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Q3(iii):
Simplify :
(iii) $\frac{11^{\frac{1}{2}}}{11^{\frac{1}{4}}}$
Solution :
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}} $
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(iv): Simplify : (iv) $7^{\frac{1}{2}} \cdot 8^{\frac{1}{2}}$
CBSE Solutions for Class 9 Mathematics Number Systems
Chapters in CBSE - Class 9 Mathematics
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