https://file.aiquickdraw.com/mathbot/file/1728403482153lws1wgjf.jpg

The problem in the image is written in Lithuanian and asks to "Uzrašykite laipsniu. Pateikite visus galimus variantus," which translates to "Write in powers. Provide all possible options." 

Here are the options listed:
a) \( 16^2 \cdot 16 \)
b) \( 16^2 \cdot 64 \)
c) \( 8^2 \cdot 64 \)
d) \( 64^2 \cdot 256 \)

The goal is likely to express each of these expressions in terms of powers of a single base.

### Let's break each one down:

#### a) \( 16^2 \cdot 16 \)
Since \( 16 = 2^4 \), this expression becomes:
\[
16^2 \cdot 16 = (2^4)^2 \cdot 2^4 = 2^8 \cdot 2^4 = 2^{12}
\]

#### b) \( 16^2 \cdot 64 \)
We have \( 16 = 2^4 \) and \( 64 = 2^6 \), so:
\[
16^2 \cdot 64 = (2^4)^2 \cdot 2^6 = 2^8 \cdot 2^6 = 2^{14}
\]

#### c) \( 8^2 \cdot 64 \)
Since \( 8 = 2^3 \) and \( 64 = 2^6 \), this becomes:
\[
8^2 \cdot 64 = (2^3)^2 \cdot 2^6 = 2^6 \cdot 2^6 = 2^{12}
\]

#### d) \( 64^2 \cdot 256 \)
We have \( 64 = 2^6 \) and \( 256 = 2^8 \), so:
\[
64^2 \cdot 256 = (2^6)^2 \cdot 2^8 = 2^{12} \cdot 2^8 = 2^{20}
\]

### Final Results in Powers of 2:
a) \( 2^{12} \)  
b) \( 2^{14} \)  
c) \( 2^{12} \)  
d) \( 2^{20} \)

Would you like more details or have any questions about this?

Here are some related questions:
1. How do we simplify exponents when multiplying numbers with the same base?
2. What are the properties of exponents that allow this kind of simplification?
3. Can you generalize this process for any base?
4. What is the difference between multiplying powers and adding them?
5. How would these expressions change if the base numbers were different?

**Tip:** When multiplying numbers with the same base, simply add the exponents together!

Uzrašykite laipsniu. Pateikite visus galimus variantus: a) 16^2 ⋅ 16, b) 16^2 ⋅ 64, c) 8^2 ⋅ 64, d) 64^2 ⋅ 256

Solve problems involving exponentiation, including 16^2 ⋅ 16, 16^2 ⋅ 64, 8^2 ⋅ 64, and 64^2 ⋅ 256. Learn how to simplify powers using the properties of exponents, suitable for students in Grades 8-10.

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