Here is the answer and explanation

Answer : Option D

Explanation :

(132)7 × (132)x = (132)11.5


=> 7 + x = 11.5


=> x = 11.5 - 7 = 4.5

Here is the answer and explanation

Answer : Option D

Explanation :

$MF#%\begin{align}&\left(\dfrac{a}{b}\right)^{x-2} = \left(\dfrac{b}{a}\right)^{x-7}\\\\

&\Rightarrow \left(\dfrac{a}{b}\right)^{x-2} = \left(\dfrac{a}{b}\right)^{-(x-7)}\\\\

&\Rightarrow x - 2 = -(x - 7)\\\\

&\Rightarrow x - 2 = -x + 7\\\\

&\Rightarrow x-2 = -x + 7\\\\

&\Rightarrow 2x = 9\\\\

&\Rightarrow x = \dfrac{9}{2} = 4.5

\end{align} $MF#%

Here is the answer and explanation

Answer : Option A

Explanation :

7^{(x - y)} = 343 = 7³

=> x - y = 3 ---------------------------(Equation 1)

7^{(x + y)} = 16807 = 7⁵

=> x + y = 5 ---------------------------(Equation 2)

(Equation 1)+ (Equation 2) => 2x = 3 + 5 = 8

=> x = ⁸⁄₂ = 4

Here is the answer and explanation

Answer : Option C

Explanation :

$MF#%\begin{align}&(0.04)^{-2.5} = \left(\dfrac{1}{.04}\right)^{2.5} = \left(\dfrac{100}{4}\right)^{2.5}

= \left(25\right)^{2.5} = \left(5^2\right)^{2.5}\\\\

&= \left(5^2\right)^{\left(\dfrac{5}{2}\right)}= 5^5 = 3125\end{align}$MF#%

Here is the answer and explanation

Answer : Option B

Explanation :

(6)^6.5 × (36)^4.5 ÷ (216)^4.5

= (6)^6.5 × [(6)²]^4.5 ÷ [(6)³]^4.5

= (6)^6.5 × (6)⁹ ÷ (6)^13.5

= (6)^{(6.5 + 9 - 13.5)}

= (6)²

Here is the answer and explanation

Answer : Option C

Explanation :

$MF#%\begin{align}&\dfrac{1}{1 + P^{(n - m)}} + \dfrac{1}{1 + P^{(m - n)}}\\\\

&= \dfrac{1}{1 + \dfrac{P^n}{P^m}} + \dfrac{1}{1 + \dfrac{P^m}{P^n}}\\\\\\\\

&= \dfrac{P^m}{P^m + P^n} + \dfrac{P^n}{P^n + P^m}\\\\\\\\

&= \dfrac{P^m + P^n}{P^m + P^n} \\\\\\\\

&= 1\end{align} $MF#%

Here is the answer and explanation

Answer : Option D

Explanation :

$MF#%\begin{align}&\left(\sqrt{x} - \dfrac{1}{\sqrt{x}}\right)^2\\\\

&= x - 2 + \dfrac{1}{x}\\\\

&= x + \dfrac{1}{x} - 2 \\\\

&= \left(8 + 3\sqrt{7}\right) + \dfrac{1}{\left(8 + 3\sqrt{7}\right)} - 2 \\\\

&= \left(8 + 3\sqrt{7}\right) + \dfrac{\left(8 - 3\sqrt{7}\right)}{\left(8 + 3\sqrt{7}\right)\left(8 - 3\sqrt{7}\right)} - 2 \\\\

&= \left(8 + 3\sqrt{7}\right) + \dfrac{\left(8 - 3\sqrt{7}\right)}{8^2 - \left(3\sqrt{7}\right)^2} - 2 \\\\

&= \left(8 + 3\sqrt{7}\right) + \dfrac{\left(8 - 3\sqrt{7}\right)}{64 - 63} - 2 \\\\

&= \left(8 + 3\sqrt{7}\right) + \dfrac{\left(8 - 3\sqrt{7}\right)}{1} - 2 \\\\

&= 8 + 3\sqrt{7} + 8 - 3\sqrt{7} - 2 \\\\

&= 14\end{align} $MF#%

$MF#%\begin{align}&\text{We got that }\left(\sqrt{x} - \dfrac{1}{\sqrt{x}}\right)^2 = 14\\\\

