4 5is equal to 1,024. Evaluate the following expressions. a. 4 6b. 4 4c. 4 3 ⋅4 2

1. Break Down the Problem

We are given the expression $4^5 = 1024$ and need to evaluate the following expressions:

1. $4^6$
2. $4^4$
3. $4^3 \cdot 4^2$

2. Relevant Concepts

We can use the property of exponents that states: $a^m \cdot a^n = a^{m+n}$ Additionally, $a^{m+1} = a^m \cdot a$.

3. Analysis and Detail

1. Calculating $4^6$: $4^6 = 4^5 \cdot 4 = 1024 \cdot 4$ $1024 \times 4 = 4096$

2. Calculating $4^4$: $4^4 = \frac{4^5}{4} = \frac{1024}{4} = 256$

3. Calculating $4^3 \cdot 4^2$: Using the property of exponents: $4^3 \cdot 4^2 = 4^{3+2} = 4^5$ $4^5 = 1024$

4. Verify and Summarize

• $4^6 = 4096$
• $4^4 = 256$
• $4^3 \cdot 4^2 = 1024$

Final Answer

1. $4^6 = 4096$
2. $4^4 = 256$
3. $4^3 \cdot 4^2 = 1024$