Problem 29 - Intermediate Algebra

Question

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

Step-by-Step Solution

Answer

The simplified form is \( \frac{3}{5a^6} \).

Step 1: Identify common terms

The expression is \( \frac{15 a^2}{25 a^8} \). Identify the common numerical factor and the powers of \( a \) in the numerator and the denominator.

Step 2: Simplify the numerical coefficients

Divide both the numerator and denominator by their greatest common factor, which is 5. So, \( \frac{15}{25} = \frac{3}{5} \).

Step 3: Simplify the variable expression

The variables in the expression are \( a^2 \) and \( a^8 \). Apply the property of exponents: \( \frac{a^m}{a^n} = a^{m-n} \). Therefore, \( \frac{a^2}{a^8} = a^{2-8} = a^{-6} \).

Step 4: Combine simplified parts

Combine the simplified numerical and variable expressions. The result is \( \frac{3}{5}a^{-6} \).

Step 5: Express with positive exponents

Rewrite \( a^{-6} \) as \( \frac{1}{a^6} \), which gives the final expression: \( \frac{3}{5a^6} \).

Key Concepts

Exponents

Exponents are a mathematical notation indicating the number of times a number, known as the base, is multiplied by itself. In expressions like \( a^2 \) or \( a^8 \), the base is \( a \) and the exponent is the number written as a superscript, indicating repeated multiplication. For example, \( a^2 \) means \( a \times a \), and \( a^8 \) means \( a \times a \times a \times a \times a \times a \times a \times a \).

When simplifying rational expressions with exponents, one useful rule is \( \frac{a^m}{a^n} = a^{m-n} \). This formula tells us that when dividing powers with the same base, you can subtract the exponents. Using this rule, \( \frac{a^2}{a^8} \) can be simplified to \( a^{2-8} = a^{-6} \). This negative exponent indicates a reciprocal. Thus, \( a^{-6} \) can be rewritten as \( \frac{1}{a^6} \), placing the expression in the denominator with a positive exponent.

Greatest Common Factor

The Greatest Common Factor (GCF) is the highest number that divides exactly into two or more numbers. In the context of simplifying rational expressions, identifying the GCF helps you divide both the numerator and the denominator by the same number, simplifying the expression.

In the expression \( \frac{15a^2}{25a^8} \), the numerical GCF is calculated by looking at the numbers 15 and 25. The number 5 is the largest number that divides both evenly (\( 15 \div 5 = 3 \) and \( 25 \div 5 = 5 \)). By dividing by the GCF, you obtain \( \frac{3}{5} \).

• This process reduces the expression to its simplest form.
• It only works if you choose a number that is a factor of both the numerator and the denominator.

Utilizing the GCF is crucial in making calculations faster and simpler.

Numerical Coefficients

Numerical coefficients are numbers that multiply a variable in an algebraic expression. For instance, in \( 15a^2 \), 15 is the numerical coefficient. These numbers play a significant role in simplifying rational expressions. To simplify, you divide the coefficients of the numerator and the denominator by the GCF.

In our expression \( \frac{15a^2}{25a^8} \), the numerical coefficients are 15 and 25. Firstly, find the GCF, which is 5, and divide both coefficients by it. This leaves you with \( \frac{3}{5} \), streamlining the expression.

• Numerical simplifications make expressions more manageable.
• They help maintain the focus on the variable component of the expression.

By reducing the coefficients, you clarify the expression, making subsequent operations easier to handle.