Please help,

1) How do you solve 2017^2017 without a calculator?

2) What is the last digit of the solution to this problem: 2017^(2017^2017)?

Thank you!

look at numbers ending in 7, such as

7^1 = 7
7^2 = 49
7^3 = 343
7^4 = 2401
7^5 = 16807
7^6 = 117649
7^7 = 823543

notice that the unit digits forms a cyclic pattern:
7, 9, 3, 1, 7, 9 3, 1, ....

the same would be true for something like 37^n
try it.

Also:

2017^1 = 2017 --- 4 digits
2017^2 = 4068289 --- 7 digits
2017^3 = 8205738913 --- 10 digits
2017^4 = 1.655097539.. x 10^13
= 1655097539...3 --- 14 digits
2017^5 = 3.338...1 x 10^16 ---- 17 digits
2017^6 = 6.7334...7 x 10^19 --- 20 digits
...
2017^15 = 3.7203... x 10^49 --- 50 digits
..
2017^30 = 1.38... x 10^99 ---- 100 digits

You could use the first pattern to establish what the last digit of 2017^(2017^2017) is.

As to "solving" 2017^2017 without a calculator, or even with a calculator would be silly.
This would fall into the category of "true" pure math.

To solve 2017^2017 without a calculator, you can use the following steps:

Step 1: Start by finding the prime factorization of 2017. 2017 is a prime number, so its prime factorization is simply 2017.

Step 2: Next, we can use the properties of exponents to simplify the calculation. We have:

2017^2017 = (2017^1008)^2 * 2017

Step 3: Evaluate (2017^1008)^2 to find the solution.

2017^1008 = (2017^504)^2

Step 4: Simplify (2017^504)^2 further.

2017^504 = (2017^252)^2

Step 5: Continue this process until we get to 2017^1. When we reach 2017^1, we can calculate it directly since we have reached a small enough exponent.

2017^1 = 2017

Step 6: Multiply all the simplified expressions together to get the final result.

2017^2017 = (2017^1008)^2 * (2017^504)^2 * (2017^252)^2 * (2017^126)^2 * (2017^63)^2 * 2017

Now let's move on to the second question.

To find the last digit of the solution to 2017^(2017^2017), we need to focus on the last digit of each number.

The last digit of 2017 is 7.

For any exponent raised to a power, the last digit of the original number is preserved. So we need to find the remainder when 2017^2017 is divided by 10.

To find the remainder when 2017^2017 is divided by 10, we can simplify the exponent by finding the remainder when 2017 is divided by 10.

The remainder when 2017 is divided by 10 is 7.

Since the last digit of 2017^2017 is 7, the last digit of the power 2017^(2017^2017) is also 7.

Sure! Let's break down each question and explain how to solve them.

1) To solve 2017^2017 without a calculator, we can use a mathematical property known as modular arithmetic. We need to find the remainder when 2017^2017 is divided by a certain number.

The key here is to observe the pattern of the last digit of the powers of 2017. We can start by finding the last digit of 2017, which is 7. Then, we can calculate the powers of 7: 7^2 = 49, 7^3 = 343, 7^4 = 2401, and so on. You will notice that the last digits repeat in a cycle: 7, 9, 3, and 1.

Since the last digit of 2017 is 7 and the cycle repeats every four powers, we need to find the remainder when 2017 divided by 4. If the remainder is 0, the last digit of 2017^2017 will be 1; if the remainder is 1, the last digit will be 7; if the remainder is 2, the last digit will be 9; and if the remainder is 3, the last digit will be 3.

To find the remainder, we divide 2017 by 4: 2017 ÷ 4 = 504 remainder 1. So, the last digit of 2017^2017 is 7.

2) To find the last digit of the solution to 2017^(2017^2017), we can follow a similar approach as in question 1 but with an extra step.

First, let's find the last digit of 2017 raised to the power of 2017. Using the same digit cycling pattern as before, we find that the last digit of 2017 to the power of 2017 is 9.

Now, we need to find the pattern of the last digit of powers of 9. Calculate the powers of 9: 9^2 = 81, 9^3 = 729, 9^4 = 6561, and so on. Again, we see a cycle repeating: 9, 1, 9, 1. Therefore, the last digit of any power of 9 will be either 9 or 1, depending on whether the exponent is odd or even, respectively.

Since the exponent 2017^2017 is an odd number, the last digit of 9 raised to this power will be 9.

Therefore, the last digit of the solution to 2017^(2017^2017) is 9.

I hope this helps! Let me know if you have any further questions.