x(1-1%2F7)x(1-1%2F8)x(1-1%2F9)x(1-1%2F10).jpeg "(1-1/6)x(1-1/7)x(1-1/8)x(1-1/9)x(1-1/10)")
Simplifying the Expression: (1-1/6)x(1-1/7)x(1-1/8)x(1-1/9)x(1-1/10)
This expression presents a pattern that allows for a much simpler solution than simply multiplying each term individually.
Understanding the Pattern
Notice that each term in the expression is in the form of (1 - 1/n), where n is an integer starting from 6 and increasing by one.
Let's expand the first few terms:
- (1 - 1/6) = 5/6
- (1 - 1/7) = 6/7
- (1 - 1/8) = 7/8
Observe that the numerator of each term is one less than the denominator of the next term. This pattern continues throughout the expression.
Simplifying the Expression
Now, let's rewrite the expression using this observation:
(1-1/6)x(1-1/7)x(1-1/8)x(1-1/9)x(1-1/10) = (5/6) x (6/7) x (7/8) x (8/9) x (9/10)
Notice that most of the terms cancel out:
(5/ 6) x (6 / 7) x (7 / 8) x (8 / 9) x (9 / 10) = 5/10
Final Result
Therefore, the simplified value of the expression (1-1/6)x(1-1/7)x(1-1/8)x(1-1/9)x(1-1/10) is 1/2.