Problem 15 - Algebra 2

Question

Simplify. Assume that no variable equals 0. $$ \frac{a^{2} n^{6}}{a n^{5}} $$

Step-by-Step Solution

Answer

The simplified expression is \( an \).

Step 1: Identify the Fraction

The expression given is a fraction: \( \frac{a^{2} n^{6}}{a n^{5}} \). Our task is to simplify this expression by canceling common terms in the numerator and the denominator.

Step 2: Simplify the Coefficient 'a'

The term \( a^2 \) in the numerator and \( a \) in the denominator can be simplified by canceling one 'a' from both, resulting in \( a \) in the numerator. The fraction now looks like: \( \frac{a n^{6}}{n^{5}} \).

Step 3: Simplify the Variable 'n'

The term \( n^6 \) in the numerator and \( n^5 \) in the denominator also can be simplified. Since \( n^6 = n^5 \cdot n \), you can cancel \( n^5 \) from both, resulting in \( n \) in the numerator. The expression now is \( an \).

Step 4: Final Answer

After simplifying, we find that the given expression reduces to \( an \).

Key Concepts

Fractions in Algebra

In algebra, fractions can include variables just like numbers. Understanding the concepts related to algebraic fractions helps in simplifying expressions neatly. An algebraic fraction is essentially the division of two expressions. For example, consider \( \frac{a^{2} n^{6}}{a n^{5}} \) where both the numerator and the denominator include variables.Working with algebraic fractions involves:

• Simplifying the expression by reducing like terms.
• Identifying common factors in the numerator and the denominator.

The core principle is to eliminate those common terms or shared factors to reduce the expression to its simplest form. This involves using techniques such as canceling common variables or numbers, which will be explained further.By mastering fractions in algebra, you lay the foundation for tackling more complex algebraic equations with confidence.

Exponent Rules

Exponents are a way to represent repeated multiplication of the same number or variable. They are crucial in algebra for simplifying expressions. Knowing the rules that govern exponents makes it easier to simplify complex algebraic fractions like \( \frac{a^{2} n^{6}}{a n^{5}} \). Here are some essential rules to remember:

• **Product of Powers Rule:** To multiply expressions with the same base, add the exponents: \( a^m \times a^n = a^{m+n} \).
• **Quotient of Powers Rule:** To divide expressions with the same base, subtract the exponents: \( \frac{a^m}{a^n} = a^{m-n} \).
• **Power of a Power Rule:** To raise a power to another power, multiply the exponents: \( (a^m)^n = a^{m \times n} \).

Using these rules, you can simplify \( a^2 \) divided by \( a \) to just \( a^{2-1} = a \), and likewise, simplify \( n^6 \) divided by \( n^5 \) to \( n^{6-5} = n \). This step-by-step application of exponent rules reduces complex expressions swiftly.

Canceling Common Factors

Canceling common factors is an essential technique in simplifying algebraic fractions. It's about eliminating terms that appear both in the numerator and the denominator, making the expression simpler.For example, in the expression \( \frac{a^{2} n^{6}}{a n^{5}} \):

• First, you have \( a^{2} \) over \( a \) where one \( a \) is common, allowing cancellation to get \( a \).
• Next, \( n^{6} \) over \( n^{5} \) means \( n^{5} \) is common, reducing it to \( n \).

The resultant expression is simplified to \( an \), after removing these common factors.This process is analogous to simplifying numeric fractions. For instance, simplifying \( \frac{6}{3} \) involves dividing both terms by the common factor 3, resulting in 2. By identifying and canceling such analogous terms, complex algebraic expressions become more manageable and straightforward.