To solve the expression \( 2^6 \times 2^4 \) in base 4, follow these steps:

### Step 1: Simplify the Expression
First, multiply the exponents since the bases are the same:
\[
2^6 \times 2^4 = 2^{6+4} = 2^{10}
\]

### Step 2: Calculate \( 2^{10} \)
Next, calculate \( 2^{10} \):
\[
2^{10} = 1024
\]

### Step 3: Convert 1024 to Base 4
Now, convert 1024 from base 10 to base 4.

To do this, divide the number by 4 repeatedly and keep track of the remainders:

1. \( 1024 \div 4 = 256 \) remainder 0
2. \( 256 \div 4 = 64 \) remainder 0
3. \( 64 \div 4 = 16 \) remainder 0
4. \( 16 \div 4 = 4 \) remainder 0
5. \( 4 \div 4 = 1 \) remainder 0
6. \( 1 \div 4 = 0 \) remainder 1

So, reading the remainders from bottom to top, \( 1024_{10} = 1000000_4 \).

### Final Answer:
\[
2^6 \times 2^4 = 1000000_4
\]

Would you like more details on any of these steps or have any questions?

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Here are 5 related questions:

1. How do you convert any base-10 number to a different base?
2. What is the general rule for multiplying powers with the same base?
3. How would you express \( 3^5 \) in base 4?
4. What is the result of \( 2^8 \times 2^3 \) in base 4?
5. How do you perform division in different number bases?

**Tip:** When multiplying powers with the same base, simply add the exponents to simplify the expression.

Learn how to solve the expression 2^6 multiplied by 2^4 in base 4 with a detailed, step-by-step solution. This problem covers key concepts like exponent multiplication and base conversion, suitable for students in Grades 7-10.

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