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Given:

We have to find the difference between \({25 \over 12}\) and \(\frac {15}8\)

Concept Used:

Concept of fractional subtraction

Calculation:

 \(\frac {25}{12}\) -  \(\frac {15}8\)

 \(\Rightarrow {{50 - 45} \over 24}\)

\(\Rightarrow {5 \over 24}\)

∴ The required difference between \({25 \over 12}\) and \(\frac {15}8\) is \({5 \over 24}\).

The given number is:

213 × 35 × 53 × 76

For each term, the perfect cubes can be obtained as:

For 2¹³ = 2⁰, 2³, 2⁶, 2⁹, 2¹²; i.e. 5 terms

For 3⁵ = 3⁰, 3³; i.e. 2 terms

For 5³ = 5⁰, 5³; i.e. 2 terms

For 7⁶ = 7⁰, 7³, 7⁶; i.e. 3 terms

So, we have the multiples of the number 213 × 35 × 53 × 76 as perfect cubes = 5 × 2 × 2 × 3 = 60 

∴ The number of multiples of the number 213 × 35 × 53 × 76 are perfect cubes or complete cubes, are 60

Concept used:

A given natural number can be expressed as 2a.3b.5c.7d...... and so on 

Where a, b, c, d ...... and so on are whole numbers

Total number of factors = (a + 1) × (b + 1) × (c + 1) × (d + 1) ...... and so on

Total number of even factors = a × (b + 1) × (c + 1) × (d + 1) ...... and so on

Calculation:

Total number of even factors = 8 × (2 + 1) × (3 + 1) × (5 + 1) 

∴ Total numbers of even factors is 576

To solve this type of question we have to follow BODMAS rule:

The given equation is:

\(16 - \left[ {5 - \left( {6 + 2\left( {7 - \overline {8 - 5} } \right)} \right)} \right]\)

⇒ 16 - [5 - (6 + 2 (7 - 3))]

⇒ 16 - [5 - (6 + 2 (4))]

⇒ 16 - [5 - (6 + 8)]

⇒ 16 - [5 - 14]

⇒ 16 - [- 9]

⇒ 25

∴ The correct answer is option 2.

Given:

\(0.\overline{63} + 0.\overline{37}\)

Calculations:

\(0.\overline{63}\) = 0.636363.........

⇒ 63/99

\( 0.\overline{37}\) = 0.373737.........

⇒ 37/99

\(0.\overline{63} + 0.\overline{37}\)

⇒ (63/99) + (37/99) =100/99

⇒1.010101....

∴The value is 1.0101.....

Given:

324 × 918 = 27^k

Formula used:

(am)n = amn

am ×  an = am + n

am = an   ⇒ m = n

Calculations:

27 = 3³

9 = 3²

⇒ 3²⁴ × (3²)⁽¹⁸⁾ = (3)^3k      ----(∵ (a^m)ⁿ = a^mn)

⇒ 3^{(24 + 36)} = 3^3k      ----(∵ a^m ×  aⁿ = a^{m + n})

⇒ 3⁶⁰ = 3^3k      ----(∵ a^m = aⁿ   ⇒ m = n)

⇒ 3k = 60 

⇒ k = 20

∴ The value of k is 20

Suppose a worker remained ideal for x days then he worked for 60-x days 

20*(60-x)-3x=280 

1200-23x=280 

23x=920 

x=40

(b) ? = √3781 × 5.36
61.5 × 5.36 =329.64 =330

Calculation:

0.5 × 0.002 = (5/10) × (2/1000) 

0.5 × 0.002 = (10/10000) = 1/1000 ⇒ 0.001

∴ The required answer is 0.001

(d) ? = 4⁷ ÷ (4²)⁴ × 4 = 4⁸ ÷ 4⁸ = 1

Given:
\(\dfrac{3^{502} - 3^{500} + 16}{3^{500} + 2}\)

Calculations:

Dividing both the numerator and denominator by 3⁵⁰⁰

\(⇒ \dfrac{3^{2} - 1 + \dfrac{16}{3^{500}}}{1 + \dfrac{2}{3^{500}}}\)

∵ \( \dfrac{16}{3^{500}}\) and \( \dfrac{2}{3^{500}}\) are very very small and thus negligible. We can put 0 in their places 

\(⇒ \dfrac{3^{2} - 1}{1}\)

⇒ 9 - 1 

⇒ 8

∴ The value of the given expression is 8

Given:

0.2 × 0.02 × 0.002 × 0.4

Calculation:

0.2 × 0.02 × 0.002 × 0.4

= 0.004 × 0.002 × 0.4

= 0.0016 × 0.002

= 0.0000032

Given:

√(√16 × √4 × 4² × 2³)

Calculation:

√(√16 × √4 × 4² × 2³)

⇒ √(4 × 2 × 16 × 8)

⇒ √(2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2)

⇒ (2 × 2 × 2 × 2 × 2)

⇒ 32

Given:

4² × 2³ × √16 × √4

Calculation:

4² × 2³ × √16 × √4

⇒ 16 × 2³ × √16 × √4

⇒ 16 × 8 × √16 × √4

⇒ 16 × 8 × 4 × √4

⇒ 16 × 8 × 4 × 2

⇒ 1,024

(a) Let the required number be x
x² – (9)³ = 567
x² = 567 + 729 = 1296

 x = √1296 = √(36)² = 36

Given

√2 = 1.414

Calculation

√20000

= √(2 × 10000)

= √2 × √10000

= 1.414 × 100

= 141.4

(b) Let the number be x
x² – (46)² = 485
x² = 485 + (46)² = 2601
x = √2601 = 51