Problem 28 - Beginning and Intermediate Algebra

Question

Simplify. Assume all variables represent nonzero real numbers. The answer should not contain negative exponents. $$\frac{36 k^{8}}{12 k^{5}}$$

Step-by-Step Solution

Answer

The simplified expression is \(3k^3\).

Step 1: Simplify the fraction

Cancel out the common factors between the numerator and the denominator: \(\frac{36 k^{8}}{12 k^{5}} = 3k^{8-5}\)

Step 2: Use the rule for dividing exponents

Apply the rule for dividing exponents to simplify the expression: \(3k^{8-5} = 3k^3\) The simplified expression is \(3k^3\).

Key Concepts

Fraction Simplification

Fraction simplification is a fundamental skill in algebra that involves reducing a fraction to its simplest form. In an algebraic fraction like \(\frac{36 k^{8}}{12 k^{5}}\), simplification involves identifying and canceling out the common factors in the numerator and the denominator. Here, both 36 and 12 have a common factor of 12. By dividing both by 12, we simplify the fraction.

• 36 divided by 12 equals 3
• 12 divided by 12 equals 1

Thus, the simplified form of the fraction is \(\frac{3}{1} = 3\). After this step, it is crucial to apply any relevant algebraic rules, such as handling exponents, to fully simplify the expression. Simplifying fractions makes complex equations easier to work with and understand. The goal is to ensure the expression is as straightforward as possible.

Exponent Rules

Understanding exponent rules is key to simplifying expressions involving powers. The exercise \(3k^{8-5}\) applies one of the basic exponent rules: the division rule for exponents. This rule states that when you divide two expressions with the same base, you subtract the exponents. Given \(k^{8}\) and \(k^{5}\), both have the base \(k\). By subtracting their exponents (8 - 5), you get \(k^{3}\). Thus, the expression \(3k^{8-5}\) simplifies to \(3k^{3}\).

• For any base \(a\), \(\frac{a^m}{a^n} = a^{m-n}\)
• If \(m = n\), remember \(a^{0} = 1\)

These rules make it easier to simplify and manipulate algebraic expressions. Keeping these rules in mind can help you tackle complex problems involving powers efficiently.

Negative Exponents Avoidance

Negative exponents can complicate algebraic expressions, especially when simplifying. Avoiding them is often a requirement, as seen in the task to simplify an expression without negative exponents. When you have a negative exponent, it signals an inverse or reciprocal. For example, \(k^{-n}\) can be rewritten as \(\frac{1}{k^n}\). To prevent negative exponents in your final answer, always simplify by reducing fractions and adjusting exponents upfront. In our example, the division of exponents naturally yields positive values \(k^{3}\), which is preferable. Here is a quick tip to avoid negative exponents:

• Ensure you simplify fractions first, reducing the chance for negative values.
• Focus on positive exponent results to maintain clarity and simplicity.

Remember, tidy and positive exponent answers often lead to clearer and more elegant mathematical expressions.