<Question>
Convert $\frac{57}{160}$ to a terminating decimal.
</Question>

<think>
<Plan_1>
Okay, so I need to convert the fraction 57 over 160 into a terminating decimal. Hmm, let me remember how to do this. I think a fraction will convert to a terminating decimal if the denominator, after simplifying the fraction, has no prime factors other than 2 or 5. Right?

So first, maybe I should check if 57 and 160 have any common factors. Let's see.

<prompt_1.1> Check if the numerator 57 and the denominator 160 have any common factors. </prompt_1.1>

Therefore, the fraction is already in its simplest form.
</Plan_1>

<Execution_1>
<execution_1.1>
The numerator is 57, which is 3 times 19, if I recall correctly. And the denominator is 160. Let me factorize 160. 160 divided by 2 is 80, divided by 2 again is 40, again by 2 is 20, again by 2 is 10, and again by 2 is 5. So 160 is 2^5 times 5. So the prime factors of 160 are 2 and 5. 

Wait, and the numerator 57 is 3 times 19. So 57 and 160 don't have any common factors other than 1, right? So the fraction 57/160 is already in its simplest form.
</execution_1.1>
</Execution_1>

<Plan_2>
Based on execution_1.1:
Therefore, to determine if it's a terminating decimal, I need to look at the denominator. Since the fraction is simplified, the denominator is 160, which factors into 2^5 * 5. 

Wait, the rule says that if the denominator has only 2s and 5s as prime factors, then the decimal terminates.
Since 160 is 2^5 * 5^1, which are both 2 and 5, that means the decimal should terminate.

So, how do I convert 57/160 into a decimal? Well, one way is to divide 57 by 160. Let me do that division step by step.

Alternatively, maybe I can adjust the denominator to make it a power of 10, since terminating decimals are related to powers of 10.

<prompt_2.1> Consider adjusting the denominator of 57/160 to make it a power of 10 and compare methods. </prompt_2.1>

Let me try the division method.

So 57 divided by 160. Let's write that as 57 ÷ 160.

Since 160 is larger than 57, the decimal will start with 0. So, we have 0. and then proceed with the division.

To perform the division, we can add a decimal point and zeros to 57, making it 57.000...

<prompt_2.2> Perform the division of 57 by 160 step by step to convert it into a decimal. </prompt_2.2>
-----

Alternatively, using the power of 10 method. Since the denominator is 2^5 *5, if we multiply numerator and denominator by 5^(4) to make the denominator 2^5 *5^5, which is (2*5)^5 = 10^5 = 100,000. Wait, let me check that.

<prompt_2.3> Use the power of 10 method to convert 57/160 into a decimal and verify the result. </prompt_2.3>
</Plan_2>

<Execution_2>
<execution_2.1>
Since 160 is 2^5 * 5, if I can multiply both the numerator and denominator by 5^(3) to make the denominator 2^5 *5^4, which would be 10^5. Wait, let me check. 

Wait, 160 is 2^5 * 5^1. To convert the denominator to 10^n, which is 2^n * 5^n, we need to have the same number of 2s and 5s. Since there are 5 2s and 1 5, we need to multiply by 5^(4) to get 5^5, making the denominator 2^5 *5^5 = 10^5. So, multiplying numerator and denominator by 5^4 (which is 625) would give us a denominator of 10^5. Then the numerator becomes 57 * 625. Let me compute that.
</execution_2.1>

<execution_2.2>
So, 160 goes into 570 three times because 160*3 is 480. Subtract 480 from 570, we get 90. Bring down the next 0, making it 900. 160 goes into 900 five times (160*5=800). Subtract 800 from 900, we get 100. Bring down the next 0, making it 1000. 160 goes into 1000 six times (160*6=960). Subtract 960 from 1000, we get 40. Bring down the next 0, making it 400. 160 goes into 400 exactly two times (160*2=320). Wait, wait, 160*2 is 320? Wait, no, 160*2 is 320, but 160*2.5 would be 400. Wait, but we are dealing with whole numbers here. So 160*2=320, subtract that from 400, we get 80. Bring down the next 0, making it 800. 160 goes into 800 exactly 5 times. So 160*5=800, subtract that, we get 0. So the division is done here.

