digit values, (0 to 3, for example), that outcome will have a probability of 0.4 = 40% when a digit is randomly chosen.

To choose the correct answer here, you only need to understand the above, then choose the answer choice that has an outcome that is defined as 4 of the digits.

___

By looking at 7 digits, you effectively run a simulation in which you do the trial 7 times. Here, you're buying 7 boxes of cereal, so you're interested in 7 trials, each with a probability of success of 40%. This further confirms that the answer choice should include the wording "look at the first 7 digits."

___

More explanation

If the first digit is a number 0-3, it means you got a toy in the first box of cereal.

If the second digit is a number 0-3, it means you got a toy in the second box of cereal.

...

If the 7th digit is a number 0-3, it means you got a toy in the 7th box of cereal.

By counting the number of digits of the first 7 digits that are in the range 0-3, you are effectively counting the number of toys you got in those 7 boxes of cereal.

For example, if the first 7 digits of the table are 9656369*, there is only 1 digit in the range 0-3. That means this purchase of 7 boxes of cereal resulted in 1 toy.

_____

In order to answer Lydia's question, many groups of 7 digits would need to be evaluated, and the ratio of 2-toy purchases to total purchases computed from those results. A trial involving 7000 digits resulted in 268 purchases out of 1000 that had exactly 2 toys, for an experimental probability of 26.8%. Using the binomial distribution, the theoretical probability is about 26.1%--a fairly good match.

___

* this number was generated by random [dot] org

The correct answer would be: C. 5.02 x 10^18

Here's how you solve it!

Since the earthquake registered is 7.4 on the scale let it represent RS=7.4

The reference intensity is 2.0 x 10^11 so let it represent RI= 2.0 x 10^11

Now you need to use the formula.

[tex]RS=log(\frac{I}{I_{r} } )[/tex]

Then we need to plug in the values for the formula

[tex]7.4=log(\frac{I}{2.0 x 10^{11} } )[/tex]

[tex]I=10^{7.4}[/tex] x [tex]2.0[/tex] x [tex]10^{11}[/tex]

[tex]I= 5.02[/tex] x [tex]10^{18}[/tex]

Hope this helps! :3

Answer:

5.02×10^18

I got it right.

Answer:

  r = 3 + sin(θ)

Step-by-step explanation:

The curve extends below the x-axis, so the added constant must be larger than the coefficient of the sine function. There are two choices matching that description.

The extent in the -y (θ=-π/2) direction looks to be about 2/3 of the extent in the +x (θ=0) direction, so we expect the appropriate equation is ...

  r = 3 + sin(θ)

Answer:

47 ft

Step-by-step explanation:

The base of the tower, the guy-wires be secured to the ground will be 73.1 feet.

Trigonometric identities are the functions that include trigonometric functions such as sine, cosine, tangents, secant, and, cot.

Sinθ = a/b

Where θ = angle to the horizontal, a = Height of the tower, and b = length of the wire.

Also, b = a/sinθ

Given: a = 56 feet, θ = 50°

Substitute these values into equation;

b = 56 /sin 50°

b = 56/0.766

b = 73.1 feet

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Let x = amount of 16% alloy, and y = amount of 30% alloy he should use.

Mixing the alloys will result in a compound weighing x + y = 100 grams.

For each gram of the 16% alloy used, 0.16 gram of titanium is contributed; similarly, for each gram of the 30% alloy used, there's a contribution of 0.3 gram. He wants to end up with an alloy of 23% titanium, or 23 grams (23% of 100), so that 0.16x + 0.3y = 23.

Solve the system:

[tex]x+y=100\implies y=100-x[/tex]

[tex]0.16x+0.3y=23\implies 0.16x+0.3(100-x)=23\implies7=0.14x[/tex]

[tex]\implies\boxed{x=50}\implies\boxed{y=50}[/tex]

Answer:

  see below

Step-by-step explanation:

The applicable rules of logarithms are ...

  log(a^b) = b·log(a)

  log(a/b) = log(a) -log(b)

___

The expression can be rewritten as ...

