Evaluate each of the following :
`(i)2^(3)xx2^(2)" "(ii)3^(5)divide3^(2)" "(iii)(5^(2))^(3)" "(iv)((3)/(4))^(3)" "(v)((2)/(3))^(-3)" "(vi)2^(4)xx2^(-4)`

Explanation

Let's evaluate each part step by step:

### (i) \(2^3 \times 2^2\)

1. **Identify the bases and exponents**: The bases are the same (2), and the exponents are 3 and 2.
2. **Use the property of exponents**: According to the property \(a^m \times a^n = a^{m+n}\), we can add the exponents.
3. **Calculate**:
\[ 2^3 \times 2^2 = 2^{3+2} = 2^5 \]
4. **Evaluate \(2^5\)**:
\[ 2^5 = 32 \]

**Final answer for (i)**: \(32\)

### (ii) \(3^5 \div 3^2\)

1. **Identify the bases and exponents**: The bases are the same (3), and the exponents are 5 and 2.
2. **Use the property of exponents**: According to the property \(a^m \div a^n = a^{m-n}\), we can subtract the exponents.
3. **Calculate**:
\[ 3^5 \div 3^2 = 3^{5-2} = 3^3 \]
4. **Evaluate \(3^3\)**:
\[ 3^3 = 27 \]

**Final answer for (ii)**: \(27\)

### (iii) \((5^2)^3\)

1. **Identify the base and exponents**: The base is 5, the first exponent is 2, and the second exponent is 3.
2. **Use the property of exponents**: According to the property \((a^m)^n = a^{m \cdot n}\), we can multiply the exponents.
3. **Calculate**:
\[ (5^2)^3 = 5^{2 \cdot 3} = 5^6 \]
4. **Evaluate \(5^6\)**:
\[ 5^6 = 15625 \]

**Final answer for (iii)**: \(15625\)

### (iv) \(\left(\frac{3}{4}\right)^3\)

1. **Identify the base and exponent**: The base is \(\frac{3}{4}\) and the exponent is 3.
2. **Use the property of exponents**: According to the property \(\left(\frac{a}{b}\right)^m = \frac{a^m}{b^m}\), we can apply the exponent to both the numerator and denominator.
3. **Calculate**:
\[ \left(\frac{3}{4}\right)^3 = \frac{3^3}{4^3} \]
4. **Evaluate**:
\[ 3^3 = 27 \quad \text{and} \quad 4^3 = 64 \]
Therefore,
\[ \frac{3^3}{4^3} = \frac{27}{64} \]

**Final answer for (iv)**: \(\frac{27}{64}\)

### (v) \(\left(\frac{2}{3}\right)^{-3}\)

1. **Identify the base and exponent**: The base is \(\frac{2}{3}\) and the exponent is -3.
2. **Use the property of exponents**: According to the property \(a^{-m} = \frac{1}{a^m}\), we can take the reciprocal.
3. **Calculate**:
\[ \left(\frac{2}{3}\right)^{-3} = \frac{1}{\left(\frac{2}{3}\right)^3} = \frac{1}{\frac{2^3}{3^3}} = \frac{3^3}{2^3} \]
4. **Evaluate**:
\[ 3^3 = 27 \quad \text{and} \quad 2^3 = 8 \]
Therefore,
\[ \frac{3^3}{2^3} = \frac{27}{8} \]

**Final answer for (v)**: \(\frac{27}{8}\)

### (vi) \(2^4 \times 2^{-4}\)

1. **Identify the bases and exponents**: The bases are the same (2), and the exponents are 4 and -4.
2. **Use the property of exponents**: According to the property \(a^m \times a^n = a^{m+n}\), we can add the exponents.
3. **Calculate**:
\[ 2^4 \times 2^{-4} = 2^{4 + (-4)} = 2^0 \]
4. **Evaluate \(2^0\)**:
\[ 2^0 = 1 \]

**Final answer for (vi)**: \(1\)

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