Putting it all together, the decimal is 0.35625.
</execution_2.2>

<execution_2.3>
The denominator is 2^5 *5^1. To make the denominator a power of 10, we need the same exponents for 2 and 5. So the higher exponent is 5 for 2. So we need to multiply by 5^(5-1) = 5^4. Wait, no. Let me think.

Wait, if the denominator is 2^5 *5^1, then to make it 10^n, we need to have both exponents equal. So the maximum exponent is 5. So we need to multiply numerator and denominator by 5^(5-1) = 5^4? Wait, no. Wait, denominator has 2^5 and 5^1. To make it 10^n, we need 2^n *5^n. So the current denominator has 2^5 and 5^1. So to make n equal to 5, we need to have 5^5. But the denominator already has 5^1, so we need to multiply by 5^(5-1) = 5^4. So denominator becomes 2^5 *5^1 *5^4 = 2^5 *5^5 = (2*5)^5 = 10^5. So yes, multiplying numerator and denominator by 5^4 (625) will convert the denominator to 10^5. 

So let's try that. 

Multiply numerator and denominator by 625:

Numerator: 57 * 625. Let's calculate that. 

57 * 600 = 34,200. 

57 * 25 = 1,425. 

So total numerator is 34,200 + 1,425 = 35,625. 

Denominator: 160 * 625 = 100,000. 

So the fraction becomes 35,625 / 100,000. 

Which is 0.35625.
</execution_2.3>
</Execution_2>

<Plan_3>
Based on execution_2.2:
Wait, let me check that again step by step.

<prompt_3.1> Verify the division of 57 by 160 step by step to ensure accuracy. </prompt_3.1>
-----

Based on execution_2.3:
But let me confirm once more. Let's do 57 ÷ 160.

<prompt_3.2> Confirm the result of converting 57/160 into a decimal using the power of 10 method. </prompt_3.2>
</Plan_3>

<Execution_3>
<execution_3.1>
First step: 57 divided by 160. 160 goes into 57 zero times. So we write 0. and then proceed. Then add a decimal point and zeros: 57.000000... 

160 into 570 (the first three digits after the decimal). 160*3=480, which is less than 570. 160*4=640, which is too much. So 3. Subtract 480 from 570: 90. Bring down the next 0: 900. 160*5=800. Subtract 800 from 900: 100. Bring down next 0: 1000. 160*6=960. Subtract 960 from 1000: 40. Bring down next 0: 400. 160*2=320. Subtract 320 from 400: 80. Bring down next 0: 800. 160*5=800. Subtract 800: 0. So the division gives 0.35625.
</execution_3.1>

<execution_3.2>
Alternatively, 160 is 1/625 of 100,000. Wait, no, 160*625=100,000. So 57*625=35,625. So 35,625 divided by 100,000 is 0.35625. 

Yes, that's correct.

So both methods give the same result. Therefore, 57/160 as a decimal is 0.35625, which is a terminating decimal.
</execution_3.2>
</Execution_3>

<Plan_4>
Based on execution_3.2:
Let me just confirm once more with another approach. Let's convert the denominator to 10 by multiplying numerator and denominator by 5^4 as before.

<prompt_4.1> Confirm the result using another approach and ensure consistency across different methods. </prompt_4.1>
</Plan_4>

<Execution_4>
<execution_4.1>
Wait, but I already did that. So that gives 35,625/100,000, which is 0.35625. 

Alternatively, maybe simplify the multiplication. Let's see 57*625. Let's break down 625 as 500 + 125. So 57*500 = 28,500. 57*125. 125 is 100 + 25. So 57*100=5,700. 57*25=1,425. So 5,700 + 1,425 = 7,125. Then 28,500 + 7,125 = 35,625. Yep, that's right. 

So 35,625 divided by 100,000. Since 100,000 is 10^5, moving the decimal point five places to the left. 35,625 becomes 0.35625. 

Everything checks out.
</execution_4.1>
</Execution_4>

<Final_answer>
Therefore, $\frac{57}{160}$ expressed as a terminating decimal is $\boxed{0.35625}$.
</Final_answer>
</think>