Q3(iv):

Simplify : (iv) $7^{\frac{1}{2}} \cdot 8^{\frac{1}{2}}$

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 $2\sqrt{14}$).