(3) $4^{240}-3^{12}$
Answer (2)
The expression 4 240 − 3 12 can be simplified to ( 2 240 − 729 ) ( 2 240 + 729 ) using exponent rules and the difference of squares formula. First, we rewrite 4 240 as 2 480 and 3 12 as 72 9 2 . Then, we apply the difference of squares factoring technique to reach the final answer.
;
Rewrite 4 240 as ( 2 2 ) 240 = 2 480 .
Rewrite 3 12 as ( 3 6 ) 2 = 72 9 2 .
Apply the difference of squares factorization: a 2 − b 2 = ( a − b ) ( a + b ) to get ( 2 240 ) 2 − ( 729 ) 2 = ( 2 240 − 729 ) ( 2 240 + 729 ) .
The simplified expression is ( 2 240 − 729 ) ( 2 240 + 729 ) .
Explanation
Initial Analysis We are asked to analyze the expression 4 240 − 3 12 . Our goal is to simplify or factorize this expression to better understand its structure.
Rewriting with Exponent Rules We can rewrite the expression using exponent rules. First, we rewrite 4 240 as ( 2 2 ) 240 = 2 480 . Next, we rewrite 3 12 as ( 3 6 ) 2 .
Calculating 3^6 Calculating 3 6 , we get 3 6 = 729 . Therefore, 3 12 = ( 3 6 ) 2 = 72 9 2 .
Rewriting the Expression Now we can rewrite the original expression as 4 240 − 3 12 = 2 480 − 72 9 2 . We can also write 2 480 as ( 2 240 ) 2 . Thus, the expression becomes ( 2 240 ) 2 − 72 9 2 .
Applying Difference of Squares We can now apply the difference of squares factorization, which states that a 2 − b 2 = ( a − b ) ( a + b ) . In our case, a = 2 240 and b = 729 . Therefore, we have ( 2 240 ) 2 − 72 9 2 = ( 2 240 − 729 ) ( 2 240 + 729 ) .
Final Result So, 4 240 − 3 12 = ( 2 240 − 729 ) ( 2 240 + 729 ) . This is the simplified form of the expression using difference of squares.
Examples
The difference of squares factorization, used here, is a fundamental concept in algebra. It's used in various applications, such as simplifying complex expressions, solving equations, and even in cryptography. For example, in cryptography, factoring large numbers into their prime factors is a crucial step in breaking certain encryption algorithms. The difference of squares can sometimes be used to factor these large numbers more easily. Also, in engineering, simplifying expressions is crucial for efficient computation and design. Factoring expressions using difference of squares can lead to more efficient algorithms and better designs.
