Finding Integral Part of Numbers of the Form a^b Where a Is Irrational

Q.

$i^2+i^4+i^6+..........(2n+1)$terms equal to:

1. $i$

2. $-i$

3. $1$

4. $-1$

Q.

Find the cube root by prime factorization of $110592$

Q.

Whats the square root of $625$?

Q.

The difference between an integer and its cube is divisible by

1. $4$

2. $6$

3. $9$

4. None of these

Q.

What is the square root of $160$?

Q. Consider the binomial expansion $R=(1+2x{)}^n=I+f,$ where $I$ is the integral part of $R$ and $f$ is the fractional part of $R$ for $n\in N$. If the sum of the coefficients of $R$ is $6561$ and $K=n+R-Rf,$ then which of the following is/are correct?

1. $K=9$ when $x=\frac{1}{\sqrt{2}}$
2. $K=7$ when $x=\frac{1}{\sqrt{2}}$
3. $5^{\text{th}}$ term is the greatest term when $x=\frac{1}{2}$
4. If $k^{\text{th}}$ term has the greatest coefficient, then the sum of all possible value(s) of $k$ is $13$

Q.

Find the square root of $324$.

Q. If $p=(8+3\sqrt{7}{)}^n$ and $f=p-\left[p\right]$, then
(where [.] denotes the greatest integer function)

1. $p^{1/n}-(1-f{)}^{1/n}=6\sqrt{7}$
2. $p(1-f)=4$
3. $p(1-f)=1$
4. $p^{1/n}+(1-f{)}^{1/n}=16$

Q. If $p=(8+3\sqrt{7}{)}^n$ and $f=p-\left[p\right]$, then the value of $p(1-f)$ is
(where [.] denotes the greatest integer function)

1. $1$
2. $2$
3. $2^n$
4. $2^{2n}$

Q. If $p=(8+3\sqrt{7}{)}^n$ and $f=p-\left[p\right]$, then
(where [.] denotes the greatest integer function)

1. $p^{1/n}-(1-f{)}^{1/n}=6\sqrt{7}$
2. $p(1-f)=4$
3. $p(1-f)=1$
4. $p^{1/n}+(1-f{)}^{1/n}=16$

Q. The last two digits of the number $(23{)}^{14}$ are

1. $01$
2. $05$
3. $03$
4. $09$

Q.

If m is a positive integer, then $\left[\right(\sqrt{3}+1{)}^{2m}]+1$, where [x] denotes greatest integer $\le x$, is divisible by

1. $2^m$
2. $2^{m+1}$
3. $2^{m+1-2}$
4. $2^{2m}$

Q. The last two digits of the number $(23{)}^{14}$ are

1. $03$
2. $09$
3. $05$
4. $01$