Find the value of each of the following: (i) 16^1/4  (ii) 625^(-3/4)

Exponential notation helps us to represent extremely large and small numbers in a simple and readable manner. Exponents or powers signifies the number of times a base (integer, fraction, decimal) is multiplied by itself.

Answer: The value of (i) 16^1/4 = 2 and (ii) 625^(-3/4) = 1/125

Let us look into the steps below to solve them.

Explanation:

(i) 16^1/4

Let's represent 16^1/4 as a base of 2.

We know that,

16 = 2⁴

Substituting in the given expression we get,

16^1/4 = (2⁴) ^1/4

According to the power of exponent rules we have,

(a^m)ⁿ = a^mn 

Thus, 

(2⁴) ^1/4 = 2^{4 × (1/4)}

(2⁴) ^1/4 = 2

(ii) 625^(-3/4)

Since, the base has a negative power we will use the exponent rules to change it to a positive power.

According to the negative property of exponents we have,

a^-m = 1/a^m

Thus, 

625^(-3/4) = (1/625) ^3/4

              = {(1/625) ^1/4}³        (Since, (a^m)ⁿ = a^mn )

              = {(1/5⁴) ^1/4}³           (Since, 625 = 5⁴)

              = {(1/5)^{4 × (1/4)}}³       (Since, (a^m)ⁿ = a^mn )

              = (1/5)³

              = 1/5³                       (Since, (a/b)^m = a^m/b^m)

              = 1/125

Thus, the value of (i) 16^1/4 = 2 and (ii) 625^(-3/4) = 1/125.