Evaluating Algebraic Expressions Using Identities

The problem asks us to evaluate a given algebraic expression. The expression looks complex, but we can simplify it by recognizing a common algebraic identity.

The given expression is:

\(\dfrac{(0.73)^3+(0.31)^3}{(0.73)^2-0.73\times0.31+(0.31)^2}\)

Let's look at the structure of the expression. It is in the form of a fraction where the numerator is a sum of cubes and the denominator is a quadratic expression involving the same terms.

Consider the algebraic identity for the sum of cubes:

\(a^3 + b^3 = (a+b)(a^2 - ab + b^2)\)

Now, let's compare this identity with our given expression.

Let \(a = 0.73\) and \(b = 0.31\).

The numerator of the given expression is \((0.73)^3 + (0.31)^3\), which is in the form \(a^3 + b^3\).

The denominator of the given expression is \((0.73)^2 - 0.73 \times 0.31 + (0.31)^2\), which is in the form \(a^2 - ab + b^2\).

So, the given expression can be written as:

\(\dfrac{a^3 + b^3}{a^2 - ab + b^2}\)

Using the identity \(a^3 + b^3 = (a+b)(a^2 - ab + b^2)\), we can rewrite the numerator:

\(\dfrac{(a+b)(a^2 - ab + b^2)}{a^2 - ab + b^2}\)

Assuming that the denominator \(a^2 - ab + b^2\) is not equal to zero, we can cancel out the common term \((a^2 - ab + b^2)\) from the numerator and the denominator.

In this case, \(a=0.73\) and \(b=0.31\). The denominator is \((0.73)^2 - (0.73)(0.31) + (0.31)^2\). Since \(a\) and \(b\) are positive, \(a^2\) and \(b^2\) are positive. While \(ab\) is positive, the entire expression \(a^2 - ab + b^2\) for real numbers \(a, b\) is always non-negative (it is equal to \((a - b/2)^2 + 3b^2/4\)). If \(a=b=0\), it is zero, but here \(a\) and \(b\) are non-zero. If \(a\) and \(b\) have the same sign, it is zero only if \(a=b=0\). If \(a\) and \(b\) have opposite signs, it is always positive. Here \(a=0.73\) and \(b=0.31\) are positive, so \(a^2 - ab + b^2 > 0\).

Therefore, we can safely cancel the term:

\(\dfrac{(a+b)\cancel{(a^2 - ab + b^2)}}{\cancel{a^2 - ab + b^2}}\)

The simplified expression is simply:

\(a+b\)

Now, substitute the values of \(a\) and \(b\) back into the simplified expression:

\(a+b = 0.73 + 0.31\)

Adding these two numbers:

\(0.73 + 0.31 = 1.04\)

Thus, the value of the given algebraic expression is \(1.04\).

Step-by-Step Evaluation

1. Identify the given expression: \(\dfrac{(0.73)^3+(0.31)^3}{(0.73)^2-0.73\times0.31+(0.31)^2}\).
2. Recognize the form \( \dfrac{a^3 + b^3}{a^2 - ab + b^2} \) with \(a = 0.73\) and \(b = 0.31\).
3. Recall the algebraic identity \(a^3 + b^3 = (a+b)(a^2 - ab + b^2)\).
4. Substitute the identity into the expression: \( \dfrac{(a+b)(a^2 - ab + b^2)}{a^2 - ab + b^2} \).
5. Cancel the common term \((a^2 - ab + b^2)\), leaving \(a+b\).
6. Substitute the values of \(a\) and \(b\) back: \(0.73 + 0.31\).
7. Calculate the sum: \(0.73 + 0.31 = 1.04\).

Revision Table: Key Algebraic Identities

Additional Information: Importance of Algebraic Identities

Algebraic identities are fundamental equations that are true for all values of the variables involved. They are powerful tools used to simplify expressions, factor polynomials, and solve equations more efficiently.

• They help in quickly simplifying complex expressions, as shown in this problem.
• Knowing identities can save significant time during exams.
• They form the basis for understanding more advanced algebraic concepts.
• Identities like the sum/difference of cubes and squares are frequently used in various fields of mathematics and science.

Mastering these basic identities is crucial for anyone studying algebra or preparing for competitive exams that involve quantitative aptitude sections.