Q3(i):

Simplify : (i) $2^{\frac{2}{3}} \cdot 2^{\frac{1}{5}}$

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 $2^{\frac{13}{15}}$.