Problem 33 - Elementary and Intermediate Algebra

Question

Simplify each rational expression. See Example 3 . $$\frac{15 a^{2}}{25 a^{8}}$$

Step-by-Step Solution

Answer

\(\frac{3}{5a^6}\)

Step 1: Identify Common Factors

First, let's look at the numerical coefficients and the powers of the variable. The coefficient 15 and 25 have a common factor of 5. Meanwhile, the powers of \(a\) can be simplified by dividing the exponents.

Step 2: Factor the Coefficients

Divide both the numerator and the denominator by their greatest common factor, which is 5. So, \(15 \div 5 = 3\) and \(25 \div 5 = 5\). Now the expression becomes \(\frac{3a^2}{5a^8}\).

Step 3: Simplify the Variable Terms

To simplify the variable terms, subtract the exponent in the numerator (2) from the exponent in the denominator (8). Thus, \(a^2\) divided by \(a^8\) becomes \(a^{2-8} = a^{-6}\).

Step 4: Rewrite with Positive Exponents

Rewrite the expression with positive exponents. Since \(a^{-6} = \frac{1}{a^6}\), the expression can be rewritten as \(\frac{3}{5a^6}\).

Key Concepts

Rational Expressions

Rational expressions are similar to fractions, but instead of just involving numbers, they can also include variables. In a rational expression, we have both a numerator and a denominator that are polynomials. When simplifying rational expressions, the main goal is to make the expression as simple as possible by reducing the numerator and the denominator to their simplest form.

To simplify a rational expression, identify any common factors between the numerator and the denominator. This can involve numbers or variables. Once identified, these common factors can be cancelled out to simplify the expression. However, it is crucial to remember that the denominator of a rational expression should never equal zero, as division by zero is undefined.

Here's a quick recap on simplifying rational expressions:

• Identify any common factors in both the numerator and the denominator.
• Simplify by cancelling out those common factors.
• Always ensure the denominator does not equal zero.

Exponent Rules

Exponent rules are fundamental when working with variables, especially in rational expressions. These rules allow us to simplify expressions where the same base is repeatedly multiplied by itself. The key rules for exponents are:

• Product of Powers: When multiplying like bases, add their exponents: \[ a^m \times a^n = a^{m+n} \]
• Quotient of Powers: When dividing like bases, subtract the exponents: \[ \frac{a^m}{a^n} = a^{m-n} \]
• Negative Exponent: A negative exponent indicates the reciprocal of that base raised to the positive exponent: \[ a^{-n} = \frac{1}{a^n} \]

Understanding and applying these rules helps simplify expressions by combining like terms efficiently. In the case of simplifying \(\frac{15 a^{2}}{25 a^{8}}\), these rules drastically reduce the complexity by eliminating powers of the variable and getting rid of negative exponents.

Simplifying Fractions

Simplifying fractions, whether numerical or algebraic, involves finding the greatest common factor (GCF) for both the numerator and the denominator. In mathematics, simplifying ensures that a fraction is reduced to its most basic form while maintaining its equivalence.

To simplify a fraction:

• Find the GCF of the numerator and the denominator.
• Divide both the numerator and the denominator by the GCF.
• Rewrite the fraction in its simplest form.

In the expression \(\frac{15 a^{2}}{25 a^{8}}\), the GCF of the coefficients 15 and 25 is 5. This means dividing both by 5, resulting in a simplified fraction of \(\frac{3 a^{2}}{5 a^{8}}\). Applying the quotient of powers rule to the variable part further simplifies the expression to \(\frac{3}{5 a^6}\), ensuring simplification is complete.