1. (132)7 × (132)? = (132)11.5. A. 3 B. 3.5 C. 4 D. 4.5

Here is the answer and explanation

Explanation :

$a^m.a^n = a^{m+n}$
(132)7 × (132)x = (132)11.5
=> 7 + x = 11.5
=> x = 11.5 - 7 = 4.5

 2. $\left(\dfrac{a}{b}\right)^{x-2} = \left(\dfrac{b}{a}\right)^{x-7}.\text{ What is the value of x ?}$ A. 3 B. 3.5 C. 4 D. 4.5

Here is the answer and explanation

Explanation :

$a^{n} = \dfrac{1}{a^{-n}}$

\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}

 3. If 7(x - y) = 343 and 7(x + y) = 16807, what is the value of x? A. 4 B. 3 C. 2 D. 1

Here is the answer and explanation

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 = 82 = 4

 4. (0.04)-2.5 = ? A. 125 B. 25 C. 3125 D. 625

Here is the answer and explanation

Explanation :

$a^{-n} = \dfrac{1}{a^n}$

\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}

 5. (6)6.5 × (36)4.5 ÷ (216)4.5 = (6)? A. 1 B. 2 C. 4 D. 6

Here is the answer and explanation

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

 6. $\dfrac{1}{1 + P^{(n - m)}} + \dfrac{1}{1 + P^{(m - n)}} = ?$ A. $2$ B. $\dfrac{1}{1 + P}$ C. $1$ D. $\dfrac{1}{P}$

Here is the answer and explanation

Explanation :

\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}

 7. $\text{If }x = \left(8 + 3\sqrt{7}\right),\text{ what is the value of }\left(\sqrt{x} - \dfrac{1}{\sqrt{x}}\right)?$ A. $\sqrt{12}$ B. $4$ C. $2$ D. $\sqrt{14}$

Here is the answer and explanation

Explanation :

$(a - b)^2 = a^2 - 2ab + b^2$

$a^2 - b^2 = (a - b)(a + b)$

\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}

\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}

 8. if 6m = 46656, What is the value of 6m-2 A. 36 B. 7776 C. 216 D. 1296

Here is the answer and explanation

Explanation :

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

$\dfrac{a^m}{a^n} = a^{m-n}$
Given that 6m = 46656


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

-------------------------------------------------------------------------
Solution 2
--------------------------------------------------------------------------
6m = 46656
646656
67776
61296
6216
636
6
Hence, 46656 = 66
Thus, we got that 6m = 66
=> m = 6
6m-2 = 6(6-2) = 64 = 1296

 9. 10222 ÷ 10220 = ? A. 10 B. 100 C. 1000 D. 10000

Here is the answer and explanation

Explanation :

$\dfrac{a^m}{a^n} = a^{m-n}$
10222 ÷ 10220 = 10222 - 220 =  102 = 100

 10. If m and n are whole numbers and mn = 196, what is the value of (m - 3)(n+1) ? A. 2744 B. 1 C. 121 D. 1331

Here is the answer and explanation

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

 11. $\dfrac{(1024)^{n/5} \times 4^{2n + 1}}{16^n \times 4^{n-1}}\text{= ?}$ A. 256 B. 64 C. 16 D. 9

Here is the answer and explanation

Explanation :

\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}

 12. $\text{If }\left(\dfrac{a}{b}\right)^{x-4} = \left(\dfrac{b}{a}\right)^{x-8}\text{, what is the value of x?}$ A. 4 B. 6 C. 8 D. 10

Here is the answer and explanation

Explanation :

$a^{n} = \dfrac{1}{a^{-n}}$

\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}

 13. If 1000.20 = x, 100.60 = y and xz = y2, what is the value of z? A. 3 B. 6 C. 4.2 D. 2.2

Here is the answer and explanation

Explanation :

$\left({a^m}\right)^n = a^{mn} = \left({a^n}\right)^m$
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


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

 14. $\left(\dfrac{x^q}{x^r}\right)^{(q + r - p)} . \left(\dfrac{x^r}{x^p}\right)^{(r + p - q)} . \left(\dfrac{x^p}{x^q}\right)^{(p + q - r)}\text{ = ?}$ A. x(a - b - c) B. .5 C. 1 D. x(a + b + c)

Here is the answer and explanation

Explanation :

\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}


= 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 

 15. $\dfrac{1}{1 + x^{(q - p)} + x^{(r - p)}} + \dfrac{1}{1 + x^{(p - q)} + x^{(r - q)}} + \dfrac{1}{1 + x^{(q - r)} + x^{(p - r)}}\text{= ?}$ A. 0 B. 1 C. .5 D. 2

Here is the answer and explanation

Explanation :

\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}

 16. 65610.14 × 65610.11 = ? A. 16 B. 9 C. 4 D. 1

Here is the answer and explanation

Explanation :

