Simplifying a Mathematical Expression

The problem asks us to evaluate the value of a given mathematical expression which is a fraction.

The expression is: \(\frac{48.3\times[(4.95)^2+4.95\times13.25]}{[(12.55)^2-(5.65)^2]\times19.8} \)

Step 1: Simplify the Numerator

The numerator is \(48.3\times[(4.95)^2+4.95\times13.25]\). We can see that \(4.95\) is a common factor inside the square brackets.

Let's factor out \(4.95\):

\((4.95)^2+4.95\times13.25 = 4.95 \times (4.95 + 13.25)\)

Now, calculate the sum inside the parenthesis:

\(4.95 + 13.25 = 18.20\)

So, the numerator becomes:

\(48.3 \times [4.95 \times 18.20]\)

\(\text{Numerator} = 48.3 \times 4.95 \times 18.20\)

Step 2: Simplify the Denominator

The denominator is \([(12.55)^2-(5.65)^2]\times19.8\). The term inside the square brackets is in the form of a difference of squares, \(a^2 - b^2\), where \(a=12.55\) and \(b=5.65\).

Recall the difference of squares formula: \(a^2 - b^2 = (a-b)(a+b)\).

Apply the formula:

\((12.55)^2 - (5.65)^2 = (12.55 - 5.65)(12.55 + 5.65)\)

Calculate the terms inside the parentheses:

\(12.55 - 5.65 = 6.90\)

\(12.55 + 5.65 = 18.20\)

So, the term inside the square brackets is \(6.90 \times 18.20\).

The denominator becomes:

\([6.90 \times 18.20] \times 19.8\)

\(\text{Denominator} = 6.90 \times 18.20 \times 19.8\)

Step 3: Rewrite the Expression with Simplified Numerator and Denominator

Substitute the simplified forms back into the original expression:

\(\frac{48.3 \times 4.95 \times 18.20}{6.90 \times 18.20 \times 19.8}\)

Step 4: Cancel Common Terms

We can see that \(18.20\) appears in both the numerator and the denominator. We can cancel this common term:

\(\frac{48.3 \times 4.95 \times \cancel{18.20}}{6.90 \times \cancel{18.20} \times 19.8} = \frac{48.3 \times 4.95}{6.90 \times 19.8}\)

Step 5: Perform Remaining Calculations

Now we need to calculate \(\frac{48.3 \times 4.95}{6.90 \times 19.8}\).

We can calculate the ratios separately:

• \(\frac{48.3}{6.90}\): This is equivalent to \(\frac{483}{69}\). Both 483 and 69 are divisible by 3 (4+8+3=15, 6+9=15). \(483 \div 3 = 161\), \(69 \div 3 = 23\). So we have \(\frac{161}{23}\). Since \(23 \times 7 = 161\), this ratio is 7.
• \(\frac{4.95}{19.8}\): This is equivalent to \(\frac{495}{1980}\). Both are divisible by 5: \(495 \div 5 = 99\), \(1980 \div 5 = 396\). So we have \(\frac{99}{396}\). Both are divisible by 9: \(99 \div 9 = 11\), \(396 \div 9 = 44\). So we have \(\frac{11}{44}\). This simplifies to \(\frac{1}{4}\) or 0.25.

Multiply the results of the ratios:

\(7 \times 0.25\)

\(7 \times 0.25 = 1.75\)

Thus, the value of the given expression is 1.75.

Revision Table: Key Concepts Used

Additional Information: Algebraic Simplification Techniques

Simplifying mathematical expressions often involves applying various algebraic techniques to make calculations easier. Some common techniques include:

• Factoring: Identifying common factors in terms and rewriting the expression in a product form. This can reveal cancellations or simpler structures.
• Using Identities: Applying algebraic identities such as the difference of squares \((a^2-b^2)\), perfect squares \((a+b)^2\), sum/difference of cubes, etc. These identities provide shortcuts for expanding or factoring expressions.
• Combining Like Terms: Adding or subtracting terms that have the same variables raised to the same powers.
• Finding Common Denominators: When dealing with sums or differences of fractions, finding a common denominator allows the numerators to be combined.
• Canceling Terms: In fractions, canceling identical factors from the numerator and denominator simplifies the expression without changing its value. This is only possible for factors, not terms connected by addition or subtraction.

Mastering these techniques is crucial for solving more complex problems efficiently.