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(e) (7)³/√?= 14-7=7

√? = 7³/7= 49

?=49²= 2401

√? + 29 = √2704 

√?+29=52

√?=52-29=23

?=23x23=529

(e) 59475/√?=195x5

√?=59475/195*5=61

61 × 61 = 3721

(d) √? = 31- 17 = 14 = √196

(d) √? + 17 = √961
or  √?  + 17 = 31
or  √?  = 31 – 17
or  √?  = 14
or ? = 14 × 14 = 196

Given-

32% of ? = 3596 - 832

Calculation-

Let x will come in place of question mark.

32% of x = 3596 - 832

⇒ x = 2764 × 100/32

⇒ x = 8637.5

∴ 8637.5 will come in place of (?)

Calculations :

(0.0579 × 2.43) / (0.007 × 17.3 × 0.8)

⇒ (579 × 243)/(7 × 173 × 8 × 10)

⇒ 140697/96880

⇒ 1.4522 

∴ 1.4522 will be the right answer 

Given - 

\(\frac 1 {4.263} = 0.2346\)

Concept used - 

(1/0.1) = 10

(1/0.01) = 100

Solution - 

\(\frac 1 {4.263} = 0.2346,\)

⇒ \(\frac 1 {0.0004263}\) = 0.2346 × 10⁴ = 2346

∴ \(\frac 1 {0.0004263}\) = 2346.

Calculation:

A. 35 ÷ 0.07 = 500

B. 2.5 ÷ 0.0005 = 5000

C. 136.09 ÷ 43.9 = 3.1

∴ B > A > C 

Given:

 \(√{768}\) × \(√{3267}\) = 44 × y

Firmulae required:

(a^m)ⁿ = a^mn

√a = a^1/2

a^m ×  b^m = (a × b)^m

Calculations:

Factors of 768 = 256 × 3

⇒ 2⁸ × 3                  (∵ 256 = 2⁸)

Factors of 3267 = 121 × 27

⇒ 11² × 3³     (∵ 121 = 11² , 3³ = 27)

⇒ (768)^1/2 × (3267)^1/2 = 44 × y

⇒ (28 × 3 × 112 × 33)^1/2 = 44 × y

⇒ (2⁸ × 3⁴ × 11² )^1/2 = 44 × y

⇒ 2^{8 × 1/2} × 3^{4 × 1/2} × 11^2 ×1/2 = 44 × y

⇒ 2⁴ × 3² × 11 = 44 × y

⇒ 16 × 9 × 11 = 44 × y

⇒ y = 9 × 4 = 36

∴ The value of y is 36

Calculation:

7 +[7 – {7of 7-( -7 ÷ 7 × 7)}]

⇒ 7 +[7 – {7of 7-( -1 × 7)}]

⇒ 7 +[7 – {7of 7-( -7)}]

⇒ 7 +[7 – {7of 7 + 7}]

⇒ 7 +[7 – {49 + 7}]

⇒ 7 +[7 – 56]

⇒ 7 +[– 49]

∴ 7 +[7 – {7of 7-( -7 ÷ 7 × 7)}] = –42

The Correct option is 3 i.e. -42

Given

43/2 × 2432/5 × 9-3/2

Calculation

43/2 × 2432/5 × 9-3/2

= (2²)^3/2 × (3⁵)^2/5 × (3²)^-3/2

= 2³ × 3² × 3^-3

= 8 × 3^-1

= 8/3

√[4 + √(44 + √10000)]

= √[4 + √(44 + 100)]

= √[4 + √144]

= √[4 + 12]

= √[16]

= 4

To solve questions of this type, follow the laws of “Surds and indices’’ given below-

Laws of Indices:-

\(1 - :\;{a^m} \times {a^n} = {a^{\left\{ {m + n} \right\}}}\)

\(2 - :\;{a^m} \div {a^n} = {a^{\left\{ {m - n} \right\}}}\)

\(3 - :\;{\left( {{a^m}} \right)^n} = {a^{mn}}\)

\(4 - :{\left( a \right)^{ - m}} = \frac{1}{{{a^m}}}\)

\(5 - :{\left( a \right)^{\frac{m}{n}}} = \sqrt[n]{{{a^m}}}\)