&\text{Hence ,}\left(\sqrt{x} - \dfrac{1}{\sqrt{x}}\right) = \sqrt{14}\end{align} $MF#%

Here is the answer and explanation

Answer : Option D

Explanation :

-------------------------------------------------------------------------
Solution 1
--------------------------------------------------------------------------

Given that 6m = 46656

$MF#%6^{m-2} = \dfrac{6^m}{6^2} = \dfrac{46656}{6^2} = \dfrac{46656}{36} = 1296$MF#%

-------------------------------------------------------------------------
Solution 2
--------------------------------------------------------------------------
6m = 46656 


6
46656
6
7776
6
1296
6
216
6
36
    6


Hence, 46656 = 66


Thus, we got that 6m = 66 


=> m = 6


6m-2 = 6(6-2) = 64 = 1296

Here is the answer and explanation

Answer : Option B

Explanation :

10222 ÷ 10220 = 10222 - 220 =  102 = 100

Here is the answer and explanation

Answer : Option D

Explanation :

mⁿ = 196

We know that 14² = 196

Hence we can take m = 14 and n = 2

(m - 3)⁽ⁿ⁺¹⁾ = (14 - 3)⁽²⁺¹⁾ = 11³ = 1331

Here is the answer and explanation

Answer : Option C

Explanation :

$MF#%\begin{align}&\dfrac{(1024)^{n/5} \times 4^{2n + 1}}{16^n \times 4^{n-1}}\\\\

&= \dfrac{(4^5)^{n/5} \times 4^{2n + 1}}{(4^2)^n \times 4^{n-1}}\\\\

&= \dfrac{4^n \times 4^{2n + 1}}{4^{2n} \times 4^{n-1}}\\\\

&= 4^{(n + 2n + 1) - (2n)-(n-1)}\\\\

&= 4^2\\\\

&= 16

\end{align} $MF#%

Here is the answer and explanation

Answer : Option B

Explanation :

$MF#%\begin{align}&\left(\dfrac{a}{b}\right)^{x-4} = \left(\dfrac{b}{a}\right)^{x-8}\\\\

&\Rightarrow \left(\dfrac{a}{b}\right)^{x-4} = \left(\dfrac{a}{b}\right)^{-(x-8)}\\\\

&\Rightarrow x-4 = -(x-8)\\\\

&\Rightarrow x-4 = -x + 8\\\\

&\Rightarrow 2x = 12\\\\

&\Rightarrow x = \dfrac{12}{2} = 6\end{align} $MF#%

Here is the answer and explanation

Answer : Option A

Explanation :

x = (100)0.2


y = (10)0.6


xz = y2


=> (1000.2)z = (100.6)2


=> 1000.2z = 10(0.6 × 2)


=> (102)0.2z = 10(0.6 × 2)


=> 10(2 × 0.2z)= 10(0.6 × 2)


=> 2 × 0.2z = 0.6 × 2(∵ powers are equal as base values are equal in both side)


=> 0.2z = 0.6

$MF#%\Rightarrow z = \dfrac{.6}{.2} = \dfrac{6}{2} = 3$MF#%

Here is the answer and explanation

Answer : Option C

Explanation :

$MF#%\begin{align}&\left(\dfrac{x^q}{x^r}\right)^{(q + r - p)} . \left(\dfrac{x^r}{x^p}\right)^{(r + p - q)}

. \left(\dfrac{x^p}{x^q}\right)^{(p + q - r)} \\\\

&= \left[x^{(q-r)}\right]^{(q + r - p)} . \left[x^{(r-p)}\right]^{(r + p - q)}

. \left[x^{(p-q)}\right]^{(p + q - r)} \end{align} $MF#%

= x[(q - r)(q + r - p) + (r - p)(r + p - q) + (p - q)(p + q - r)] 


= x[(q - r)(q + r)- p(q - r) + (r - p)(r + p) - q(r - p) + (p - q)(p + q) - r(p - q)]


