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Let's go through the problems step by step.

### 1. (a) \( \left( \frac{1}{2} \right)^0 \)

Any non-zero number raised to the power of 0 equals 1. Thus:
\[
\left( \frac{1}{2} \right)^0 = 1
\]

### 1. (b) \( 1^0 + (-3)^0 \)

Both \(1^0\) and \((-3)^0\) follow the same rule for exponents of 0:
\[
1^0 = 1 \quad \text{and} \quad (-3)^0 = 1
\]
Thus:
\[
1^0 + (-3)^0 = 1 + 1 = 2
\]

### 2. (a) \( \left( \frac{1}{3} \right)^{-2} \times (-2)^{-3} \)

1. For \( \left( \frac{1}{3} \right)^{-2} \), we apply the negative exponent rule:
   \[
   \left( \frac{1}{3} \right)^{-2} = \left( \frac{3}{1} \right)^2 = 9
   \]
2. For \( (-2)^{-3} \), we apply the same rule for negative exponents:
   \[
   (-2)^{-3} = \frac{1}{(-2)^3} = \frac{1}{-8} = -\frac{1}{8}
   \]

Now multiply the results:
\[
9 \times -\frac{1}{8} = -\frac{9}{8}
\]

### 2. (b) \( \left( \frac{3}{2} \right)^{-4} \times \left( \frac{1}{4} \right)^3 \)

1. For \( \left( \frac{3}{2} \right)^{-4} \), apply the negative exponent rule:
   \[
   \left( \frac{3}{2} \right)^{-4} = \left( \frac{2}{3} \right)^4 = \frac{16}{81}
   \]
2. For \( \left( \frac{1}{4} \right)^3 \):
   \[
   \left( \frac{1}{4} \right)^3 = \frac{1}{64}
   \]

Now multiply the results:
\[
\frac{16}{81} \times \frac{1}{64} = \frac{16 \times 1}{81 \times 64} = \frac{16}{5184} = \frac{1}{324}
\]

### 3. (a) \( 6^{-2} \times 6^{-1} \div 6^0 \)

Using exponent rules:
1. \( 6^{-2} \times 6^{-1} \) means you add the exponents:
   \[
   6^{-2} \times 6^{-1} = 6^{-2 - 1} = 6^{-3}
   \]
2. Since \( 6^0 = 1 \), dividing by \( 6^0 \) doesn't change the value:
   \[
   6^{-3} \div 1 = 6^{-3}
   \]
Thus:
\[
6^{-3} = \frac{1}{6^3} = \frac{1}{216}
\]

### 3. (b) \( (2^{-2})^{-3} \div (-2)^{-6} \times (2^{-3})^4 \)

1. For \( (2^{-2})^{-3} \), multiply the exponents:
   \[
   (2^{-2})^{-3} = 2^{6}
   \]
2. For \( (-2)^{-6} \), the negative exponent rule gives:
   \[
   (-2)^{-6} = \frac{1}{(-2)^6} = \frac{1}{64}
   \]
3. For \( (2^{-3})^4 \), multiply the exponents:
   \[
   (2^{-3})^4 = 2^{-12}
   \]

Now combine:
\[
\frac{2^6}{\frac{1}{64}} \times 2^{-12} = 2^6 \times 64 \times 2^{-12} = 64 \times 2^{-6} = \frac{64}{64} = 1
\]

Let me know if you have any questions, or would like more details.

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Here are 5 related questions you can explore:

1. What is the general rule for handling negative exponents?
2. How do you simplify expressions when multiplying terms with the same base?
3. How does the zero exponent rule work for different bases?
4. What are some real-world applications of exponentiation?
5. How do fractional exponents relate to roots (like square roots)?

**Tip:** Always combine like bases by adding or subtracting their exponents when multiplying or dividing.

Find the values of the following expressions without using a calculator (1-4). Leave your answers in fractions if necessary.

Learn how to solve problems involving zero exponents, negative exponents, and fraction exponentiation. Includes detailed solutions for expressions like (1/2)^0 and combining negative exponents.

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