Simplify :
(i) `2^(2//3).2^(1//5)`
(ii) `((1)/(3^(3)))^(7)`
(iii) `(11^((1)/(2)))/(11^((1)/(4)))`
(iv) `7^(1//2).8^(1//2)`

Answer

Let's simplify each expression step by step.

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

1. **Use the property of exponents**: When multiplying two powers with the same base, we can add the exponents.
\[ 2^{\frac{2}{3}} \cdot 2^{\frac{1}{5}} = 2^{\frac{2}{3} + \frac{1}{5}} \]

2. **Find a common denominator**: The least common multiple (LCM) of 3 and 5 is 15. We will convert the fractions to have a common denominator.
\[ \frac{2}{3} = \frac{10}{15}, \quad \frac{1}{5} = \frac{3}{15} \]

3. **Add the fractions**:
\[ \frac{10}{15} + \frac{3}{15} = \frac{13}{15} \]

4. **Combine the exponents**:
\[ 2^{\frac{2}{3} + \frac{1}{5}} = 2^{\frac{13}{15}} \]

**Final answer**: \( 2^{\frac{13}{15}} \)

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### (ii) Simplify \( \left(\frac{1}{3^3}\right)^7 \)

1. **Use the power of a power property**: When raising a power to another power, we multiply the exponents.
\[ \left(\frac{1}{3^3}\right)^7 = \frac{1^7}{(3^3)^7} = \frac{1}{3^{3 \cdot 7}} \]

2. **Calculate the exponent**:
\[ 3 \cdot 7 = 21 \]

3. **Write the final expression**:
\[ \frac{1}{3^{21}} \]

**Final answer**: \( \frac{1}{3^{21}} \)

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

1. **Use the property of exponents**: When dividing two powers with the same base, we subtract the exponents.
\[ \frac{11^{\frac{1}{2}}}{11^{\frac{1}{4}}} = 11^{\frac{1}{2} - \frac{1}{4}} \]

2. **Find a common denominator**: The LCM of 2 and 4 is 4.
\[ \frac{1}{2} = \frac{2}{4} \]

3. **Subtract the fractions**:
\[ \frac{2}{4} - \frac{1}{4} = \frac{1}{4} \]

4. **Combine the exponents**:
\[ 11^{\frac{1}{2} - \frac{1}{4}} = 11^{\frac{1}{4}} \]

**Final answer**: \( 11^{\frac{1}{4}} \)

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### (iv) Simplify \( 7^{\frac{1}{2}} \cdot 8^{\frac{1}{2}} \)

1. **Use the property of exponents**: When multiplying two powers with different bases but the same exponent, we can combine them under a single exponent.
\[ 7^{\frac{1}{2}} \cdot 8^{\frac{1}{2}} = (7 \cdot 8)^{\frac{1}{2}} \]

2. **Multiply the bases**:
\[ 7 \cdot 8 = 56 \]

3. **Combine under the exponent**:
\[ (7 \cdot 8)^{\frac{1}{2}} = 56^{\frac{1}{2}} \]

**Final answer**: \( 56^{\frac{1}{2}} \)

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