Quiz-1

\frac{3^{-2}X81^{\frac{3}{4}}}{729^{\frac{-1}{3}}}

\frac{\frac{1}{9}X(3^{4})^{\frac{3}{4}}}{(9^{3})^{\frac{-1}{3}}}

\frac{\frac{1}{9}X27}{9^{-1}}

\frac{1}{9}X27X9

Here, 121 = 11²

Hence value of a = 11 and b = 2 can be considered.
Therefore, the value of (a–1)^b + 1 = (11–1)²⁺¹ = 10³

\left (\frac{27^2}{4^{-3}} \right )^{-5/6} = \left (\frac{(3^{3})^2}{(2^2)^{-3}} \right )^{-5/6}

=\left (\frac{3^6}{2^{-6}} \right )^{-5/6} = \left (3^{6}\times 2^{6} \right )^{-5/6}

=6^{6\times (-5/6)}=6^{-5}=\left ( \frac{1}{6} \right )^5=\frac{1}{7776}

3^(x-y) = 27 = 3³      ⇒ x - y = 3 ....(i)

3^(x+y) = 243 = 3⁵      ⇒ x + y = 5 ....(ii)

On solving (i) and (ii), we get x = 4.

(\sqrt{8})^{1/3} = (8^{1/2})^{1/3}= 8^{1/6}

=(2^3)^{1/6}= 2^{1/2}= \sqrt{2}

√2ⁿ =64      ⇒ 2^n/2 = 64 = 2⁶
⇒ n/2=6         ⇒ n=12

7^(x-y) = 117,649 = 7⁶

⇒ x - y = 6 ---------------------------(Equation 1)

4^(x+y) = 16,384= 4⁷

⇒ x + y = 7 ---------------------------(Equation 2)

(Equation 1)+ (Equation 2) ⇒ 2x = 6 + 7 = 13

∴ x = 13⁄2 = 6.5

6^{m-2}=6^{m}.6^{-2}
a^{-m}=\frac{1}{a^{m}}
6^{m-2}=\frac{6^{m}}{6^{2}}=\frac{6^{m}}{36}
6^{m-2}=\frac{46656}{36}=1296

5^{2x}= 15625

5^{2x}= 5^{6}

2x=6

x=3

x^{2}= 3^{2}=9

\left (\frac{4^{-4}}{5^{2}.25^{3}} \right )^{1/4} = \left (\frac{(2^2)^{-4}}{5^{2}.5^{6}} \right )^{1/4}

=\left (\frac{2^{-8}}{5^{8}} \right )^{1/4} = \left (2^{-8}\times 5^{-8} \right )^{1/4}

=10^{-8\times (1/4)}=10^{-2}=\left ( \frac{1}{10} \right )^2=\frac{1}{100}