Evaluate Numerical Expression Using Algebraic Identities

We are asked to find the value of the given numerical expression:

\[ \frac{2.5\times 2.5+0.3\times 0.3+5\times 0.3}{1.6\times 1.6+0.6\times 0.6-1.2\times1.6} \]

Let's analyze the numerator and the denominator separately.

Analyzing the Numerator

The numerator is \(2.5\times 2.5+0.3\times 0.3+5\times 0.3\).

This can be written as \( (2.5)^2 + (0.3)^2 + 5 \times 0.3 \).

We can rewrite \(5 \times 0.3\) as \(2 \times 2.5 \times 0.3\). Let's check: \(2 \times 2.5 = 5\), so \(2 \times 2.5 \times 0.3 = 5 \times 0.3 = 1.5\). This matches the term in the numerator.

So the numerator is \( (2.5)^2 + (0.3)^2 + 2 \times 2.5 \times 0.3 \).

This expression is in the form \(a^2 + b^2 + 2ab\), which is the expansion of the algebraic identity \( (a+b)^2 \).

Here, \( a = 2.5 \) and \( b = 0.3 \).

Therefore, the numerator is \( (2.5 + 0.3)^2 = (2.8)^2 \).

Analyzing the Denominator

The denominator is \(1.6\times 1.6+0.6\times 0.6-1.2\times1.6\).

This can be written as \( (1.6)^2 + (0.6)^2 - 1.2 \times 1.6 \).

We can rewrite \(1.2 \times 1.6\) as \(2 \times 0.6 \times 1.6\). Let's check: \(2 \times 0.6 = 1.2\), so \(2 \times 0.6 \times 1.6 = 1.2 \times 1.6\). This matches the term in the denominator.

So the denominator is \( (1.6)^2 + (0.6)^2 - 2 \times 1.6 \times 0.6 \).

This expression is in the form \(c^2 + d^2 - 2cd\), which is the expansion of the algebraic identity \( (c-d)^2 \).

Here, \( c = 1.6 \) and \( d = 0.6 \).

Therefore, the denominator is \( (1.6 - 0.6)^2 = (1.0)^2 \).

Evaluating the Expression

Now we can rewrite the original expression using the simplified numerator and denominator:

\[ \frac{(2.8)^2}{(1.0)^2} \]

Calculate the values:

• Numerator: \( (2.8)^2 = 2.8 \times 2.8 \)
• Denominator: \( (1.0)^2 = 1.0 \times 1.0 \)

Let's calculate the numerator:

\( 2.8 \times 2.8 \)

Since there are two decimal places in total (one in 2.8 and one in 2.8), the result has two decimal places: \( 7.84 \).

Now, let's calculate the denominator:

\( (1.0)^2 = 1.0 \times 1.0 = 1 \).

So the expression becomes:

\[ \frac{7.84}{1} = 7.84 \]

Conclusion

The value of the given expression \(\frac{2.5\times 2.5+0.3\times 0.3+5\times 0.3}{1.6\times 1.6+0.6\times 0.6-1.2\times1.6}\) is \(7.84\).

Revision Table: Key Algebraic Identities

Additional Information on Algebraic Identities

Algebraic identities are equations that are true for all possible values of the variables they contain. They are useful tools for simplifying expressions, solving equations, and factoring polynomials.

Besides the square of sum and difference, other fundamental identities include:

• Difference of Squares: \( a^2 - b^2 = (a-b)(a+b) \)
• Cube of a Sum: \( (a+b)^3 = a^3 + 3a^2b + 3ab^2 + b^3 \)
• Cube of a Difference: \( (a-b)^3 = a^3 - 3a^2b + 3ab^2 - b^3 \)
• Sum of Cubes: \( a^3 + b^3 = (a+b)(a^2 - ab + b^2) \)
• Difference of Cubes: \( a^3 - b^3 = (a-b)(a^2 + ab + b^2) \)

Recognizing patterns that match these identities is a key skill in simplifying complex mathematical expressions and solving problems efficiently, especially in exams like competitive tests.