Question

Please answer this question correctly with perfect steps will be marked as BRAINliest ♥ answer for All please (a, b & c)

Answer 1

(a)
L.H.S: 1/4 * (2^n)
1/4 = 1/2^2 = 2^(-2)... {property:1/a^n = a^(-n) }
Now, L.H.S = 2^(-2) * 2^n = 2^n * 2^(-2) = 2^(n-2).... {property: a^m * a^(-n) = a^(m-n)}
L.H.S = R.H.S... Hence, verified.

(b)
L.H.S: 4^(n-1) = 4^n * 4^(-1).... {property: a^m * a^(-n) = a^(m-n)}
                        = 4^n/4 = 1/4 * (4^n)
L.H.S = R.H.S, hence verified [R.H.S can also be taken to prove the identity or property of exponents]

(c)
L.H.S: 25(5^(n-2)) = 5^2 * [5^n * 5^(-2)]... {property: a^m * a^(-n) = a^(m-n)}
                              = 5^2 * 5^n/5^2 = 5^n
L.H.S = R.H.S, hence verified

List of properties used:
a^m * a^n = a^(m+n)
a^m * a^(-n) = a^(m-n)
1/a^m = a^(-m)