\(6 - :{\left( a \right)^0} = 1\)

Calculations:

4⁴ × (2⁴)² ÷ (8³ × 3² × 4³)

\(⇒ {2^8} \times \frac{{{2^8}}}{{{2^9} \times {3^2} \times {2^6}}}\)

\(⇒ \frac{{{2^{8 + 8 - 9 - 6}}}}{{{3^2}}}\)

⇒ 2/9

∴ The correct answer is 2/9 

Given:

0.001 + 1.01 + 0.11

Calculations:

Solving the given terms,

⇒ 0.001 + 1.01 + 0.11

⇒ 1.12 + 0.001

⇒ 1.121

∴ The value of 0.001 + 1.01 + 0.11 is 1.121

Given:

We have to find the value of (243)0.16 × (243)0.04 

Concept Used:

a^m × aⁿ = a^{(m + n)}

Calculation:

(243)0.16 × (243)0.04

⇒ (3⁵)^0.16 × (3⁵)^0.04

⇒ (3⁵)^{(0.16 + 0.04)}

⇒ (3⁵)^0.2

⇒ (3⁵)^(1/5)

⇒ 3

∴ The required value of the given expression is 3.

Formula used: 

a^m × aⁿ = a^{m + n}

(a × b)^m = a^m × b^m

Calculation: 

By using formula

⇒ (2 × 3)² × 3^3x = (3³)^{3x + 2} × 2²

⇒ 2² × 3² × 3^3x = 3^{9x + 6} × 2²

⇒ 3^{3x + 2} = 3^{9x + 6}

Now, ccompairing power

⇒ 3x + 2 = 9x + 6 

⇒ 6x = -4

⇒ x = -4/6

⇒ x = -2/3

∴ The required value of 'x' is -2/3. 

Given:

Jewish has chocolates = ¼ of 500

He distributed = 1/5 of 1/4 of 500

Calculation:

Chocolates he has = ¼ of 500

⇒ ¼ × 500 = 125 chocolates

Chocolates he distributed = 1/5 of 125

⇒ 1/5 × 125 = 25 Chocolates

⇒ Chocolates left with him = 125 – 25 = 100

∴ 100 chocolates left with Jewish.

The correct option is 3 i.e.100.

(c) ? = 4275 ÷ 496 × 21²
4275 x (21)² /496 = 4275 x 441/495 = 3808 =3810
 

(c) ? = 14¹³ × 14³ = 14^{13 + 3} = 14¹⁶

d) 69 × 69 = 4761
68 × 68 = 4624
Clearly, 4624 < 4700 < 4761
Hence, 61 should be added to 4700 to make it a perfect square.

(d) ? = (10)²⁴ × (10)^–21
= (10)^24–21 = 10³ = 1000

(c) ? = √876 ×20.6 + 165.34
29 × 21 + 165
= 609 + 165 = 774 = 775

(e) x + y = 23 and  xy = 126
Now x² + y² = (x + y)2 – 2xy
(23)² – 2 × 126
529 – 252
x² + y² = 277

(a) Given, x + y = 20 and xy = 84
x² + y² = (x + y)² – 2xy
= (20)² – 2 × 84
= 400 – 168 = 232

(a) 49 + √? = 87
or = 87 – 49 = 38
or ? = (38)² = 1444

(e) √? = 17 + 22 = 39
? = 39 × 39
= 1521

(a) ? = 44³ × 16³ – 18678² 

= 348913664 – 348867684 

= 45980

Let the total amount be Rs. X the

x/14 - x/18 = 80

2x/126 = 80

x/63 = 63 x 80

= 5040

Hence the total amount is 5040.

Let the cost of a pen and a pencil be Rs. X and Rs. Y respectively.

Then, 2x + 3y = 86 ….(i) and 4x + y =112. 

Solving (i) and (ii), we get: x = 25 and y = 12. 

Cost of a pen =Rs. 25 and the cost of a pencil =Rs. 12.

Given:

√ 24649

Calculation:

√ 24649

⇒ √ 157 × 157

⇒ √ (157)2

⇒ 157

∴ The Square Root of 24649 is 157.