[tex]\log{\dfrac{\sqrt[3]{2-x}}{3x}}=\log{\sqrt[3]{2-x}}-\log{3x}=\dfrac{1}{3}\log{(2-x)}-\log{(3x)}[/tex]

The expression can be written as a sum or difference as:

[tex]$$\boxed{\frac{1}{3} \log(2-x) - \log(3x)}$$[/tex]

To write the expression [tex]$\log(\frac{\sqrt[3]{2-x}}{3x})$[/tex] as a sum or difference, we can use the following logarithmic identities:

[tex]$\log(a/b) = \log(a) - \log(b)$[/tex]

[tex]$\log(a^n) = n \log(a)$[/tex]

First, we can use the first identity to split the logarithm of the fraction into two logarithms:

[tex]$$\log(\frac{\sqrt[3]{2-x}}{3x}) = \log(\sqrt[3]{2-x}) - \log(3x)$$[/tex]

Next, we can use the second identity to expand the logarithm of the cube root:

[tex]$$\log(\sqrt[3]{2-x}) = \log((2-x)^{1/3}) = \frac{1}{3} \log(2-x)$$[/tex]

Substituting this back into the first expression, we get:

[tex]$$\log(\frac{\sqrt[3]{2-x}}{3x}) = \frac{1}{3} \log(2-x) - \log(3x)$$[/tex]

Therefore, the expression can be written as a sum or difference as:

[tex]$$\boxed{\frac{1}{3} \log(2-x) - \log(3x)}$$[/tex]

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Answer: The following statement is true about indexes and scales: They rank-order the units of analysis in terms of specific variables.

Indexes provides with a way to make a complex measure that iterate consequence for multiple rank-ordered related questions or statements.  

Scale is a type of complex measurement that is combined of several items that have a empirical structure among them.

Answer: 25%

Step-by-step explanation:

Given : A rectangular dartboard has an area of 648 square centimeters.

The triangular part of the dartboard has an area of 162 square centimeters.

If we assume that the dart lands in the rectangle, then the probability that it lands inside the triangle will be :-

[tex]\dfrac{\text{Area of triangular part}}{\text{Area of rectangular dart board}}\\\\=\dfrac{162}{648}=0.25[/tex]

In percent, [tex]0.25\times100=25\%[/tex]

Hence, the required probability = 25%

Answer:

Probability = 0.2381

Step-by-step explanation:

A security alarm requires a four digit code by using 0 - 9 and the digits cannot be repeated.

First we calculate how many codes can be made.

Combination = [tex]^{n}p_{r}[/tex]

Where n = 10 and r = 4

[tex]^{n}p_{r}[/tex] = [tex]\frac{10!}{(10-4)!}[/tex]

         = [tex]\frac{10!}{6!}[/tex]

         = 10 × 9 × 8 × 7

        = 5,040 combinations.

Now we have to find the probability that the code contains only odd numbers. So in 0-9 the odd numbers are = 1, 3, 5, 7, 9

There are 5 odd numbers and we have to make a code of 4 numbers.

Therefore, On first place there are 5 options and in second place 4 options, in third place there are 3 options and in fourth place we have only 2 options.

5 × 4 × 3 × 2  = 120 combinations.

Total combinations of odd numbers are 120.

Then the probability that the code contains only odd numbers is

[tex]p=\frac{120}{5040}[/tex] = 0.0238095 ≈  0.02381

Probability = 0.2381

The approximate probability that the code contains only odd numbers is 0.0238.

The probability of the code containing only odd numbers can be found by determining the number of possible combinations of four odd digits out of the total number of possible combinations of four digits.

To calculate this, we first count the number of odd digits from 0 to 9, which is 5. Then, we determine the number of combinations of 4 digits that can be formed from the 5 odd digits, which is 5C4 or 5.

The total number of possible combinations of four digits without repetition is 10C4 or 210. Therefore, the probability of the code containing only odd numbers is 5/210 or approximately 0.0238.

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Answer:

68% of the pizzas are delivered between 24 and 30 minutes

Step-by-step explanation:

First we calculate the Z-scores

We know the mean and the standard deviation.

