Q2(i):

Find : (i) $9^{\frac{3}{2}}$

Initial Setup & Given Expression

We are tasked with evaluating the exponential expression with a rational exponent:

$9^{\frac{3}{2}}$

To solve this, we can utilize the fundamental laws of exponents. There are two mathematically rigorous pathways to arrive at the solution: Prime Factorization (Power of a Power Rule) and Radical Conversion. Both methods are detailed below to ensure comprehensive understanding.

Method 1: Prime Factorization (Power of a Power Rule)

Step 1: Prime Factorization of the Base

The base of the expression is $9$. We first express $9$ in its prime factorized form. [Per the Fundamental Theorem of Arithmetic, every integer greater than $1$ can be represented uniquely as a product of prime numbers].

$9 = 3 \times 3 = 3^2$

Step 2: Applying the Laws of Exponents

Substitute the prime factorized form back into the original expression:

$9^{\frac{3}{2}} = (3^2)^{\frac{3}{2}}$

Next, we apply the Power of a Power Property of exponents, which states that for any non-zero base $a$ and rational exponents $m$ and $n$:

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

Applying this property to our expression:

$(3^2)^{\frac{3}{2}} = 3^{2 \times \frac{3}{2}}$

Step 3: Simplifying the Exponent

Multiply the exponents. The integer $2$ in the numerator cancels out the denominator $2$ in the fractional exponent:

$2 \times \frac{3}{2} = \frac{2 \times 3}{2} = 3$

Thus, the expression simplifies to a base raised to an integer power:

$3^3$

Step 4: Final Evaluation

Expand the cubic expression to find the final numerical value:

$3^3 = 3 \times 3 \times 3 = 27$

Method 2: Radical Conversion (Fractional Exponent Rule)

[Per the definition of rational exponents, an expression of the form $a^{\frac{m}{n}}$ can be rewritten as the $n$-th root of $a$ raised to the $m$-th power]:

$a^{\frac{m}{n}} = (\sqrt[n]{a})^m$

Applying this theorem to $9^{\frac{3}{2}}$, where $a = 9$, $m = 3$, and $n = 2$:

$9^{\frac{3}{2}} = (\sqrt[2]{9})^3$

Evaluate the principal square root of $9$:

$\sqrt{9} = 3$

Substitute this back into the expression:

$(3)^3 = 27$

Both analytical pathways yield the exact same scalar quantity, verifying the integrity of the result.

Visualizing the Mathematical Pathways

The following diagram illustrates the structural equivalence of both methods used to evaluate the rational exponent.

Final Solution: $9^{\frac{3}{2}} = 27$