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Q3(i):
Simplify :
(i) $2^{\frac{2}{3}} \cdot 2^{\frac{1}{5}}$
Solution :
Step 1: Initial Setup & Identification of Variables
We are tasked with simplifying the following mathematical expression involving fractional exponents:
$2^{\frac{2}{3}} \cdot 2^{\frac{1}{5}}$
By analyzing the expression, we identify the following components:
- Base ($a$): Both terms share a common base of $2$.
- First Exponent ($m$): $\frac{2}{3}$
- Second Exponent ($n$): $\frac{1}{5}$
Step 2: Application of the Laws of Exponents
To simplify the multiplication of two exponential terms with the same base, we apply the Product Rule for Exponents. [Per the fundamental laws of exponents for real numbers], when multiplying two expressions with identical bases, their exponents are added together while the base remains unchanged.
The algebraic theorem is stated as:
$a^m \cdot a^n = a^{m+n}$
Substituting our identified variables into this theorem yields:
$2^{\frac{2}{3}} \cdot 2^{\frac{1}{5}} = 2^{\left(\frac{2}{3} + \frac{1}{5}\right)}$
Step 3: Algebraic Manipulation of Fractional Exponents
We must now evaluate the sum of the fractional exponents: $\frac{2}{3} + \frac{1}{5}$.
To add fractions with different denominators, we must first determine their Least Common Multiple (LCM). The denominators are $3$ and $5$. Since both are prime numbers, their LCM is simply their product:
$\text{LCM}(3, 5) = 3 \times 5 = 15$
Next, we convert each fraction into an equivalent fraction with a denominator of $15$:
- For the first fraction: Multiply the numerator and denominator by $5$.
$\frac{2}{3} = \frac{2 \times 5}{3 \times 5} = \frac{10}{15}$ - For the second fraction: Multiply the numerator and denominator by $3$.
$\frac{1}{5} = \frac{1 \times 3}{5 \times 3} = \frac{3}{15}$
Step 4: Final Simplification
Now, substitute the equivalent fractions back into the exponent sum and evaluate:
$\frac{2}{3} + \frac{1}{5} = \frac{10}{15} + \frac{3}{15} = \frac{10 + 3}{15} = \frac{13}{15}$
Replacing this sum back into our exponential expression gives the final simplified form:
$2^{\left(\frac{2}{3} + \frac{1}{5}\right)} = 2^{\frac{13}{15}}$
Final Solution: The simplified form of the expression is $2^{\frac{13}{15}}$.
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(ii): Simplify : (ii) $(\frac{1}{3^3})^7$
- Q3(iii): Simplify : (iii) $\frac{11^{\frac{1}{2}}}{11^{\frac{1}{4}}}$
- 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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