Problem 21 - Elementary Algebra

Question

Simplify \(\frac{x^{5 n+6}}{x^{4}} .\)

Step-by-Step Solution

Answer

Question: Simplify the expression \(\frac{x^{5n+6}}{x^4}\). Answer: \(x^{5n+2}\)

Step 1: 1. Identify the expression to simplify

We are given the expression \(\frac{x^{5n+6}}{x^4}\) and want to simplify it.

Step 2: 2. Apply laws of exponents

When dividing expressions with the same base and different exponents, we subtract the exponent of the denominator from the exponent of the numerator: \((a^m) \div (a^n) = a^{(m-n)}\). Apply this rule to our expression: \(x^{(5n+6)-4} = x^{5n+2}\).

Step 3: 3. Write the final simplified expression

We have simplified the given expression to \(x^{5n+2}\), which is the final answer.

Key Concepts

Laws of Exponents

Understanding the laws of exponents is crucial for simplifying algebraic expressions involving powers. Exponents, also called indices or powers, tell us how many times a base number is multiplied by itself. For instance, if you see a term like x³, it means x * x * x. But when we combine terms with exponents through multiplication or division, we apply specific rules to simplify them.

One of the fundamental laws is the quotient rule, which applies when dividing terms with the same base. According to this rule, \( \frac{a^m}{a^n} = a^{m-n} \) where a is the base and m and n are the exponents. By subtracting the exponent of the denominator from the exponent of the numerator, you're essentially 'reducing' the number of times you multiply the base in the numerator by the times you would have divided it in the denominator.

This law keeps the expression neat and manageable, which is especially helpful when dealing with variables and algebraic expressions. Not only does this reduce complexity, but it also makes it possible to further solve equations or compare expressions, which would be much harder with lengthy repeated multiplication.

Algebraic Notation

Algebraic notation is the system we use to represent numbers, operations, and relationships. It includes variables, constants, the operational symbols such as plus, minus, multiply, and divide, as well as the use of parentheses and exponents. The beauty of this notation lies in its universality and precision — it allows us to convey complex mathematical ideas in a compact and understandable form.

Part of mastering algebraic notation is becoming comfortable with variables, which stand in for unknown values. When we work with expressions like \( \frac{x^{5n+6}}{x^4} \) we're using x as a placeholder for any number. Also, being adept with this notation means understanding the order of operations. When simplifying an expression, we perform calculations in a specific sequence: parentheses, exponents, multiplication and division from left to right, and finally addition and subtraction from left to right.

Clear and correct algebraic notation is key to avoiding mistakes and it supports logical thought processes, which are invaluable for solving advanced problems in mathematics and related fields.

Simplifying Exponents

Simplifying exponents is the process of reducing expressions with powers to their simplest form. Besides using the laws of exponents, this may involve combining like terms, which are terms that have the same variable raised to the same power. Simplification makes complex expressions easier to manage and is often necessary to perform more operations or to solve equations.

As we saw in the example from the textbook, the expression \( \frac{x^{5n+6}}{x^4} \) can be simplified using the quotient rule to \( x^{5n+2} \). This is far more straightforward than the original expression, which could be daunting with its higher power and addition within the exponent.

When simplifying exponents, always be on the lookout for opportunities to apply the laws of exponents. For instance, if exponents are added or subtracted, see if they belong to terms with the same base. If so, you can usually combine them into a single term. Simplification ensures you're working with the most efficient form of the expression, ultimately leading to clearer understanding and faster, error-free calculations.