Simplifying a Complex Algebraic Expression

We are asked to simplify the following expression:

\(\frac{(375+125)^2-(125-375)^2}{375\times375-125\times125}\)

This expression looks complicated with numbers, but we can simplify it significantly by using algebraic identities. Let's represent the numbers with variables to see the structure more clearly.

Let \(a = 375\) and \(b = 125\).

The expression can be rewritten as:

\(\frac{(a+b)^2-(b-a)^2}{a^2-b^2}\)

Analyzing the Numerator

The numerator is \((a+b)^2-(b-a)^2\). We know that \((b-a)^2 = (-(a-b))^2 = (a-b)^2\). So the numerator is actually \((a+b)^2 - (a-b)^2\).

We can use the algebraic identity: \((x+y)^2 - (x-y)^2 = 4xy\).

Using this identity with \(x=a\) and \(y=b\), the numerator simplifies to \(4ab\).

Analyzing the Denominator

The denominator is \(a^2-b^2\).

We can use the algebraic identity: \(x^2 - y^2 = (x-y)(x+y)\).

Using this identity with \(x=a\) and \(y=b\), the denominator simplifies to \((a-b)(a+b)\).

Simplifying the Entire Expression

Now substitute the simplified numerator and denominator back into the expression:

\(\frac{\text{Numerator}}{\text{Denominator}} = \frac{4ab}{(a-b)(a+b)}\)

Substituting the Values of a and b

Now, substitute \(a = 375\) and \(b = 125\) back into the simplified expression:

\(\frac{4 \times 375 \times 125}{(375-125)(375+125)}\)

Calculate the values in the denominator:

• \(375 - 125 = 250\)
• \(375 + 125 = 500\)

So the expression becomes:

\(\frac{4 \times 375 \times 125}{250 \times 500}\)

Now, let's simplify this fraction by cancelling common factors. Notice that \(250 = 2 \times 125\) and \(500 = 4 \times 125\) or \(500 = 2 \times 250\).

\(\frac{4 \times 375 \times 125}{250 \times 500} = \frac{4 \times 375 \times 125}{(2 \times 125) \times (4 \times 125)}\)

Cancel the 4 and 125 from the numerator and denominator:

\(\frac{4 \times 375 \times 125}{(2 \times 125) \times (4 \times 125)} = \frac{375}{2 \times 125} = \frac{375}{250}\)

Now simplify the fraction \(\frac{375}{250}\). Both numbers are divisible by 25.

• \(375 \div 25 = 15\)
• \(250 \div 25 = 10\)

The fraction becomes \(\frac{15}{10}\).

Both numbers are divisible by 5.

• \(15 \div 5 = 3\)
• \(10 \div 5 = 2\)

The simplified fraction is \(\frac{3}{2}\).

Final Answer

The simplified value of the given expression is \(\frac{3}{2}\).

Revision Table: Key Algebraic Identities for Simplification

Understanding key algebraic identities is crucial for simplifying such expressions quickly and accurately. The identities used here are fundamental.

• Square of Sum: \((x+y)^2 = x^2 + 2xy + y^2\)
• Square of Difference: \((x-y)^2 = x^2 - 2xy + y^2\)
• Difference of Squares: \(x^2 - y^2 = (x-y)(x+y)\)
• Derived Identity: \((x+y)^2 - (x-y)^2 = (x^2 + 2xy + y^2) - (x^2 - 2xy + y^2) = x^2 + 2xy + y^2 - x^2 + 2xy - y^2 = 4xy\)

Additional Information: Applying Identities in Math Problems

Using algebraic identities is a powerful technique in mathematics, especially in competitive exams and algebra simplification problems. They help in converting complex expressions into simpler forms, making calculations easier. Recognizing patterns like the difference of squares or the difference of squared sums/differences allows you to bypass lengthy arithmetic calculations. In this specific problem, recognizing \((a+b)^2 - (a-b)^2\) as \(4ab\) and \(a^2 - b^2\) as \((a-b)(a+b)\) was key to simplifying the problem efficiently. Always look for opportunities to apply these standard identities when dealing with squares or differences in algebraic or numerical expressions.