? = √(13)⁴ = √13 x 13 x 13 x 13
Apply the rule of square root.

Correct Option: B

? = √(13)⁴ = √13 x 13 x 13 x 13
Apply the rule of square root.
= 13 x 13 = 169

? = 1/ ( 1 + x^{b - a} + x^{c - a} ) + 1/ ( 1 + x^{a - b} + x^{c - b} ) + 1 / ( 1 + x^{b - c} + x^{a - c} )
Apply the law of algebra and solve the given equation.
⇒ ? =1/ ( 1 + x^b/x^a + x^c/x^a ) + 1/ ( 1+ x^a/x^b + x^c/x^b ) + 1/ ( 1 + x^b/x^c + x^a/x^c )

Correct Option: B

? = 1/ ( 1 + x^{b - a} + x^{c - a} ) + 1/ ( 1 + x^{a - b} + x^{c - b} ) + 1 / ( 1 + x^{b - c} + x^{a - c} )
Apply the law of algebra and solve the given equation.
⇒ ? =1/ ( 1 + x^b/x^a + x^c/x^a ) + 1/ ( 1+ x^a/x^b + x^c/x^b ) + 1/ ( 1 + x^b/x^c + x^a/x^c )
⇒ ? = x^a/ ( x^a+x^b+x^c ) + x^b/ ( x^b + x^a+ x^c ) + x^c/ ( x^c+x^b+x^a )
⇒ ? = ( x^a + x^b + x^c ) / ( x^a + x^b + x^c ) = 1

Let us assume
P = √1 + a + √1 - a
Square on both side and put the value of a in solved equation.

Correct Option: D

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

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}
and
if p^X = p^Y then X will be equal to Y. means X = Y;

Correct Option: D

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

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

Correct Option: C

(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