Problem 58 - Algebra 1: Concepts and Skills

Question

Simplify the expression. The simplified expression should have no negative exponents. (Lesson 8.4). $$ x^{5} \cdot \frac{1}{x^{4}} $$

Step-by-Step Solution

Answer

The simplified expression with no negative exponents is \(x\).

Step 1: Identify the laws of exponents

Knowing the law that states when dividing expressions with the same base, you subtract the exponents: \(a^n / a^m = a^{n-m}\), one can simplify the expression by reducing the power of x.

Step 2: Apply the law of exponents

Applying the exponent rule, the power of x reduces to \(x^{5-4} = x^1\).

Step 3: Simplify the expression

In maths, \(x^1\) is simply written as \(x\). Thus, the simplified expression with no negative exponents is \(x\).

Key Concepts

Laws of Exponents

The laws of exponents are powerful tools in algebra that help us work with exponential expressions. One such rule is: when you divide two expressions with the same base, you subtract the exponents. This is expressed mathematically as: \[a^n / a^m = a^{n-m}\] Here, \(a\) is the base, and \(n\) and \(m\) are the exponents. This rule simplifies the expression by decreasing the exponent of the base rather than handling the complex multiplication directly.

• For example, \(x^5 \cdot \frac{1}{x^4}\) simplifies by subtracting \(4\) from \(5\), resulting in \(x^{5-4} = x^1\).
• These rules make calculations easier and expressions more manageable.

Understanding these rules enables quick and accurate simplifications, crucial for solving more complex problems.

Negative Exponents

In mathematics, negative exponents can initially seem daunting, but they're quite straightforward once you understand their function. A negative exponent indicates that you take the reciprocal of the base raised to the positive of that exponent.

• For example, \(x^{-n}\) can be rewritten as \(\frac{1}{x^n}\).

This means that negative exponents allow us to flip base expressions over a fraction line. However, when an expression asks you to have no negative exponents, it implies simplifying using positive powers only.
In the example expression \(x^5 \cdot \frac{1}{x^4}\), there is no negative exponent because when simplified using the laws of exponents, it directly results in \(x^1\), which is a positive exponent. Understanding negative exponents adds to your ability to manipulate and simplify expressions efficiently.

Algebraic Manipulation

Algebraic manipulation refers to the techniques used to transform and simplify algebraic expressions through the application of mathematical principles, which include laws of exponents, combining like terms, and distributing factors. Each operation performed should aim to enhance simplicity and clarity. When simplfying \(x^5 \cdot \frac{1}{x^4}\), algebraic manipulation was employed:

• By applying laws of exponents, we logicaly converted a division of similar bases into subtraction of exponents \((5 - 4)\).
• The result, \(x^1 = x\), is a more elegant and simplified version of the original expression.

Mastering algebraic manipulation is essential, as it enables problem-solving by reshaping complex expressions into understandable and solvable forms. With practice, you can seamlessly execute these transformations to enhance your mathematical toolkit.