$32^4+33^{2^2}-3^{3^{2^{5^1}}}$
Answer (2)
To evaluate the expression 3 2 4 + 3 3 2 2 − 3 3 2 5 1 , first calculate the individual powers. After substituting the calculated values into the original expression, the final result is − 1853020186617344 .
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Evaluate the exponents: 2 2 = 4 , 5 1 = 5 , 2 5 = 32 , and calculate 3 32 .
Calculate 3 2 4 = 1048576 and 3 3 4 = 1185921 .
Substitute the calculated values into the expression: 3 2 4 + 3 3 4 − 3 32 = 1048576 + 1185921 − 1853020188851841 .
Perform the arithmetic to obtain the final result: − 1853020186617344 .
Explanation
Initial Analysis We are asked to evaluate the expression 3 2 4 + 3 3 2 2 − 3 3 2 5 1 . To do this, we need to follow the order of operations (PEMDAS/BODMAS), which means we evaluate exponents first.
Simplifying Exponents First, let's simplify the exponents. We have:
2 2 = 4 5 1 = 5 2 5 = 32 3 32 = 1853020188851841
Rewriting the Expression Now we can rewrite the expression as:
3 2 4 + 3 3 4 − 3 3 2 5 1 = 3 2 4 + 3 3 4 − 3 3 2 5 = 3 2 4 + 3 3 4 − 3 3 32
Since we already calculated 3 32 , we have:
3 2 4 + 3 3 4 − 3 32
Calculating Powers Next, we calculate 3 2 4 and 3 3 4 :
3 2 4 = ( 3 2 2 ) 2 = ( 1024 ) 2 = 1048576 3 3 4 = ( 3 3 2 ) 2 = ( 1089 ) 2 = 1185921
Substituting Values Now we can substitute these values back into the expression:
3 2 4 + 3 3 4 − 3 32 = 1048576 + 1185921 − 1853020188851841
Final Calculation Finally, we perform the addition and subtraction:
1048576 + 1185921 − 1853020188851841 = 2234497 − 1853020188851841 = − 1853020186617344
Final Answer Therefore, the final answer is:
− 1853020186617344
Examples
Understanding large numbers and exponents is crucial in many fields, such as cryptography and computer science. For example, when generating encryption keys, we often use very large prime numbers raised to large powers. The difficulty in factoring these large numbers is what keeps our data secure. This problem demonstrates how quickly numbers can grow with exponentiation, which is a fundamental concept in these applications.
