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Q3(iv):
Simplify :
(iv) $7^{\frac{1}{2}} \cdot 8^{\frac{1}{2}}$
Solution :
Given Expression & Theoretical Foundation
We are tasked with simplifying the following mathematical expression involving rational exponents:
$7^{\frac{1}{2}} \cdot 8^{\frac{1}{2}}$
Upon analyzing the expression, we identify two distinct bases ($a = 7$ and $b = 8$) raised to an identical rational power ($m = \frac{1}{2}$).
Step 1: Invoking the Relevant Law of Exponents
To simplify the product of two different bases raised to the same exponent, we apply the Multiplicative Law of Exponents for Identical Powers. [Per the fundamental axioms of real number exponents], this law states:
$a^m \cdot b^m = (a \cdot b)^m$
This theorem holds true for any positive real numbers $a$ and $b$, and any rational number $m$.
Step 2: Algebraic Substitution and Simplification
By substituting our specific values into the established theorem ($a = 7$, $b = 8$, and $m = \frac{1}{2}$), we consolidate the bases inside a single parenthesis:
$7^{\frac{1}{2}} \cdot 8^{\frac{1}{2}} = (7 \cdot 8)^{\frac{1}{2}}$
Next, we perform the arithmetic multiplication within the parentheses:
$7 \cdot 8 = 56$
Substituting this product back into the expression yields:
$56^{\frac{1}{2}}$
Step 3: Radical Conversion and Deep Simplification (Analytical Depth)
While $56^{\frac{1}{2}}$ is the simplified exponential form, it is mathematically rigorous to understand its equivalent radical form. [By the definition of rational exponents], an exponent of $\frac{1}{2}$ denotes the principal square root of the base:
$x^{\frac{1}{2}} = \sqrt{x}$
Therefore, our expression can be written as:
$\sqrt{56}$
To ensure the radical is in its simplest form, we extract the largest perfect square factor from the radicand (56). We perform prime factorization:
- $56 = 4 \cdot 14$
- $56 = 2^2 \cdot 14$
Applying the product property of radicals ($\sqrt{x \cdot y} = \sqrt{x} \cdot \sqrt{y}$):
$\sqrt{56} = \sqrt{4 \cdot 14} = \sqrt{4} \cdot \sqrt{14} = 2\sqrt{14}$
Final Solution: The simplified expression is $56^{\frac{1}{2}}$ (which can also be rigorously expressed in radical form as $2\sqrt{14}$).
More Questions from Class 9 Mathematics Number Systems EXERCISE 1.5
- Q1(i): Find : (i) $64^{\frac{1}{2}}$
- Q1(ii): Find : (ii) $32^{\frac{1}{5}}$
- Q1(iii): Find : (iii) $125^{\frac{1}{3}}$
- 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) $\frac{11^{\frac{1}{2}}}{11^{\frac{1}{4}}}$
CBSE Solutions for Class 9 Mathematics Number Systems
Chapters in CBSE - Class 9 Mathematics
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