hectictar

We could do it pretty quickly by applying the exponent rules....but if that's confusing or if you're not super comfortable using exponent rules, we can go through the first part like this:

First let's just look at  (pq³)³  :

     \({(pq^3)^3}\ =\ (pq^3)(pq^3)(pq^3)\)

     Imagine replacing every q³ with qqq. How many p's would there be? 3. How many q's would there be? 9. So...

     \( {(pq^3)^3}\ =\ p^3q^9\)

Next let's look at  (4p²q)²  :

     \((4p^2q)^2\ =\ (4p^2q)(4p^2q)\)

     In this case, how many p's are there? 4. How many q's are there? 2. How many 4's are there? 2. So...

     \({(4p^2q)^2}\ =\ 16p^4q^2\)

Now let's look at  (2pq²)³  :

     \((2pq^2)^3\ =\ (2pq^2) (2pq^2) (2pq^2)\)

     Again let's ask how many p's, q's , and 2's there are to get that...

     \( {(2pq^2)^3}\ =\ 8p^3q^6\)

And so...

\(\dfrac{(pq^3)^3(4p^2q)^2}{(2pq^2)^3}\ =\ \dfrac{(p^3q^9)(16p^4q^2)}{8p^3q^6}\)

There are 3 p's in the first set of parenthesees in the numerator and 4 p's in the second. That makes 7 p's total in the numerator. There are 9 q's in the first set of parenthesees in the numerator and 2 q's in the second. That makes 11 q's total in the numerator. I'm not going to write explanations for these next steps... so feel free to ask if anything confuses you!!

\(\dfrac{(pq^3)^3(4p^2q)^2}{(2pq^2)^3}\ =\ \dfrac{(p^3q^9)(16p^4q^2)}{8p^3q^6}\\~\\ \phantom{\dfrac{(pq^3)^3(4p^2q)^2}{(2pq^2)^3}}\ =\ \dfrac{16p^7q^{11}}{8p^3q^6} \\~\\ \phantom{\dfrac{(pq^3)^3(4p^2q)^2}{(2pq^2)^3}}\ =\ \dfrac{16}{8}\cdot\dfrac{p^7}{p^3}\cdot\dfrac{q^{11}}{q^6}\\~\\ \phantom{\dfrac{(pq^3)^3(4p^2q)^2}{(2pq^2)^3}}\ =\ 2\cdot p^4\cdot q^5\\~\\ \phantom{\dfrac{(pq^3)^3(4p^2q)^2}{(2pq^2)^3}}\ =\ 2p^4q^5\)

Anytime you don't know how to handle an exponent, it can help to rewrite it as repeated multiplication....I used to always do this until I became super familiar with exponent rules...and I still have to do it sometimes!!

Hope this helps!

\(\phantom{=\quad}\dfrac{x^5y^3}{xy^6}\\~\\ {=\quad}\dfrac{x\ x\ x\ x\ x\ \cdot\ y\ y\ y}{x\ \cdot\ y\ y\ y\ y\ y\ y}\\~\\ {=\quad}\dfrac{x\ x\ x\ x\ \not x\ \cdot\ \not y\ \not y\ \not y}{\not x\ \cdot\ y\ y\ y\ \not y\ \not y\ \not y}\\~\\ {=\quad}\dfrac{x\ x\ x\ x}{y\ y\ y}\\~\\ {=\quad}\dfrac{x^4}{y^3} \)

P.S. I like your profile picture

The primes less than 50 are:  2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47

We can find the answer using trial and error..

43² - 19²  =  1488

(First I tried  47² - 2²  then  47² - 3²  etc.  until 47² - 29²

Then I tried  43² - 23²  and then  43² - 19² which worked)

Haha thanks!!