The mean is:

[tex]\mu=27[/tex]

The standard deviation is:

[tex]\sigma=3[/tex]

The Z-score formula is:

[tex]Z = \frac{x-\mu}{\sigma}[/tex]

For [tex]x=24[/tex] the Z-score is:

[tex]Z_{24}=\frac{24-27}{3}=-1[/tex]

For [tex]x=30[/tex] the Z-score is:

[tex]Z_{30}=\frac{30-27}{3}=1[/tex]

Then we look for the percentage of the data that is between [tex]-1 <Z <1[/tex] deviations from the mean.

According to the empirical rule 68% of the data is less than 1 standard deviations of the mean.  This means that 68% of the pizzas are delivered between 24 and 30 minutes

Using the empirical rule, approximately 68% of pizzas are delivered between 24 and 30 minutes, as this range falls within one standard deviation of the mean delivery time, which is normal distribution practice.

The question is about the percentage of pizzas delivered within a certain time frame, assuming a normal distribution of delivery times. The empirical rule, also known as the 68-95-99.7 rule, states that for a normal distribution:

In this case, the mean delivery time is 27 minutes and the standard deviation is 3 minutes. Thus, using the empirical rule, about 68% of pizzas should be delivered between 24 minutes (27 - 3) and 30 minutes (27 + 3).

Answer:

y = 2

Step-by-step explanation:

I am assuming there was some info that got left out of this that states somewhere along the line that this is right triangle inscribed in a circle or something like that.  That means that angle R is a right angle.  Therefore,

53y - 16 = 90 so

53y = 106 and

y = 2

The value of y is 2.

Using the Inscribed angle theorem, in a semi-circle, the inscribed arc measures 180° for which inscribed angle in semi-circle will be half of 180° i.e. the inscribed angle in semi-circle will be right-angle i.e. 90°.

Here As PQ crosses the center of the circle M. so PQ ia the diameter.

the measure of the arc PRQ is 180°.

then using inscribed angle theorem, ∠PRQ will be half of 180°.

So, ∠PRQ =90°

Given, ∠PRQ= 53y-16°

⇒90°=53y-16°

⇒53y=90°+16°=116°

⇒y=116°/53°

⇒y=2

Therefore the value of y is 2.

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Answer:

  11, 7, 44

Step-by-step explanation:

Let x represent the first number. Then the second is (x-4) and the third is (4x). Their sum is ...

  x +(x -4) +(4x) = 62

  6x = 66 . . . . . . . . . . . . add 4, collect terms

  x = 11 . . . . . . . . . divide by 6

  x -4 = 7 . . . . . . . find the second number

  4x = 44 . . . . . . . find the third number

The three numbers are 11, 7, and 44.

Answer: 1088 :)

Step-by-step explanation:

The distance from San Juan to Miami is 1088 miles.

A triangle is a polygon with three sides. The sum of angles in a triangle is 180 degrees. Types of triangles include: scalene, right triangle, isosceles and equilateral triangle.

The law of sine would be used to determine the distance.

(a / sin a) = (b / sin b) = (c / sin c)

(a / sin 63) = [960 / (180 - 62 - 63)]

a / sin 63 = 960 / sin 54

a = (sin 63 x 960) / sin 54

a = 1088 miles

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Answer:

Karen spent $10, Maggie spent $30 and Jasmine spent $20

Step-by-step explanation:

let's call Karen's money spent 'x'

Maggie therefore is 3x

And Jasmine is 2x

6x=$60

x=$10

Now we substitute this back in

so Karen spent $10

Maggie spent $30

And Jasmine spent $20

Answer:

[tex]2^53^5[/tex]

Step-by-step explanation:

So if we compare [tex]2^5 \cdot 3^7[/tex] to [tex]2^7 \cdot 3^5[/tex], we should see the most amount of factors of 2 that they have in common is 5 and the most amount of factors of 3 that they have in common is 5.

If you aren't sure on the number of factors of 2 and 3 they have in common you could write it all out:

[tex]2^53^7=(2)(2)(2)(2)(2)\cdot\text{ }(3)(3)(3)(3)(3)(3)(3[/tex]

[tex]2^73^5=(2)(2)(2)(2)(2)(2)(2)\text{ }(3)(3)(3)(3)(3)[/tex]

So if I were able to circle each pair of 2's they had in common I would circle 5 pairs.

If were able to circle each pair of 3's they had in common I would circle 5 pairs.