Simplify the given mathematical expression involving nested square roots and fractions.
Solution
Let's solve the given expression step by step. The expression appears as:
(a+1−a−1a+1+a−1−a+1+a−1a+1−a−1):aa+1
Step 1: Simplify each fraction in the numerator
First term:
a+1−a−1a+1+a−1
Multiply numerator and denominator by the conjugate of the denominator:
a+1−a−1a+1+a−1⋅a+1+a−1a+1+a−1
This results in:
(a+1)2−(a−1)2(a+1+a−1)2
The denominator simplifies as:
(a+1)2−(a−1)2=a+1−(a−1)=2
The numerator expands as:
(a+1+a−1)2=(a+1)+2(a+1)(a−1)+(a−1)=2a+2a2−1
Thus, the first term becomes:
22a+2a2−1=a+a2−1
Second term:
a+1+a−1a+1−a−1
Following the same process:
a+1+a−1a+1−a−1⋅a+1−a−1a+1−a−1
The denominator is again:
(a+1)2−(a−1)2=2
The numerator expands as:
(a+1−a−1)2=(a+1)−2(a+1)(a−1)+(a−1)=2a−2a2−1
Thus, the second term becomes:
22a−2a2−1=a−a2−1
Step 2: Simplify the numerator
The numerator now becomes:
(a+a2−1)−(a−a2−1)
Simplify:
a+a2−1−a+a2−1=2a2−1
Step 3: Include the denominator
The entire expression now becomes:
aa+12a2−1
Simplify by multiplying by the reciprocal of the denominator:
2a2−1⋅a+1a=a+12aa2−1
Final Answer:
a+12aa2−1
Let me know if you'd like any further clarifications or details!
Related Questions:
What are the restrictions on a for this expression to be valid?
How can we rationalize the denominator of the final result?
What happens to the expression as a→∞?
How does this result change if a is negative?
Can this expression be simplified further for specific values of a?
Tip:
Always check the domain of the variables when dealing with square roots to