Simplifying the Algebraic Expression

The question asks for the value of \(k\) in the equation \(\frac{(2.53)^3+(2.47)^3}{25.3×25.3-624.91+24.7×24.7} = 5× 10^k\).

Let's analyze the given expression:

The numerator is in the form \(a^3 + b^3\), where \(a = 2.53\) and \(b = 2.47\).

The formula for the sum of cubes is: \(a^3 + b^3 = (a+b)(a^2 - ab + b^2)\).

Using this formula, the numerator can be written as:

\((2.53 + 2.47)((2.53)^2 - (2.53)(2.47) + (2.47)^2)\)

Let's calculate the sum \(a+b\):

\(a + b = 2.53 + 2.47 = 5.00 = 5\)

So the numerator is \(5 \times ((2.53)^2 - (2.53)(2.47) + (2.47)^2)\).

Now let's look at the denominator: \(25.3 \times 25.3 - 624.91 + 24.7 \times 24.7\).

Notice the relationship between the numbers in the denominator and the numerator:

• \(25.3 = 2.53 \times 10\)
• \(24.7 = 2.47 \times 10\)

Let's substitute these into the denominator:

\((2.53 \times 10) \times (2.53 \times 10) - 624.91 + (2.47 \times 10) \times (2.47 \times 10)\)

\(= (2.53)^2 \times 100 - 624.91 + (2.47)^2 \times 100\)

\(= 100 \times (2.53)^2 - 624.91 + 100 \times (2.47)^2\)

Now let's examine the term \(624.91\).

Let's calculate the product \(25.3 \times 24.7\):

\(25.3 \times 24.7 = (25 + 0.3) \times (25 - 0.3)\)

This is in the form \((x+y)(x-y) = x^2 - y^2\).

\(25.3 \times 24.7 = 25^2 - (0.3)^2 = 625 - 0.09 = 624.91\).

So, \(624.91 = 25.3 \times 24.7 = (2.53 \times 10) \times (2.47 \times 10) = 100 \times (2.53 \times 2.47)\).

Substituting this back into the denominator:

Denominator \(= 100 \times (2.53)^2 - 100 \times (2.53 \times 2.47) + 100 \times (2.47)^2\)

We can factor out 100 from all terms:

Denominator \(= 100 \times [(2.53)^2 - (2.53 \times 2.47) + (2.47)^2]\)

Now let's rewrite the original expression with the factored numerator and denominator:

Expression \( = \frac{5 \times ((2.53)^2 - (2.53)(2.47) + (2.47)^2)}{100 \times [(2.53)^2 - (2.53 \times 2.47) + (2.47)^2]}\)

We can see that the term \((2.53)^2 - (2.53)(2.47) + (2.47)^2\) is present in both the numerator and the denominator. Assuming this term is not zero (which it isn't, as \(a^2-ab+b^2\) for non-equal real numbers a,b is generally non-zero), we can cancel it out.

Expression \( = \frac{5}{100}\)

Now simplify the fraction:

\(\frac{5}{100} = \frac{5}{10^2}\)

Using the rule of exponents \(\frac{1}{a^n} = a^{-n}\), we get:

\(\frac{5}{10^2} = 5 \times 10^{-2}\)

The problem states that the value of the expression is equal to \(5 \times 10^k\).

We found the value of the expression to be \(5 \times 10^{-2}\).

Comparing the two forms:

\(5 \times 10^{-2} = 5 \times 10^k\)

Since the bases (5 and 10) are equal, the exponents must be equal.

Therefore, \(k = -2\).

Revision Table: Key Concepts and Formulas

Additional Information on Simplifying Expressions

Simplifying algebraic expressions often involves recognizing common formulas and factoring patterns. In this problem, the key was identifying the numerator as a sum of cubes and relating the denominator terms to the base numbers in the numerator by considering powers of 10.

• Pattern Recognition: Always look for patterns like \(a^2 - b^2\), \(a^3 + b^3\), \(a^3 - b^3\), or perfect squares/cubes within the expression.
• Factorization: Factoring is a crucial technique to simplify complex fractions by canceling out common terms in the numerator and denominator.
• Decimal Manipulation: Expressions involving decimals can often be simplified by recognizing them as multiples of base numbers or powers of 10. For example, \(25.3\) is \(2.53 \times 10\).
• Exponent Rules: Understanding how to work with exponents (like \(10^2\) and \(10^{-2}\)) is essential for writing numbers in scientific notation or the required form.

Practicing various types of expressions helps in quickly identifying the appropriate method for simplification.