Problem 14 - Elementary and Intermediate Algebra

Question

Divide the monomials. See Example 1. $$ \frac{a^{12}}{a^{8}} $$

Step-by-Step Solution

Answer

\( \frac{a^{12}}{a^8} = a^4 \)

Step 1: Understand the Division of Monomials

When dividing two monomials of the same base, you subtract the exponent of the denominator from the exponent of the numerator. This follows the rules of exponents.

Step 2: Identify the Exponents

In this problem, the base is the same in both the numerator and the denominator, which is 'a'. We need to focus on their exponents. The exponent of 'a' in the numerator is 12, and in the denominator, it is 8.

Step 3: Subtract the Exponents

According to the rule of exponents for division, subtract the exponent in the denominator from the exponent in the numerator: \[ 12 - 8 \]

Step 4: Solve the Exponentiation

Calculate the subtraction: \[ 12 - 8 = 4 \] Thus, the result of the division is: \[ a^4 \]

Step 5: Write the Final Answer

The division of the monomials \( \frac{a^{12}}{a^8} \) results in \( a^4 \). This is the simplified form.

Key Concepts

Rules of Exponents

When working with monomials, understanding the rules of exponents is crucial. Monomials are algebraic expressions consisting of a single term, which may include numbers or variables raised to a power. Rules of exponents help us handle the powers of these variables effectively.
One essential rule is that when you divide two expressions with the same base, you subtract their exponents. This rule can be formulated as follows:

• If you have an expression like \( \frac{x^m}{x^n} \), the result is \( x^{m-n} \).
• The bases must match for this rule to apply.

By using this rule, you maintain the properties of the base and simplify the expression, making calculations easier.

Subtraction of Exponents

Subtraction of exponents is a key technique in simplifying division problems involving exponents. This technique stems directly from the rules of exponents discussed above.
To subtract exponents straightforwardly:

• Identify the exponents: Look at the exponents of the numerator and the denominator.
• Ensure the bases are the same: Only subtract exponents when dealing with the same base, like "a" in both \( a^{12} \) and \( a^8 \).
• Calculate \( m - n \): Using the previous example, subtract the exponent in the denominator (8) from the exponent in the numerator (12).
• The resulting expression is now simpler, expressed as \( a^4 \).

This simplification process helps us reach a more manageable and useful form of the expression.

Simplifying Expressions

Simplifying an algebraic expression means reducing it to its simplest form, where no further simplification is possible. For monomials, simplifying often involves applying the rules of exponents.
To simplify the expression \( \frac{a^{12}}{a^8} \), apply the exponent subtraction rule. This leads to:

• Subtract the exponent of the denominator (8) from the exponent of the numerator (12) to get \( a^{12-8} \), which simplifies to \( a^4 \).
• Ensure there are no more like terms or further simplifications needed. In this case, since the base and exponent are now in simplest form, we conclude with \( a^4 \).

Simplification helps to solve problems more efficiently and understand the relationships within algebraic equations better. It strips away unnecessary complexity, giving you a clearer and more concise result.