1/1 + x^{b - a} + x^{c - a} + 1/1 + x^a-b + x^{c - b} + 1/1 + x^{b - c}+ x^{a - c} =?

Correct Answer: ?3

Explanation:

Let us assume
P = ?1 + a + ?1 - a
Square on both side,
P ² = ( ?1 + a + ?1 - a ) ²
Use the formula ( a + b ) ² = a ² + b ² + 2ab
we will get
? P ² = (1 + a) + (1- a) + 2 ?1 + a x ?1 - a
? P ² = (1 + a) + (1- a) + 2 ?(1 + a ) x ( 1 - a )
? P ² = 2 + 2 ?1 - a²
? P ² = 2 (1 + ?1 - a²)
Now put the value of a which is given in question;
? P ² = 2 (1 + ?1 - ( ?3/2 )²)
? P ² = 2(1 + ?1 - 3/4 )
? P ² = 2(1 + ?1/4 )
? P ² = 2(1 + 1/2) = 2 x 3/2 = 3
? P= ?3
? (?1 + a + ?1 - a) = ?3

Correct Answer: 2

Explanation:

Given equation is below
(p/q)^{n - 1} = (q/p)^{n - 3}
Apply the law of Fractional Exponents and Laws of Exponents
a ^{- m} = 1/a ^m
or
a^m = 1/a ^{- m}
(p/q)^{n - 1} = (q) ^{- (n - 3)} x 1/ (p) ^{- (n - 3)}
(p/q)^{n - 1} = (p) ^{3 - n} x 1/ (q) ^{3 - n}
(p/q)^{n - 1} = (p/q) ^{3 - n}
if p^X = p^Y then X will be equal to Y. means X = Y;
? n - 1 = 3 - n
? 2n = 4
? n = 2

Correct Answer: 0.25

Explanation:

(1/5 )^3a = 0.008 = 8/1000 = 1/125 = (1/5 )³
Apply the Algebra Law and Solve the equation.
? 3a = 3
? a = 1
? ( 0.25 )^a = ( 0.25 )¹ = 0.25

Correct Answer: 47

Explanation:

? = ?103823
? = ?(47) ³
Apply the Law of Algebra
? = [(47)³]^1/3
? = 47^{3 x 1/3} = 47
? = 47

Correct Answer: 748.6875

Explanation:

? = [ (11)³ x (6)² ] x 1/4³
? = [ 11³ x 6² ] x 1/4³
Solve the equation by algebra law.
?= [ 1331 x 36 ] x 1/64 =1331 x 36 / 64
? = 748.6875

Correct Answer: 14

Explanation:

Given that, a = 2 + ? 3
Then, 1/a = 2 - ?3 [ by conjugate property ]
Conjugate law is below,
if x = p + ? q
then 1/x = p - ? q
Apply the law of algebra in above equation.
Now, we have, a ² + a ^-2
= ( a + 1/a ) ² - 2
= ( 2 + ? 3 + 2 - ? 3) ² - 2
= (4) ² - 2 = 16 - 2 = 14

Correct Answer: irrational unless n is the mth power of an integer

Explanation:

If m and n are natural numbers, then m?n is irrational unless n is m^th power of an integer.

Correct Answer: 4

Explanation:

Given that ,
m = 7 - 4?3
Then I/m = 1/7 - 4?3......................(1)

Now multiply and divide with 7 + 4?3 in Eq. (1)
We will get,
1/m = ( 1/7 - 4?3 ) x ( 7 + 4?3 / 7 + 4?3)
?1/m = ( 7 + 4?3 )/ ( 7 - 4?3 ) x ( 7 + 4?3
?1/m = ( 7 + 4?3 ) / ( 7² - ( 4?3)² )
?1/m = ( 7 + 4?3 ) / ( 49 - 4 x 4 x 3 )
?1/m = ( 7 + 4?3 ) /49 - 48
?1/m = ( 7 + 4?3 ) / 1
?1/m = 7 + 4?3
? m + 1/m = 14
? ( m + 1/m ) + 2 = 14 + 2 = 16
? (?m + 1/?m)² = ( m + 1/m )+ 2
? (?m + 1/?m)² = 16
? (?m + 1/?m)² = 4²
? (?m + 1/?m) = 4

Correct Answer: 12

Explanation:

?2ⁿ = 64 ? 2ⁿ = (64)²
? 2ⁿ = (2 × 2 × 2 × 2 × 2 × 2)²
? 2ⁿ = (26)² ? 2ⁿ = 26 × 2 = 212
? n = 12.

Correct Answer: 1

Explanation:

If any given number (suppose x) is a rational number other than zero, then x⁰ = 1.