Melody

Your question doesn't make sense, (not to me anyway)  I think something is missing.

Hi Teja2,

It is nice to meet you.

It is easier for me to answer myself than it is for me to check your answer.

So that is the only reason I am answering here

\(4^{10}*8^{20}*16^{30}\\ =2^{20}*2^{60}*2^{120}\\ =2^{200}\)

there you go, if you got it wrong then I did too.   

Give yourself a point!

\(\displaystyle\left(\frac{1}{a}+\frac{1}{b}\right)\left(\frac{1}{b}-\frac{1}{a}\right)\\ \displaystyle\left(\frac{1}{b}+\frac{1}{a}\right)\left(\frac{1}{b}-\frac{1}{a}\right)\\ = \displaystyle\left(\frac{1}{b}\right)^2-\left(\frac{1}{a}\right)^2\\\)

For the smallest value you want the absolute value of  b to be as big as possible,  and the absolute value of a to be as small as possible but neither can be 0

b=3,  a= -1

1/9  -  1 = -8/9

What a clever little Claire.

You get all your homework done for you and you don't have to learn a thing.

Let the sequence be      a-d,  a,  a+d

So 

\(x^3-6x^2-13x-k=(x-(a-d))\;x\;(x-(a+d))\\ x^3-6x^2-13x-k=x\;((x-a)+d))\;(x-a)-d))\\ x^3-6x^2-13x-k=x\;((x-a)^2-d^2)\\ etc\)

Simplify it down and equaet coefficients.

Answered.

Now, I expect a polite response from you Jmaster10.

Thank you would be nice but more than just thank you would be better.  

I mean if you want to correct something or ask something, that would be good.

I have never studied modulo mathematics except by looking at the odd question on this forum.

My setting out will be very poor but my logic should be ok.

\((ab)(ab)^{-1}\equiv 1\mod n\\ so\\ 2ab\equiv 1\;\;\mod n\\ 2\equiv \frac{1}{ab}\;\;\mod n\\~\\ \frac{1}{a+b}(\frac{1}{a}+\frac{1}{b})\mod n\\~\\ \equiv \frac{1}{a+b}(\frac{a+b}{ab})\mod n\\~\\ \equiv \frac{a+b}{ab(a+b)}\mod n\\~\\ \equiv \frac{1}{ab} \mod n\\~\\ \equiv 2 \mod n\\~\\ \)

Well you can say    

John is playing.                       "John" is a noun.

He is playing                          "He" is a pronoun

A pronoun can be used in place of nouns.

Pronouns are words like     he,she, it, him, her, ..

I just found this.

It is the same for numerals and pronumerals.   Pronumerals (letters) can be used in place of a number.

for example in the maths expression     5y        y is a pronumeral.