(0.49)⁴ x (0.343)⁴ ÷ (0.2401)⁴ x = (70 ÷ 100)^{? + 3} ?

Correct Answer: 9/4

Explanation:

Simplify 2³ to 8
= 5/8 x 8/5÷4/9
By using simple algebraic rule: a ÷ b/c = a x c/b
= 5/8 x 8/5 x 9/4
= (5×8×9)/(?8×5×4)
= (5×9)/(?5×4)
= (9)/(?4)
= 9/4

Correct Answer: 63/8

Explanation:

(a² + 1/a²) = 17/4
subtracts 2 and Add 2 from above equation, we will get
? a² + 1/a² - 2 + 2 = 17/4
? a² + 1/a² - 2a x 1/a + 2 = 17/4
Now apply the formula, x² + x² - 2xy
? (a - 1/a)² + 2 = 17/4
? (a - 1/a)² = 17/4 - 2
? (a - 1/a)² = (17 - 8)/4
? (a - 1/a)² = 9/4
? (a - 1/a) = ?9/4
(a -1/a) = 3/2
After that, cubing On both sides, we get
(a -1/a)³ = (3/2)³
Apply the formula (x ? y)³ = x³ ? 3x²y + 3xy² ? c³
? (x ? y)³ = x³ ? 3xy(x - y) ? y³
? a³ - 1/a³ - 3 x a x 1/a (a - 1/a) = 27/8
? a³ - 1/a³ = 27/8 + 3 x (3/2)
? a³ - 1/a³ = 27/8 + 9/2
? (a³ - 1/a³) = 63/8

Correct Answer: 1

Explanation:

Given that, a + 1/b = 1
? a = (1 - 1/b) = (b - 1)/b
? 1/a = b/(b - 1) [reciprocal] ..........................(1)
and b + 1/c = 1
? 1/c = (1 - b)
? c = 1/(1 -b) .........................(2)
? c + 1/a ..........................(3)
put the value of c from equ. (2) and 1/a from equ. (1) in equ. (3)
? c + 1/a = 1/(1 -b) + b/(b -1)
? c + 1/a = 1/(1 -b) - b/(1 - b)
? c + 1/a = (1 - b)/(1 - b)
? c + 1/a = 1

Correct Answer: 75.5

Explanation:

(1 + 1/2) (1 + 1/3) (1 + 1/4) .... (1 + 1/150)
= 3/2 x 4/3 x 5/4 x ... 151/150
= (1/2) x 151
= 151/2 = 75.5

Correct Answer: 2

Explanation:

Given equation is,
a + 1/a = ?3 .............................(i)
On squaring both sides
? a² + 1/a² + 2 = 3
? a² + 1/a² = 1 ..............................(ii)

Now, multiplying Equation (i) and (ii), we get
(a + 1/a) (a² + 1/a²) = ?3
? a³ + a/a² + a²/a + 1/a³ = ?3
? a³ + 1/a³ + ( a + 1/a ) = ?3
now put the value of a + 1/a in above equation,
? a³ + 1/a³ + ?3 = ?3 [from Eq. (i)]
? a³ + 1/a³ = 0
? ( a⁶ + 1 ) /a³ = 0
? a⁶ + 1 = 0 x a³
? a⁶ + 1 = 0
? a⁶ = -1
? a⁶ - 1/a⁶ + 2
put the value of a⁶
= (-1)⁶ - 1/(-1)⁶ + 2
= 1 - 1 + 2 = 2

Correct Answer: 5

Explanation:

We know that, a³ + b³ + c³ - 3abc = (a + b + c ) ( a² + b² + c² - ab - bc - ca )
? (a + b + c) = ( a³ + b³ + c³ - 3abc ) / ( a² + b² + c² - ab - bc - ca ) .......................(i)
Given that in question,
(2.247)³ + (1.730)³ + (1.023)³ - 3 x 2.247 x 1.730 x 1.023 / (2.247)² + (1.730)² + (1.023)² - (2.247 x 1.730) - (1.730 x 1.023) - (2.247 x 1.023)
Apply the formula given in Equation (i) and we will get,
= ( a + b + c )
= ( 2.247 + 1.730 + 1.023) [ from eq. (i) ]
= 5.000 = 5

Correct Answer: 13

Explanation:

Let a = 8.73 and b = 4.27
Then, given expression =( a³ + b³ ) / ( a² - ab + b² )
As we Know the Algebra formula,
a³ + b³ = (a + b) (a² - ab + b²)
Apply the above algebra formula in given above expression,
= (a + b) (a² - ab + b²) / ( a² - ab + b² )
= (a + b)
= 8.73 + 4.27 = 13

Correct Answer: 34

Explanation:

Given that,
x = ( ?2 + 1 ) / ( ?2 - 1 )
Expanding given equation by multiplying and dividing with ?2 + 1
we will get,
? x = ( ( ?2 + 1 ) / ( ?2 - 1 ) ) x ( ( ?2 + 1 ) / ( ?2 + 1 ) )
? x = ( ?2 + 1 ) x ( ?2 + 1 ) / ( ?2 - 1 ) x ( ?2 + 1 )
? x = ( ?2 + 1 ) ² / ( ?2 - 1 ) x ( ?2 + 1 ) [ Use the formula (A + B)(A - B) = A² - B² ]
? x = ( ?2 + 1 ) ² / ( ?2² - 1² ) [ Use the formula (A + B)² = A² + 2AB + B² ]
? x = ( 2 + 1 + 2?2 ) / (2 - 1)
? x = 3 + 2?2 .....................(i)
and given in question x - y = 4 ?2
? y = x - 4?2
Put the value of x [ from Equation (i) ]
? y = 3 + 2?2 - 4?2
? y = 3 - 2?2
Now, x² + y² = (3 + 2?2)² + (3 - 2?2)
Use the Algebra formula and solve the equation.
[ Use the formula (A + B)² = A² + 2AB + B² ]
[ Use the formula (A - B)² = A² - 2AB + B² ]

? x² + y² = 9 + 8 + 12?2 + 9 + 8 - 12?2
? x² + y² = 34

Correct Answer: 5/16

Explanation:

Given expression can be written as
= 1/(1 x 4) + 1/(4 x 7) + 1/(7 x 10) + 1/(10 x 13) + 1/(13 x 16)
= 1/3{ 3/(1 x 4) + 3/(4 x 7) + 3/(7 x 10) + 3/(10 x 13) + 3/(13 x 16) }
= 1/3{ (4 - 1)/(1 x 4) + (7 - 4)/(4 x 7) + (10 - 7 )/(7 x 10) + (13 - 10)/(10 x 13) + (16 - 13/(13 x 16) }
= 1/3[(1 - 1/4) + (1/4 - 1/7) + (1/7 - 1/10) + (1/10 - 1/13) + (1/13 - 1/16)]
= 1/3[1 - 1/4 + 1/4 - 1/7 + 1/7 - 1/10 + 1/10 - 1/13 + 1/13 - 1/16]
= 1/3[1 - 1/16]
= 1/3[ (16 - 1)/16]
= 1/3[ (15)/16]
= 1 x 15/ 3 x 16
= 5/16

Correct Answer: -2/a

Explanation:

Given, x + a/x = 1
? x² + a = x .................(i)
? x² - x = -a ................(ii)
Now, ( x² + x + a ) / ( x³ - x²)
( x² + a + x ) / ( x³ - x²)
= ( x + x )/ x(x² - x) [from Eq. (i)]
= 2x/x(-a) [from Eq. (ii) ]
= 2/-a
= -2/a