Answer :

GinnyAnswer
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.

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