Simplifying the Expression: (x^22x/2x^2+82x^2/84x+2x^2x^3)(11/x2/x^2)
This article will guide you through the process of simplifying the complex expression:
(x^22x/2x^2+82x^2/84x+2x^2x^3)(11/x2/x^2)
Let's break down the steps to achieve a simplified form.
Step 1: Simplifying the First Expression
First, we need to simplify the expression within the first set of parentheses:
(x^22x/2x^2+82x^2/84x+2x^2x^3)

Combine like terms:
 (x^2 + 2x^2  x^3)  (2x/2x^2)  (2x^2/84x) + 8

Simplify fractions:
 (3x^2  x^3)  (1/x)  (x^2/42x) + 8

Rearrange the terms:
 x^3 + 3x^2  x^2/(42x)  1/x + 8
Step 2: Simplifying the Second Expression
Next, we simplify the expression within the second set of parentheses:
(11/x2/x^2)

Find a common denominator:
 (x^2/x^2  x/x^2  2/x^2)

Combine terms:
 (x^2x2)/x^2
Step 3: Multiplying the Simplified Expressions
Now, we multiply the simplified expressions from Steps 1 and 2:
(x^3 + 3x^2  x^2/(42x)  1/x + 8) * (x^2x2)/x^2

Distribute:
 (x^5 + x^4 + 2x^3 + 3x^4  3x^3  6x^2  x^2(x^2x2)/x^2(42x) + (x+2)/x^3 + 8x^2  8x  16)/x^2

Simplify by canceling common factors:
 (x^5 + 4x^4  x^3  6x^2  (x^2x2)/(42x) + (x+2)/x^3 + 8x^2  8x  16)/x^2
Step 4: Further Simplification (Optional)
We can simplify the expression further by finding a common denominator for the remaining terms and combining them. However, this process can be quite involved. The expression obtained in Step 3 is already in a relatively simplified form.
Conclusion
By breaking down the original expression into smaller parts and simplifying them systematically, we arrive at a simplified form:
(x^5 + 4x^4  x^3  6x^2  (x^2x2)/(42x) + (x+2)/x^3 + 8x^2  8x  16)/x^2
This expression can be further simplified depending on the desired level of detail and complexity.