x(1-1%2F3)x(1-1%2F4)x(1-1%2F5)x(1-1%2F6)x(1-1%2F7).jpeg "(1-1/2)x(1-1/3)x(1-1/4)x(1-1/5)x(1-1/6)x(1-1/7)")
Simplifying a Product of Fractions
Let's analyze the expression: (1-1/2)x(1-1/3)x(1-1/4)x(1-1/5)x(1-1/6)x(1-1/7).
This expression consists of a product of several fractions. Each fraction is of the form (1-1/n), where 'n' is a consecutive integer starting from 2.
Simplifying each fraction:
- (1-1/2) = 1/2
- (1-1/3) = 2/3
- (1-1/4) = 3/4
- (1-1/5) = 4/5
- (1-1/6) = 5/6
- (1-1/7) = 6/7
Finding the product:
Now we have: (1/2) x (2/3) x (3/4) x (4/5) x (5/6) x (6/7)
Notice that most of the numerators and denominators cancel out:
(1/ 2) x (2 / 3) x (3 / 4) x (4 / 5) x (5 / 6) x (6 / 7) = 1/7
The Result
Therefore, the value of the expression (1-1/2)x(1-1/3)x(1-1/4)x(1-1/5)x(1-1/6)x(1-1/7) is 1/7.
This pattern highlights a useful technique for simplifying expressions with fractions. By observing cancellations, we can often arrive at a simplified result.