Q3(ii):

Simplify : (ii) $(\frac{1}{3^3})^7$

Initial Setup & Given Expression

We are tasked with simplifying the following exponential expression:

$ E = \left(\frac{1}{3^3}\right)^7 $

To simplify this expression rigorously, we will apply the fundamental Laws of Exponents. There are two primary analytical pathways to arrive at the solution. Both methods are detailed below to ensure comprehensive understanding.

Method 1: Applying the Negative Exponent Rule First

Step 1: Transform the Fractional Base

[Per the Reciprocal Law of Exponents], any non-zero base raised to a positive power in the denominator can be expressed as a negative exponent in the numerator. The governing axiom is:

$ \frac{1}{a^n} = a^{-n} $

Applying this theorem to the inner fraction of our expression:

$ \frac{1}{3^3} = 3^{-3} $

Step 2: Substitute and Apply the Power of a Power Rule

Substitute the transformed base back into the original expression:

$ E = \left(3^{-3}\right)^7 $

[Per the Power of a Power Property], when an exponential term is raised to another exponent, the exponents are multiplied. The governing axiom is:

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

Applying this property to our expression yields:

$ E = 3^{-3 \cdot 7} $

Step 3: Arithmetic Simplification

Perform the multiplication in the exponent:

$ -3 \cdot 7 = -21 $

$ E = 3^{-21} $

Method 2: Applying the Power of a Quotient Rule First

Step 1: Distribute the Exponent

[Per the Power of a Quotient Rule], an exponent applied to a fraction is distributed to both the numerator and the denominator. The governing axiom is:

$ \left(\frac{a}{b}\right)^n = \frac{a^n}{b^n} $

Applying this to our expression:

$ E = \frac{1^7}{(3^3)^7} $

Step 2: Simplify Numerator and Denominator

Since $1$ raised to any finite power is $1$ ($1^7 = 1$), the numerator simplifies immediately. For the denominator, we again apply the Power of a Power Property $(a^m)^n = a^{m \cdot n}$:

$ E = \frac{1}{3^{3 \cdot 7}} $

$ E = \frac{1}{3^{21}} $

Step 3: Convert to a Negative Exponent

Using the Reciprocal Law of Exponents $\left(\frac{1}{a^n} = a^{-n}\right)$ to express the final answer without a fraction:

$ E = 3^{-21} $

Visual Representation of the Exponent Laws

The following diagram illustrates the sequential application of the exponent rules used in Method 1.

Final Solution: The simplified expression is $3^{-21}$ (which can also be written as $\frac{1}{3^{21}}$).