Q1(iii):

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

Initial Setup & Given Expression

We are tasked with evaluating the exponential expression $125^{\frac{1}{3}}$.

[Per the definition of rational exponents, an expression of the form $a^{\frac{1}{n}}$ represents the $n$-th principal root of the real number $a$. Therefore, evaluating $125^{\frac{1}{3}}$ is algebraically equivalent to finding the principal cube root of $125$, denoted as $\sqrt[3]{125}$].

Step 1: Prime Factorization of the Base

To simplify the expression systematically, we first determine the prime factorization of the base, $125$. We divide $125$ by its smallest prime factor, which is $5$:

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

Thus, the prime factorization is $125 = 5 \times 5 \times 5 = 5^3$.

[By the Fundamental Theorem of Arithmetic, every integer greater than $1$ can be represented uniquely as a product of prime numbers, up to the order of the factors].

Step 2: Geometric Visualization (The Cube Root)

Geometrically, finding the cube root of $125$ is equivalent to finding the side length $s$ of a perfect cube whose total volume is $125$ cubic units. The relationship is defined by the volume formula $V = s^3$.

Step 3: Substitution into the Original Expression

Substitute the prime factored form of the base back into the given expression:

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

Step 4: Application of the Laws of Exponents

To simplify $(5^3)^{\frac{1}{3}}$, we apply the Power of a Power Property.

[The Power of a Power Property states that $(a^m)^n = a^{m \cdot n}$. This law dictates that when an exponential term is raised to another power, the exponents must be multiplied].

Applying this theorem:

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

Step 5: Simplification of the Exponent

Perform the arithmetic multiplication in the exponent:

$3 \cdot \frac{1}{3} = \frac{3}{3} = 1$

Thus, the expression simplifies to:

$5^1$

[By the identity property of exponents, any non-zero real number raised to the power of $1$ is the number itself, meaning $a^1 = a$].

$5^1 = 5$

Final Solution: 5