AzizHusain

40: 1,2,4,5,8,10,20,40

56: 1,2,4,7,8,14,28,56

8 would be the greatest common factor.

15 - (-25)

Two negatives make a plus, +.

So, 15 - (-25) becomes 15 + (+25) = 15 + 25 = 40.

Are you trying to find a function that satisfies this condition?

If so, f(x,y) = x^3 - x.

Check: f(6,4) = 6^3 - 6 = 216 - 6 = 210.

Y=7-x

x-y=-3

For this one, use substitution -->

x - y = - 3 --> x - (7 - x) = -3 --> x - 7 + x = -3 -->

2x - 7 = -3

2x = 4

x = 2

y = 7 - 2 = 5

x = 2; y = 5.

************************************************************************************************************************************

X+2y=11

x+2y=14

For this one, use elimination -->

Subtract both equations: This yields 0 = -3 [x - x cancel out & 2y - 2y cancels out as well] which does not make sense...therefore, this system has no solution.

3.) The product of 3 less than twice x and 2 more than the quantity 3 times x...we can think of this as:

Product of (2x - 3) and (3x + 2)...3 less than twice x is the same as twice x - 3 or 2x - 3 & 2 more than the quantity 3 times x is the same as 3 times x plus 2 = 3x + 2 -->

(2x - 3)*(3x + 2)

Now, FOIL: first, outside, inside, & last -->

first: 6x^2

outside: 4x

inside: -9x

last: -6

Add all the terms: 6x^2 + 4x - 9x - 6...lastly, simplify by collecting "like" terms -->

D, 6x^2 - 5x - 6.

4.) [(5x(x+1)^2) / (x^3(x^3 + 9x))] * [(x^2 + 9) / (x + 1)]

Notice, we can get rid of the (x+1)'s leaving us with:

5x(x+1) / x^3(x^3 + 9x) * [(x^2 + 9)]

5x(x+1) / x^3*x(x^2 + 9) * [(x^2 + 9)]

5x(x+1) / x^4(x^2 + 9) * [(x^2 + 9)]

5x(x+1) / x^4

Get rid of the x in 5x by taking away 1 x from x^4:

C, 5(x+1) / x^3.

64) (x^2/x-11)^1/3

we can simplify this by doing the following:

(x^2)^1/3 / (x-11)^1/3

x^(2/3) / (x-11)^(1/3).

66) (x^-2/3/y^-1/3)^15

Since we have negative exponents, we can change their positions as follows:

[y^(1/3)/x^(2/3)]^15

Now, we can further simplify by doing:

[y^(1/3)^15] / [x^(2/3)^15]

y^5 / x^30/3

y^5 / x^10

$${\frac{{{\mathtt{y}}}^{{\mathtt{5}}}}{{{\mathtt{x}}}^{{\mathtt{10}}}}}$$.

72) f(x)=3x, g(x)=x^2

f(g(x))...we basically input whatever g(x) is into any x in f(x)

f(x^2) = 3x^2.

g(f(x))...we basically input whatever f(x) is into any x in g(x)

g(3x) = (3x)^2 = 9x^2.

74) f(x)=3x^2+2, g(x)=2x

f(g(x)) = f(2x) = 3[(2x)^2]+2 = 3(4x^2)+2 = 12x^2 + 2.

g(f(x)) = g(3x^2+2) = 2(3x^2+2) = 6x^2+4.

f(x)=2x+5 & g(x)=x^2-3x+2

56) -3f(x)*g(x)

-3(2x+5)*(x^2-3x+2)

(-6x-15)*(x^2-3x+2)

-6x^3+18x^2-12x-15x^2+45x-30

-6x^3+3x^2+33x-30

There are no restrictions to the x values as seen above...therefore, the domain is all real numbers.

58) 5f(x)/g(x)

5(2x+5)/(x^2-3x+2)

10x+25/(x^2-3x+2)

Here, note that we have a polynomial in the denominator...therefore, we have to consider what values make the denominator equal to 0, which will ultimately produce something that is undefined...

(x^2-3x+2) = (x-2)(x-1)...set both of these equal to 0 -->

(x-2) = 0 & (x-1) = 0

x = 2 and 1...now we know that x cannot equal 2 and 1...so our domain is restricted to all real numbers except for x = 2 & x = 1.

Life is meant to be something mysterious! We just explore new places, share moments with others, store our own memories, keep ourselves busy, and lots more!

However, people sometimes say "You are wasting life." But, life is just so complex so as long as we are doing something, we are not wasting it...but then we get into all the different perspectives....

Also, we have currencies, jobs, travel, and the feeling of interconnectedness which binds us together....then, we have hierarchies where people rule others and people behave in specific manners so as to avoid getting punished and sent to jail...so, we get some laws/morality rules enforced...

Inventions and constructing objects as well as math functions, numbers, and the various subjects we study in school are all part of life. 

An interesting example: freedom = free doom [hidden in one word]...

Life is just all over the place. We don't know what to expect, but, sometimes we can predict what will happen. Other times, we just have to go with the flow and do something creative to keep ourselves on a path that makes us productive and keeps us from questioning other aspects of life or something.....lots more to comment on but the question about the meaning of life is interesting...what can we learn and what more is there...will we ever find the answer? In 300,000 years????? Or will climate change's deadly consequences result in our demise and what will sprout after that?

$${{\mathtt{22}}}^{{\mathtt{2}}}$$ = 22*22 = 484.