Problem 17 - Intermediate Algebra

Question

Simplify. Write answers using positive exponents. \(\frac{33 a^{2} b^{2}}{44 a^{4} b^{2}}\)

Step-by-Step Solution

Answer

The simplified expression is \( \frac{3}{4a^2} \).

Step 1: Simplify the Fraction Coefficients

Divide the coefficients 33 and 44. Both can be divided by their greatest common divisor, which is 11. \[ \frac{33}{44} = \frac{3}{4} \]

Step 2: Cancel Out Common Terms in the Numerator and Denominator

Both the numerator and the denominator have an \(a^2\) term and a \(b^2\) term that can be cancelled out:\[ \frac{a^2 b^2}{a^4 b^2} = \frac{a^2}{a^4} \times \frac{b^2}{b^2} \]This results in: \[ \frac{1}{a^{4-2}} \]since \(a^2\) cancels out \(a^4\), and \(b^2\) cancels out itself.

Step 3: Simplify Exponents

Subtract the exponents for the \(a\) term:\[ a^{4-2} = a^2 \]Combining with the fractions:\[ \frac{3}{4 a^2} \]

Step 4: Final Expression

The simplified expression, with only positive exponents, is:\[ \frac{3}{4a^2} \]

Key Concepts

Positive Exponents

Positive exponents are fundamental in algebra and signify how many times the base is multiplied by itself. An expression like \(a^2\) means \(a\) times \(a\), or \(a \times a\). When it comes to simplifying expressions, ensuring all exponents are positive is vital for clear and correct results.

When we simplify expressions, we must convert any negative exponents to positive. For example, \(a^{-3} = \frac{1}{a^3}\). The negative exponent indicates the reciprocal of the base raised to the positive exponent. Therefore, whenever you encounter negative exponents, flip the base to the denominator or numerator, depending on where it originated, and change the exponent to positive.

Understanding this concept helps ensure clarity in final expressions, as many mathematical conventions, such as standard form, prefer positive exponents.

Greatest Common Divisor

The greatest common divisor (GCD) is the largest number that divides two or more integers without leaving a remainder. In our expression, we start simplifying fractions by finding the GCD of the coefficients of the numerator and the denominator. Here, the coefficients 33 and 44 share a common divisor, which is 11.

1. **Identify common factors:** List factors of each number. For 33, they are \(1, 3, 11, 33\). For 44, they are \(1, 2, 4, 11, 22, 44\).
2. **Select the highest common factor:** The largest common factor they both share is 11.
3. **Divide to simplify:** Divide both the numerator and the denominator by the GCD. So, \( \frac{33}{44} = \frac{33 \div 11}{44 \div 11} = \frac{3}{4} \).

Using the GCD simplifies the problem greatly, eliminating unnecessary complexity from the coefficients.

Canceling Terms

Canceling terms involves removing identical terms from the numerator and the denominator, reducing the expression while maintaining its value. When dealing with expressions like polynomials, look for common factors in both parts.

For instance, in \( \frac{a^2 b^2}{a^4 b^2} \), both the numerator and the denominator share \(a^2\) and \(b^2\). We can cancel them by dividing them out of both parts, which simplifies significantly. Here's how:

• Simplify \( \frac{a^2 b^2}{a^4 b^2}\) to \( \frac{a^2}{a^4} \cdot \frac{b^2}{b^2} \).
• Since \(b^2\) is in both the numerator and the denominator, it cancels completely to 1.
• So, the expression becomes \( \frac{1}{a^{4-2}} \), simplifying the \(a\) terms.

This process helps in reducing fractions to their simplest form efficiently.

Fraction Simplification

Fraction simplification requires dividing terms until the fraction is in its simplest form. This means no common factors exist between the numerator and denominator, making the fraction irreducible.

Follow these steps for effective simplification:

• Start by simplifying numerical coefficients using GCD, as shown previously.
• Remove common variables or terms present in both parts of the expression, as they "cancel out."
• Reassess the fraction to confirm it can't be reduced further.

In our solved problem, after simplifying the coefficients with GCD and canceling the like terms, we've reached \( \frac{3}{4a^2} \). Always double-check to confirm no further simplification possible, ensuring the expression is in its final, simplest form.