Q3(iii):

Simplify : (iii) $\frac{11^{\frac{1}{2}}}{11^{\frac{1}{4}}}$

Given Expression & Initial Setup

We are tasked with simplifying the following exponential expression:

$ \frac{11^{\frac{1}{2}}}{11^{\frac{1}{4}}} $

Here, we identify the components of the expression:

• Base ($a$): $11$
• Numerator Exponent ($m$): $\frac{1}{2}$
• Denominator Exponent ($n$): $\frac{1}{4}$

Step 1: Identifying the Relevant Law of Exponents

To simplify the division of two exponential terms with the same base, we apply the Quotient Law of Exponents. [Per the fundamental laws of algebra for real numbers], when dividing like bases, we subtract the exponent of the denominator from the exponent of the numerator.

The formal theorem states:

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

Condition: $a > 0$ and $a \neq 1$. Since our base is $11$, this condition is perfectly satisfied.

Step 2: Applying the Quotient Law

Substituting our specific values ($a = 11$, $m = \frac{1}{2}$, $n = \frac{1}{4}$) into the formula, we get:

$ \frac{11^{\frac{1}{2}}}{11^{\frac{1}{4}}} = 11^{\left(\frac{1}{2} - \frac{1}{4}\right)} $

Step 3: Simplifying the Fractional Exponent

We must now perform the arithmetic operation on the exponents: $\frac{1}{2} - \frac{1}{4}$.

[By the rules of rational number subtraction], we must first find a common denominator for the fractions. The Least Common Multiple (LCM) of the denominators $2$ and $4$ is $4$.

• Convert $\frac{1}{2}$ to an equivalent fraction with a denominator of $4$:
  $ \frac{1 \times 2}{2 \times 2} = \frac{2}{4} $

Now, substitute the equivalent fraction back into the exponent expression:

$ \frac{2}{4} - \frac{1}{4} = \frac{2 - 1}{4} = \frac{1}{4} $

Step 4: Formulating the Final Expression

Replacing the simplified exponent back onto our base of $11$, we obtain the final simplified form:

$ 11^{\frac{1}{4}} $

This can also be written in radical notation as the fourth root of $11$, or $\sqrt[4]{11}$, but standard exponential simplification leaves it in fractional exponent form unless otherwise specified.

Final Solution: $ 11^{\frac{1}{4}} $