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3 years ago

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Solve the equation (linear equation)

D%29%5E%7B3x%7D" id="TexFormula1" title="8^{2x+7} = (\frac{1}{32})^{3x}" alt="8^{2x+7} = (\frac{1}{32})^{3x}" align="absmiddle" class="latex-formula">

1 answer:

Minchanka [31]3 years ago

8 0

Answer: $x=-1$

Step-by-step explanation:

By the negative exponent rule, you have that:

$(\frac{1}{a})^n=a^{-n}$

By the exponents properties, you know that:

$(m^n)^l=m^{(nl)}$

Therefore, you can rewrite the left side of the equation has following:

$(\frac{1}{8})^{-(2x+7)}=(\frac{1}{32})^{3x}$

Descompose 32 and 8 into its prime factors:

$32=2*2*2*2*2=2^5\\8=2*2*2=2^3$

Rewrite:

$(\frac{1}{2^3})^{-(2x+7)}=(\frac{1}{2^5})^{3x}$

Then:

$(\frac{1}{2})^{-3(2x+7)}=(\frac{1}{2})^{5(3x)}$

As the base are equal, then:

$-3(2x+7)=5(3x)$

Solve for x:

$-6x-21=15x\\-21=15x+6x\\-21=21x\\x=-1$

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WHAT IS THE REMAINDER WHEN <img src="https://tex.z-dn.net/?f=32%5E%7B37%5E%7B32%7D%20%7D" id="TexFormula1" title="32^{37^{32} }"

Feliz [49]

Recall Euler's theorem: if $\gcd(a,n) = 1$ , then

$a^{\phi(n)} \equiv 1 \pmod n$

where $\phi$ is Euler's totient function.

We have $\gcd(9,32) = 1$ - in fact, $\gcd(9,32^k)=1$ for any $k\in\Bbb N$ since $9=3^2$ and $32=2^5$ share no common divisors - as well as $\phi(9) = 6$ .

Now,

$37^{32} = (1 + 36)^{32} \\\\ ~~~~~~~~ = 1 + 36c_1 + 36^2c_2 + 36^3c_3+\cdots+36^{32}c_{32} \\\\ ~~~~~~~~ = 1 + 6 \left(6c_1 + 6^3c_2 + 6^5c_3 + \cdots + 6^{63}c_{32}\right) \\\\ \implies 32^{37^{32}} = 32^{1 + 6(\cdots)} = 32\cdot\left(32^{(\cdots)}\right)^6$

where the $c_i$ are positive integer coefficients from the binomial expansion. By Euler's theorem,

$\left(32^{(\cdots)\right)^6 \equiv 1 \pmod9$

so that

$32^{37^{32}} \equiv 32\cdot1 \equiv \boxed{5} \pmod9$

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2 years ago

An Internet and cable-television supplier surveyed a random sample of their customers. The results are shown in the table.

IrinaVladis [17]

For the statement "....A 4-column table with 3 rows. The first column has no label with entries internet, cable television, or total. The second column is labeled satisfied with entries ..." statement about the two-way frequency table that is true is About one-fourth of the cable-television customers are not satisfied. Option D.

This is further explained below.

<h3>What is a two-way frequency table?</h3>

Generally, A two-way table may be used to show the frequency of occurrence of two categories. A row represents a single category, whereas a column represents a different category.

In conclusion, There are 3,141 customers in total, of which 2,032 are internet customers and 1,109 are cable customers.

Read more about the two-way frequency table

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7 0

1 year ago

The table below shows the number of marbles of different colors in a bag:

noname [10]

There are a total of 5+8+2=15 marbles. There's a 5/15 chance of drawing a red marble in the first trial. Then, there are 4 red marbles and 14 total marbles left, so afterwards, there's a probability of 4/14 that the second marble is red. The probability overall is found by multiplying these together, so the first answer is the best.

There are 2+6+7=15 cakes. There's a 6/15 chance that the first cake selected will be a pineapple cake, and then a 2/14 chance that the second cake selected will be a strawberry cake, by the same logic as above. This is equal to 12/210. The first answer is the best option.

3 0

3 years ago

Read 2 more answers

Give the equation of the line that is perpendicular to y= (1/2)x -1 and passes through the point (3,4)

nexus9112 [7]

Answer: y = -2x + 10

Step-by-step explanation:

Two lines are said to be perpendicular if the product of their slope = -1, that is , if $m_{1}$ is the slope of the first line and $m_{2}$ is the slope of the second line , if the two lines are perpendicular , then $m_{1}$ $m_{2}$ = -1

The equation of the line given :

y = 1/2x - 1

comparing with the equation of line in slope intercept form

y = mx + c , where m is the slope and c is the y - intercept , it implies that

$m_{1}$ = 1/2

Therefore :

$m_{2}$ = -2

To find the equation of the line with slope -2 and passes through the point ( 3 , 4 ) we will use the formula

$y-y_{1}$ = m ( $x-x_{1}$ )

y - 4 = -2 ( x - 3 )

y - 4 = -2x + 6

y = -2x + 6 + 4

y = -2x + 10

7 0

3 years ago

Many utility companies promote energy conservation by offering discount rates to consumers who keep their energy usage below cer

aleksley [76]

Answer:

a. 0.1681 = 16.81% probability that all five qualify for the favorable rate.

b. 0.5283 = 52.83% probability that at least four qualify for the favorable rates

Step-by-step explanation:

For each Puerto Rico resident, there are only two possible outcomes. Either they qualify for discounted rates, or they do not. The probability of a person in the sample qualifying for discounted rates is independent of any other person in the sample. This means that the binomial probability distribution is used to solve this question.

Binomial probability distribution

The binomial probability is the probability of exactly x successes on n repeated trials, and X can only have two outcomes.

$P(X = x) = C_{n,x}.p^{x}.(1-p)^{n-x}$

In which $C_{n,x}$ is the number of different combinations of x objects from a set of n elements, given by the following formula.

$C_{n,x} = \frac{n!}{x!(n-x)!}$

And p is the probability of X happening.

70% of the island residents of Puerto Rico have reduced their electricity usage sufficiently to qualify for discounted rates.

This means that $p = 0.7$

Five residential subscribers are randomly selected from San Juan, Puerto Rico

This means that $n = 5$

a. All five qualify for the favorable rate

This is P(X = 5). So

$P(X = x) = C_{n,x}.p^{x}.(1-p)^{n-x}$

$P(X = 5) = C_{5,5}.(0.7)^{5}.(0.3)^{0} = 0.1681$

0.1681 = 16.81% probability that all five qualify for the favorable rate.

b. At least four qualify for the favorable rates

This is

$P(X \geq 4) = P(X = 4) + P(X = 5)$

So

$P(X = x) = C_{n,x}.p^{x}.(1-p)^{n-x}$

$P(X = 4) = C_{5,4}.(0.7)^{4}.(0.3)^{1} = 0.3602$

$P(X = 5) = C_{5,5}.(0.7)^{5}.(0.3)^{0} = 0.1681$

$P(X \geq 4) = P(X = 4) + P(X = 5) = 0.3602 + 0.1681 = 0.5283$

0.5283 = 52.83% probability that at least four qualify for the favorable rates

5 0

3 years ago