= x[(q2 - r2) - p(q - r) + (r2 - p2)  - q(r - p) + (p2 - q2) - r(p - q)] 


= x[(q2 - r2 + r2 - p2 + p2 - q2) - p(q - r) - q(r - p) - r(p - q)] 


= x[0 - p(q - r) - q(r - p) - r(p - q)]


= x[- p(q - r) - q(r - p) - r(p - q)]


= x[-pq + pr -qr + pq - rp + rq ]


= x[0]


= 1 

Here is the answer and explanation

Answer : Option B

Explanation :

$MF#%\begin{align}&\dfrac{1}{1 + x^{(q - p)} + x^{(r - p)}} + \dfrac{1}{1 + x^{(p - q)} + x^{(r - q)}}

+ \dfrac{1}{1 + x^{(q - r)} + x^{(p - r)}}\\\\\\\\

&= \dfrac{1}{1 + \dfrac{x^q}{x^p} + \dfrac{x^r}{x^p}} + \dfrac{1}{1 + \dfrac{x^p}{x^q}

+ \dfrac{x^r}{x^q}} + \dfrac{1}{1 + \dfrac{x^q}{x^r} + \dfrac{x^p}{x^r}}\\\\\\\\

&= \dfrac{x^p}{x^p + x^q + x^r} + \dfrac{x^q}{x^q + x^p + x^r} + \dfrac{x^r}{x^r + x^q + x^p}\\\\\\\\

&= \dfrac{x^p + x^q + x^r}{x^p + x^q + x^r}\\\\

&= 1 \end{align} $MF#%

Here is the answer and explanation

Answer : Option B

Explanation :

65610.14 × 65610.11 = 6561(0.14 + 0.11)= 6561(0.25) 


= 6561(1/4)  = (38)(1/4) = 38/4 = 32 = 9

Here is the answer and explanation

Answer : Option B

Explanation :

(2 × 4 × 5)5n = (40)5n = (405)n = (40n)5

Here is the answer and explanation

Answer : Option A

Explanation :

$MF#%\begin{align}&\left(\sqrt{3}\right)^n = 6561\\\\

&\Rightarrow \left(\sqrt{3}\right)^n = \left(\sqrt{3}\right)^{16}\\\\

&\Rightarrow n = 16\\\\\\\\

&(n)^{3/2} = (16)^{3/2} = \sqrt{16^3} = 16 \sqrt{16} = 16 \times 4 = 64\end{align} $MF#%

Here is the answer and explanation

Answer : Option D

Explanation :

$MF#%\begin{align}&5^x \times 2^3 = 36\\\\

&\Rightarrow 5^x = \dfrac{36}{2^3}\\\\

&5^{(x+1)} = 5^x × 5 = \dfrac{36}{2^3} × 5 = \dfrac{36 \times 5}{2 \times 2 \times 2}

= \dfrac{9 \times 5}{2} = 22.5\end{align} $MF#%

Here is the answer and explanation

Answer : Option D

Explanation :

$MF#%\begin{align}&36^{120} = (36 \times x )^{40}\\\\

&\Rightarrow 36^{120} = 36^{40} \times x^{40}\\\\

&\Rightarrow x^{40} = \dfrac{36^{120}}{36^{40}} = 36^{(120 - 40)} = 36^{80}\\\\

&\Rightarrow (\sqrt{x}^2)^{40} = 36^{80}\\\\

&\Rightarrow (\sqrt{x})^{80} = 36^{80}\\\\

&\Rightarrow \sqrt{x} = 36\\\\

&\Rightarrow x = 36^2 = 6^4\end{align} $MF#%

Here is the answer and explanation

Answer : Option C

Explanation :

  3
6561
  3
2187
  3
729
  3
243
  3
81
  3
27
  3
9
    3


Hence, 6561 = 38


(6561)(1/2) + (6561)(1/4) + (6561)(1/8)


= (38)(1/2) + (38)(1/4) + (38)(1/8)


= (3)(8/2) + (3)(8/4) + (3)(8/8)


= (3)4 + (3)2 + (3)1


= 81 + 9 + 3


= 93

Here is the answer and explanation

Answer : Option A

Explanation :