Thanks! xD

b² - a² > 0   so  b² > a²  and so since a, b are positive,  b > a

c² - b² > 0   so  c² > b²  and so since c, b are positive,   c > b

So   a < b < c
 

And since a, b, and c are consecutive positive odd integers:

b  =  a + 2

c  =  a + 4

Now we can find  a  using this equation:

b² - a²  =  344

(a + 2)²  -  a²  =  344

(a + 2)(a + 2) - a²  =  344

a² + 4a + 4 - a²  =  344

4a + 4  =  344

4a  =  340

a  =  85

Since  a = 85,  b = 87  and  c = 89

And so   c² - b²  =  89² - 87²  =  352

Solve  3x + y  =  15  for  y  to get  y = 15 - 3x

Let

z(x)  =  x² + y²  =  x² + (15 - 3x)²

So we want to find what value of  x  minimizes  z(x)

First let's find values of  x  which make  z'(x) = 0

z'(x)   =   2x + 2(15 - 3x)(-3)   =   2x - 90 + 18x   =   20x - 90  =  0  ⇒  x = 4.5

Now let's check whether the graph is concave up or down when  x = 4.5  using the second derivative test:

z''(x)  =  20   ⇒   z''(4.5)  =  20  >  0   so the graph is concave up

Since   z'(4.5)  =  0   and   z''(4.5) > 0  ,  we can say a min occurs when x = 4.5

When   x = 4.5,   y  =  15 - 3(4.5)  =  1.5

And   x² + y²   =   (4.5)² + (1.5)²   =   22.5

We can draw a right triangle where the hypotenuse is the slant height and the legs are the radius and the height of the cone. Then if we can find the slant height and the radius of the base of the cone, we can use the Pythagorean theorem to find the height.

The slant height of the cone is the same as the radius of the sector, which is 18

Now we need to find the radius of the base of the cone.

The circumference of the base is the same as the length of the sector, which we can find like this:

\(\frac{\text{arc length of sector}}{\text{circumference of circle}}=\frac{288^\circ}{360^\circ} \\~\\ \frac{\text{arc length of sector}}{2\pi(\text{radius of circle})}=\frac{288^\circ}{360^\circ} \\~\\ \frac{\text{arc length of sector}}{2\pi(18)}=\frac{288^\circ}{360^\circ} \\~\\ \text{arc length of sector}=\frac{288^\circ}{360^\circ} \cdot 2\pi(18)\\~\\ \text{arc length of sector}=28.8\pi\)   (The circle here is the circle from which the sector is taken)

And so

circumference of base   =   28.8 pi

radius of base   =   circumference of base  /  (2 pi)   =   28.8 pi / (2 pi)   =   14.4

Now that we know the radius of the base of the cone and we know the slant height of the cone, we can use the Pythagorean theorem to find the height of the cone.

(radius of base)² + (height)²   =   (slant height)²

                                                                               Plug in what we know...

14.4²  +  ( height )²   =   18²

                                                        Subtract  14.4²  from both sides

( height )²   =   18² - 14.4²

                                                        Simplify the right side

( height )²   =   116.64

                                                        Take the positive square root of both sides

height   =   10.8

Let  a  and  b  be the roots. Then we can make these two equations:

ab  =  18     and     a  =  2b

Substituting  2b  in for  a  into the first equation, we get:

(2b)b  =  18

2b²  =  18

b²   =   9

b   =   ± 3

Which means.....

a   =   2 * (± 3)

a   =   ± 6

So  3  and  6  is one possible solution,  and the other possible solution is  -3  and  -6  

We want to find the value of  t  when the height is  22.  So let's plug in  22  for  h  and solve for  t .

22  =  -t² + 9t + 2

                                   Subtract  22  from both sides of the equation

0  =  -t² +9t - 20

                                   Multiply both sides of the equation by  -1

0  =  t² - 9t + 20

                                   Now we want to factor the right side. What two numbers add to  -9  and multiply to  20 ?  -4  and  -5

0  =  (t - 4)(t - 5)

                                   Set each factor equal to zero and solve for  t

t - 4 = 0      or      t - 5 = 0

 t  =  4                   t = 5

The rock first reaches  22  meters at time 4 seconds.

Then it goes above 22 meters and comes back down to 22 meters at time 5 seconds.

So it stays above 22 meters for  5 seconds - 4 seconds  =  1 second