$a^m.a^n = a^{m+n}$

$\left({a^m}\right)^n = a^{mn} = \left({a^n}\right)^m$
65610.14 × 65610.11 = 6561(0.14 + 0.11)= 6561(0.25)
= 6561(1/4)  = (38)(1/4) = 38/4 = 32 = 9

 17. What is the value of (2 × 4 × 5)5n A. 25n + 45n + 55n B. (405)n C. (40)5n D. (40n)5

Here is the answer and explanation

Explanation :

$\left({a^m}\right)^n = a^{mn} = \left({a^n}\right)^m$
(2 × 4 × 5)5n = (40)5n = (405)n = (40n)5

 18. $\text{If }\left(\sqrt{3}\right)^n = 6561\text{, }(n)^{3/2}\text{ = ?}$ A. $64$ B. $64\sqrt{3}$ C. $16\sqrt{3}$ D. $16$

Here is the answer and explanation

Explanation :

$a^{p/q} = \sqrt[q]{a^p}$

\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}

 19. 5x × 23 = 36. 5(x+1) = ? A. 22 B. 21 C. 20.5 D. 22.5

Here is the answer and explanation

Explanation :

$a^m.a^n = a^{m+n}$

\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}

 20. $36^{120} = (36 \times x)^{40}\text{. What is the value of x?}$ A. 44 B. 44 C. 62 D. 64

Here is the answer and explanation

Explanation :

$\left(ab\right)^n = a^nb^n$

$\dfrac{a^m}{a^n} = a^{m-n}$

$\left({a^m}\right)^n = a^{mn} = \left({a^n}\right)^m$

\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}

 21. (6561)(1/2) + (6561)(1/4) + (6561)(1/8) = ? A. 98 B. 86 C. 93 D. 81

Here is the answer and explanation

Explanation :

  36561
32187
3729
3243
381
327
39
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

 22. $\sqrt{3^n} = 6561. \quad 3^{\sqrt{n}}\text{ = ?}$ A. 81 B. 9 C. 16 D. 25

Here is the answer and explanation

Explanation :

  36561
32187
3729
3243
381
327
39
3
Hence, 6561 = 38


\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}

 23. If 5(a + b) = 5 × 25 × 125 , what is (a + b)2 A. 12 B. 16 C. 34 D. 36

Here is the answer and explanation

Explanation :

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

=> (a + b) = 6

=> (a + b)2 = 62 = 36

 24. (7-1 - 11-1) + (7-1 + 11-1) = ? A. 2 × 7-1 B. 2 × 11-1 C. 14 D. 22

Here is the answer and explanation

Explanation :

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

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

= 7-1 + 7-1

= 2 × 7-1

 25. (5)1.25 × (12)0.25 × (60)0.75 = ? A. 420 B. 260 C. 200 D. 300

Here is the answer and explanation

Explanation :

$\left(ab\right)^n = a^nb^n$

$a^m.a^n = a^{m+n}$
(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

 26. 36 × 36 × 36 × 36 = 6? A. 10 B. 8 C. 4 D. 6

Here is the answer and explanation

Explanation :

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

 27. What is the value of (7-14 - 7-15) ? A. 6 × 7-15 B. 6 × 7-14 C. 7 × 7-15 D. 7 × 7-14

Here is the answer and explanation

Explanation :

$a^{-n} = \dfrac{1}{a^n}$

$a^{n} = \dfrac{1}{a^{-n}}$

\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}

 28. If 3(n + 4) - 3(n + 2) = 8, What is the value of n? A. 0 B. -1 C. -2 D. 2

Here is the answer and explanation

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

 29. $\text{If }2^x = \sqrt[7]{1024}\text{, what is the value of }x \text{ ?}$ A. None of these B. -7/10 C. 10/7 D. 7/10

Here is the answer and explanation

Explanation :

$a^{p/q} = \sqrt[q]{a^p}$

\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}



 30. $\left[\left(3^{-2} - 5^{-2}\right)^{17} ÷ \left(3^{-2} - 5^{-2}\right)^{18} \right]^{1/2}\text{=?}$ A. $\dfrac{15}{4}$ B. $\dfrac{4}{15}$ C. $\dfrac{7}{4}$ D. $\dfrac{4}{7}$

Here is the answer and explanation

Explanation :

\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}

Harpreet
2015-02-14 09:59:31
Good One !! More chapters like Ratio & Proportions , Sequence & Series and some statistics topics like Central tendency, Dispersion , Correlation & regression should be added.
pooja
2014-03-10 18:35:52
it's very nice method each que.each farmula are used
so that's very good simplification thank you very much..........
S.Sathiyendraan
2013-11-25 07:04:03
It is a very useful to my PGD Entrance Exam
Prabha
2013-07-25 11:35:55

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