a3+b3 Formula: This algebraic identity can be factored into (a + b)(a^2 – ab + b^2). Understanding the a^3 + b^3 formula is essential in simplifying expressions and solving equations involving cubes. For example, if a=2 and b=3, then a^3 + b^3 = 2^3 + 3^3 = 8 + 27 = 35. This formula not only aids in computation but also enhances problem-solving skills in algebra, making it a fundamental concept for students and professionals alike.
Question: What is the a3+b3 formula:
- (a+b)(a2-ab+b2)
- (a-b)(a2+ab+b2)
- a3+b3-3ab(a+b)
- a3+b3+3ab(a+b)
Answer: (A) (a+b)(a2-ab+b2)
a3+b3 Formula Solution:
- Start with the following identity:
(a+b)3 = a3 + 3a2b + 3ab2 + b3
- Expand the left-hand side of the equation using the perfect cube factorization method:
(a+b)3 = (a+b)(a2+2ab+b2)
- Subtract a3 and b3 from both sides of the equation:
(a+b)3 - a3 - b3 = (a+b)(a2+2ab+b2) - a3 - b3
- Factor out a common factor of (a+b) on the right-hand side of the equation:
(a+b)(a2+2ab+b2) - a3 - b3 = (a+b)(a2+2ab+b2 - a2 - ab - b2)
- Simplify the expression on the right-hand side of the equation:
(a+b)(a2+2ab+b2 - a2 - ab - b2) = (a+b)(a2-ab+b2)
Therefore, the formula for a3+b3 is (a+b)(a2-ab+b2).
Prime Numbers 1 to 100
Prime numbers are natural numbers greater than 1 that have no positive divisors other than 1 and themselves. In the range from 1 to 100, the prime numbers include 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, and 97. These numbers play a crucial role in various fields, including mathematics, cryptography, and computer science. Understanding prime numbers from 1 to 100 can help build a foundation for more complex mathematical concepts and problem-solving skills.
