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Q1(iii):
Find :
(iii) $125^{\frac{1}{3}}$
Solution :
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
More Questions from Class 9 Mathematics Number Systems EXERCISE 1.5
- Q1(i): Find : (i) $64^{\frac{1}{2}}$
- Q1(ii): Find : (ii) $32^{\frac{1}{5}}$
- Q2(i): Find : (i) $9^{\frac{3}{2}}$
- Q2(ii): Find : (ii) $32^{\frac{2}{5}}$
- Q2(iii): Find : (iii) $16^{\frac{3}{4}}$
- Q2(iv): Find : (iv) $125^{-\frac{1}{3}}$
- Q3(i): Simplify : (i) $2^{\frac{2}{3}} \cdot 2^{\frac{1}{5}}$
- Q3(ii): Simplify : (ii) $(\frac{1}{3^3})^7$
- Q3(iii): Simplify : (iii) $\frac{11^{\frac{1}{2}}}{11^{\frac{1}{4}}}$
- Q3(iv): Simplify : (iv) $7^{\frac{1}{2}} \cdot 8^{\frac{1}{2}}$
CBSE Solutions for Class 9 Mathematics Number Systems
Chapters in CBSE - Class 9 Mathematics
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