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Thank you David Aziz and Honga.  

You are being too obtuse for me I am afraid David.

Here the factor 5y², on the right, has an additional prime. One side of the equation has an odd number and the other has an even number. Composite numbers cannot have both an even and odd number of prime factors because all composite numbers have a unique prime factorization.

$$x^2=5y^2$$

additional prime to what?

One side has an odd number do you mean of factors?  

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Umm.  Is this what you are saying?

x^2 has an odd number of factors where as 5y^2 has an even number of factors.

This is impossible because they are the same number and must have the same number of factors

Therefore there are no integers x and y that meet this criterion.

Therefore sqrt5 is not rational

Therefore sqrt5 is irrational.

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The equation is set up as attempt to balance, with a constant, the ratio of two hypothetical integers. The constant extends, in diminishing value, into infinity. In other words, no integers exist that define this ratio.

It is the middle ground of light and shadow.

~~D~~

We know the last two digits can only result in a number lower than 100, we know multiplication between two large numbers is basically an accumalation of simple multiplications. In this case we have (1000+100+20+2)*(1000+100+20+2), since we can only allow numbers smaller than 100 we can drop 1000 and 100. We get (20+2)*(20+2) which results in 400+40+40+4, 400 is greater than 100, so 40+40+4=84. 84 are the last two digits. Hope this helps

We have two unknown variables x=amount of students, y=needed money.

We know x*25+210=y and x*32-84=y, hence x*25+210=x*32-84, solve for x. 

Now x is known we can calculate y with any of the two given formulas.

Good luck.

I don't understand David,  could you explain a little more please?

The object falls 5 meters changing E(potential) to E(kinetic), then it goes back up changing E(kinetic) to E(potential). Still it hasn't reached its previous height so there will still be E(kinetic) left. The difference in height from beginning to end is 3 meters: 5 meters (beginning)- 2 meters (end)=3 meters.

Test this by assuming √5 is rational. If it is rational then it can be expressed in lowest terms as x/y.

$$\ Let \ (\frac{x}{y})^2 = 5 \\\
\hspace{20pt}\ x^2 / y^2 \ = 5 \\\
\hspace{20pt}\ x^2 = 5y^2 \\$$

Here the factor 5y², on the right, has an additional prime. One side of the equation has an odd number and the other has an even number. Composite numbers cannot have both an even and odd number of prime factors because all composite numbers have a unique prime factorization.

These two factors “compete” for balance into infinitesimal infinity.

This is an irrational number, a surd. As CPhill might say, something that cannot be heard or uttered.

~~D~~

$$\frac{6\times10^{-6}}{-3\times 10^0}\\\\
=\frac{6\times10^{-6}}{-3\times 1}\\\\
=-2\times10^{-6}\\\\$$

Use the site calculator on the home page. 

Yes the square root of 5 is an irrational number.

Here is a proof (by contradiction)

$$I will assume that $\sqrt5$ is a rational number.\\
This means that it can be expressed as $\dfrac{a}{b}$ where $a\; and \;b$ \\are relatively prime integers (that means they have no common factors other than 1)\\
let $\sqrt5=\frac{a}{b}}$ Where a and b are relatively prime. \\
$\dfrac{a^2}{b^2}=5$\\\\
$a^2=5b^2$\\\\
Hence $a^2$ is a multiple of 5\\
Hence $a$ is a multiple of 5\\
so $a=5c$ for some integer c\\
$5b^2=a^2=(5c)^2=25c^2$\\
Dividing through by 5 this means\\
$b^2=5c^2$ \\
Hence $b^2$ is a multiple of 5\\
Hence $b$ is a multiple of 5\\\\$$

$$Now we have found that a and b must both be multiples of 5\\
This is a contradiction since we started with the premise that there existed an a and b such that a and b are relatively prime.\\
Therefore $\sqrt5$ cannot be a rational number.\\
Therefore $\sqrt5$ is an irrational number.$$

Animoz has asked for more help which I am happy to provide. (Next time could you please ask here on the forum.  Thank you.   

$$3(s+8) \\
= 3*s + 3*8 \\
= 3s+24\\\\
-4(2s-5)\\
=-4*2s -4*-5\\
=-8s--20\\
=-8s+20\\\\
so\\
3(s+8)-4(2s-5)\\
=3s+24-8s+20\\
=3s-8s+24+20\\
=-5s+48$$

Why don't you try these by yourself,  post the answers and I will see if you get them right.

If you choose to do this then please post the question with the working and the answer thank you.

1) 6(x+9)

2) 12(m-4)

3)  -3(t+5)

4)  -6x(x+3y)

5)  5(x-3)-2(m-4)

Maybe it is thousands of digits over thousands of digits, then terminates.

How do we know we cannot write it that way?

You start by expanding the brackets independantly BUT remember the minus sign belongs with the 4.

Give it a go and I'll check what you've done.

You start by finding the distance between the 2. That is the radius.

Then plug the centre point and the radius into the formula.

Magic, its done.

(X-h)^2+(y-k)^2=r^2

Thank you Alan, Aziz and David :)

2,500,712 / 1,263 = 1979 condos and 1235.00$ left over.

(6 x 10^-6) / (-3 x 10^0)

= -2 x 10^-6

Like this?

(23456 * 654)^-23

= 15340224^-23

= 1.53 x 10^-30

Yep, if you can't write it as a fraction with only integers, it's irrational.

Now its very clear .. thank you david 

 . . . what about 6 , 7 and 8 . . .

Yes, they are correct. I double checked them.

~~D~~

but what about 6 , 7 and 8 

are there correct 

fine .. my last question .. why you put the heights 3 m !