  3
6561
  3
2187
  3
729
  3
243
  3
81
  3
27
  3
9
    3


Hence, 6561 = 38

$MF#%\begin{align}&\text{Given that }\sqrt{3^n} = 6561 \\\\

&\sqrt{3^n} = 3^{8}\\\\

&3^n = 3^{16}\\\\

&\Rightarrow n = 16\\\\

&3^{\sqrt{n}} = 3^{\sqrt{16}} = 3^{4} = 81\end{align} $MF#%

Here is the answer and explanation

Answer : Option D

Explanation :

5^{(a + b)} = 5 × 25 × 125 = 5¹ × 5² × 5³ = 5^{(1 + 2 + 3)} = 5⁶

=> (a + b) = 6

=> (a + b)² = 6² = 36

Here is the answer and explanation

Answer : Option A

Explanation :

(7^-1 - 11^-1) + (7^-1 + 11^-1)

= 7^-1 + 7^-1 - 11^-1 + 11^-1

= 7^-1 + 7^-1

= 2 × 7^-1

Here is the answer and explanation

Answer : Option D

Explanation :

(5)1.25 × (12)0.25 × (60)0.75


=(5)1.25 × (12)0.25 × (12 × 5)0.75


= (5)1.25 × (12)0.25 × (12)0.75 × (5)0.75


= (5)(1.25 + 0.75) × (12)(0.25 + 0.75)


= (5)2 × (12)1


= 25 × 12


= 300

Here is the answer and explanation

Answer : Option B

Explanation :

36 × 36 × 36 × 36 = 6² × 6² × 6² × 6² = 6^(2+2+2+2) = 6⁸

Here is the answer and explanation

Answer : Option A

Explanation :

$MF#%\begin{align}&\left(7^{-14} - 7^{-15}\right) = \dfrac{1}{7^{14}} - \dfrac{1}{7^{15}} = \dfrac{7}{7^{15}} - \dfrac{1}{7^{15}}\\\\

&= \dfrac{7 - 1}{7^{15}} =\dfrac{6}{7^{15}} = 6 × 7^{-15}\end{align} $MF#%

Here is the answer and explanation

Answer : Option C

Explanation :

3^{(n + 4)} - 3^{(n + 2)} = 8

3^{(n + 2 + 2)} - 3^{(n + 2)} = 8

3^{(n + 2)} × 3⁽²⁾ - 3^{(n + 2)} = 8

3^{(n + 2)}[3⁽²⁾ - 1] = 8

3^{(n + 2)} × 8 = 8

3^{(n + 2)} = 1

=> n + 2 = 0

=> n = -2

Here is the answer and explanation

Answer : Option C

Explanation :

$MF#%\begin{align}&2^x = \sqrt[7]{1024}\\\\

&\Rightarrow 2^x = \sqrt[7]{2^{10}}\\\\

&\Rightarrow 2^x = 2^{10/7}\\\\

&\Rightarrow x = 10/7\end{align} $MF#%

Here is the answer and explanation

Answer : Option A

Explanation :

$MF#%\begin{align}&\left[\left(3^{-2} - 5^{-2}\right)^{17} ÷ \left(3^{-2} - 5^{-2}\right)^{18} \right]^{1/2}\\\\

&= \left[\left(3^{-2} - 5^{-2}\right)^{(17 - 18)} \right]^{1/2}\\\\

&= \left[\left(3^{-2} - 5^{-2}\right)^{(-1)} \right]^{1/2}\\\\

&= \left[\dfrac{1}{\left(3^{-2} - 5^{-2}\right)} \right]^{1/2}\\\\

&= \left[\dfrac{1}{\left(\dfrac{1}{3^2} - \dfrac{1}{5^2}\right)} \right]^{1/2}\\\\

&= \left[\dfrac{3^2 × 5^2}{\left(5^2 - 3^2\right)} \right]^{1/2}\\\\

&= \left[\dfrac{3^2 × 5^2}{16} \right]^{1/2}\\\\

&= \dfrac{3 \times 5}{4}\\\\

&= \dfrac{15}{4}\end{align} $MF#%