Q2(iv):

Find : (iv) $125^{-\frac{1}{3}}$

Given Expression & Initial Setup

We are tasked with evaluating the exponential expression $125^{-\frac{1}{3}}$. To solve this, we will systematically apply the fundamental laws of exponents and prime factorization.

Step 1: Prime Factorization of the Base

The base of the given expression is $125$. To simplify expressions involving fractional exponents, the standard procedure is to express the base as a power of its prime factors [Per the Fundamental Theorem of Arithmetic].

Dividing $125$ by its smallest prime factor ($5$):

• $125 \div 5 = 25$
• $25 \div 5 = 5$
• $5 \div 5 = 1$

Thus, the prime factorization is:

$125 = 5 \times 5 \times 5 = 5^3$

Step 2: Substitution into the Original Expression

We substitute the prime factored form of the base back into the original expression.

$125^{-\frac{1}{3}} = (5^3)^{-\frac{1}{3}}$

Step 3: Application of the Power of a Power Rule

[Per the Laws of Exponents], specifically the Power of a Power Property, we know that for any non-zero real number $a$ and rational numbers $m$ and $n$:

$(a^m)^n = a^{m \cdot n}$

Applying this theorem to our expression, we multiply the inner integer exponent ($3$) by the outer fractional exponent ($-\frac{1}{3}$):

$(5^3)^{-\frac{1}{3}} = 5^{3 \times \left(-\frac{1}{3}\right)}$

Step 4: Simplifying the Exponent

Perform the multiplication in the exponent:

$3 \times \left(-\frac{1}{3}\right) = -\frac{3}{3} = -1$

Substituting this back yields:

$5^{-1}$

Step 5: Application of the Negative Exponent Rule

[Per the Negative Exponent Rule], a negative exponent indicates the reciprocal of the base raised to the corresponding positive exponent. Mathematically, for any non-zero real number $a$:

$a^{-n} = \frac{1}{a^n}$

Applying this to our simplified expression:

$5^{-1} = \frac{1}{5^1} = \frac{1}{5}$

Final Solution: The value of $125^{-\frac{1}{3}}$ is $\frac{1}{5}$.