Of course! Let's solve the given expression step by step.
We have the expression:
[tex]\[
\frac{99813 \times 99815 + 1}{99814^2}
\][/tex]
### Step 1: Simplify the Numerator
The numerator is [tex]\(99813 \times 99815 + 1\)[/tex].
Notice that [tex]\(99813\)[/tex] and [tex]\(99815\)[/tex] are very close to [tex]\(99814\)[/tex]. Specifically:
[tex]\[
99813 = 99814 - 1
\][/tex]
[tex]\[
99815 = 99814 + 1
\][/tex]
So we have:
[tex]\[
99813 \times 99815 + 1 = (99814 - 1) \times (99814 + 1) + 1
\][/tex]
### Step 2: Use Algebraic Identity
We recognize that this has the form of the difference of squares:
[tex]\[
(a - 1)(a + 1) = a^2 - 1^2 = a^2 - 1
\][/tex]
Here, [tex]\(a = 99814\)[/tex]. Thus:
[tex]\[
(99814 - 1)(99814 + 1) = 99814^2 - 1
\][/tex]
Therefore:
[tex]\[
99813 \times 99815 + 1 = (99814^2 - 1) + 1 = 99814^2
\][/tex]
### Step 3: Simplify the Fraction
The expression now becomes:
[tex]\[
\frac{99814^2}{99814^2}
\][/tex]
### Step 4: Simplify the Division
Any number divided by itself is 1:
[tex]\[
\frac{99814^2}{99814^2} = 1
\][/tex]
### Final Result
The final result of the expression is:
[tex]\[
1.0
\][/tex]
This follows from our detailed step-by-step simplification and computation.
