Answer

Verified

447.9k+ views

**Hint**: General binomial expansion of \[{{(a+b)}^{n}}={}^{n}{{c}_{r}}{{(a)}^{n-r}}{{(b)}^{r}}\] where

\[{}^{n}{{c}_{r}}=\dfrac{n!}{(n)!(n-r)!}\]

Here \[{}^{n}{{c}_{r}}\] term is always a rational number, so we want the terms \[{{(a)}^{n-

r}}and{{(b)}^{r}}\] to be rational.

**:**

__Complete step-by-step answer__For making these terms rational, we have to make \[n-r\] and \[r\] as an integer, because non

integer power to an integer number can never be an integer.

So basically, we want to make the powers of a and b as integers.

We are given a binomial expression as \[{{({{9}^{\dfrac{1}{4}}}+{{8}^{ \dfrac{1}{6}}})}^{1000}}\]

Using formula \[{{(a+b)}^{n}}={}^{n}{{c}_{r}}{{(a)}^{n-r}}{{(b)}^{r}}\], here r varies from 0 to n

Expanding,

\[{{({{9}^{\dfrac{1}{4}}}+{{8}^{\dfrac{1}{6}}})}^{1000}}={}^{1000}{{c}_{r}}{{({{9}^{\dfrac{1}{4}}})}^{1000

-r}}{{({{8}^{\dfrac{1}{6}}})}^{r}}\], similarly r varies from 0 to 1000

now as we know that \[9={{3}^{2}}\] and \[8={{2}^{3}}\] so we can replace their value in above

equation\[\] \[\]

we can write \[{{9}^{\dfrac{1}{4}}}={{({{3}^{2}})}^{\dfrac{1}{4}}}\] and \[{{8}^{

\dfrac{1}{6}}}={{({{2}^{3}})}^{\dfrac{1}{6}}}\]

Using property \[{{({{x}^{a}})}^{b}}={{x}^{ab}}\]

We can write \[{{9}^{\dfrac{1}{4}}}=({{3}^{\dfrac{2}{4}}})={{3}^{\dfrac{1}{2}}}...(2)\] and \[{{8}^{

\dfrac{1}{6}}}=({{2}^{\dfrac{3}{6}}})={{2}^{\dfrac{1}{2}}}.....(3)\]

Substituting equation (2) and (3) in equation (1)

\[{{({{9}^{\dfrac{1}{4}}}+{{8}^{

\dfrac{1}{6}}})}^{1000}}={}^{1000}{{c}_{r}}{{({{3}^{\dfrac{1}{2}}})}^{1000-

r}}{{({{2}^{\dfrac{1}{2}}})}^{r}}\]

Again, using property \[{{({{x}^{a}})}^{b}}={{x}^{ab}}\]

We can write it as \[{{({{9}^{\dfrac{1}{4}}}+{{8}^{

\dfrac{1}{6}}})}^{1000}}={}^{1000}{{c}_{r}}({{3}^{\dfrac{1000-r}{2}}})({{2}^{\dfrac{r}{2}}})\]

\[\begin{align}

\to {}^{1000}{{c}_{r}}({{3}^{500-\dfrac{r}{2}}})({{2}^{\dfrac{r}{2}}}) \\

\\

\end{align}\]

Now as we know that \[{}^{1000}{{c}_{r}}\] is integer so we just want powers of 2 and 3 to be

integer to give rational term, and looking carefully we just want \[\dfrac{r}{2}\] to be integer ,

because \[500-\dfrac{r}{2}\] will also become integer if \[\dfrac{r}{2}\] is integer.

Now what if we take \[\dfrac{r}{2}\] as non integer for ex r=3 , then we can see that it becomes

\[{{2}^{\dfrac{3}{2}}}\] so it’s clearly not an rational number

We just want \[\dfrac{r}{2}\] to be integer and our r varies from 0 to 1000

So \[\dfrac{r}{2}\] will be integer whenever r will be multiple of 2

Values of r = 0,2,4….1000

Which is equals to \[\dfrac{1000}{2}+1=501\]

Hence 501 terms are rational.

**Note**: You can do some mistake while expanding the binomial expression or In using the property

\[{{({{x}^{a}})}^{b}}={{x}^{ab}}\] correctly, Convince yourself that power of a integer number must be

an integer to give a rational number , if you have a doubt cross check it by putting any non-

integer number in power of any integer number for ex-

\[{{2}^{\dfrac{1}{3}}}\] or \[{{3}^{\dfrac{3}{8}}}\] they can’t be a rational number.

Recently Updated Pages

How many sigma and pi bonds are present in HCequiv class 11 chemistry CBSE

Mark and label the given geoinformation on the outline class 11 social science CBSE

When people say No pun intended what does that mea class 8 english CBSE

Name the states which share their boundary with Indias class 9 social science CBSE

Give an account of the Northern Plains of India class 9 social science CBSE

Change the following sentences into negative and interrogative class 10 english CBSE

Trending doubts

Difference Between Plant Cell and Animal Cell

Which are the Top 10 Largest Countries of the World?

Fill the blanks with the suitable prepositions 1 The class 9 english CBSE

Differentiate between homogeneous and heterogeneous class 12 chemistry CBSE

Give 10 examples for herbs , shrubs , climbers , creepers

Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE

How do you graph the function fx 4x class 9 maths CBSE

Change the following sentences into negative and interrogative class 10 english CBSE

The Equation xxx + 2 is Satisfied when x is Equal to Class 10 Maths