[ (11)³ x (6)²] ÷ (4)³ =?

Given that ,(16)⁹ ÷ (16)⁴ x (16) ³ = (16)^?
Apply the law of Fractional Exponents and Laws of Exponents
(a^m) x (aⁿ) = a^{m + n}-------------------1
a^m ÷ aⁿ = a^{m ? n}----------------2
? (16)⁹ ÷ (16)⁴ x (16) ³ = (16)^?
? (16)^? = (16)^{9 + 3 - 4}
? ? = 12 - 4 = 8

? = [(12)^-2]²/[(12)²]^-2
Apply the Algebra Law,
(a^m)ⁿ = a^mn
= (12) ^{- 4}/(12) ^{- 4}
= 1

16^3/2 + 16^-3/2
Apply the law of Fractional Exponents and Laws of Exponents
a^m x aⁿ = a^m+n
a^-m = 1/a^m
?16^3/2 + 1/16^3/2
?(16^1/2)³ + 1/(16^1/2)³
?(4^{2 x 1/2})³ + 1/(4^{2 x 1/2})³
?4³ + 1/4³
? 64 + 1/64
? (64 x 64+ 1)/64
= (4096+1)/64
= 4097/64

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

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

(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

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

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

? = 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