Problem 56 - Algebra 1: Concepts and Skills

Question

Rewrite the expression with positive exponents. $$ \frac{2}{x^{-5}} $$

Step-by-Step Solution

Answer

The expression \( \frac {2}{x^{-5}} \) with positive exponents is \( 2x^{5} \).

Step 1: Identify the Term with a Negative Exponent

The given expression is \( \frac {2}{x^{-5}} \). Here, \(x^{-5}\) is the term with a negative exponent.

Step 2: Apply the Rule of Negative Exponents

Using the rule of negative exponents, the term \(x^{-5}\) can be rewritten as \(\frac {1}{x^{5}}\). So the given expression becomes \( \frac {2}{1/x^{5}} \).

Step 3: Simplification

The expression is simplified by removing the division by a fraction. This is achieved by multiplying the numerator and the denominator of the expression by \(x^{5}\) resulting to \(2x^{5}\).

Key Concepts

Positive Exponents

In mathematics, exponents represent repeated multiplication of a base number. When a number is raised to a positive exponent, it signifies the base is multiplied by itself the number of times indicated by the exponent. For instance, if you have the base number 3 raised to the power of 4, it means you multiply 3 by itself 4 times: \(3^4 = 3 \times 3 \times 3 \times 3 = 81\).

Positive exponents are straightforward and the default mechanism in algebra.

• They indicate standard multiplication of the base number.
• They rarely require special rules or transformations like negative exponents.
• Positive exponents help in simplifying and solving algebraic expressions efficiently.

In the original exercise, the objective is to transform the expression so that all exponents are positive. This makes calculations easier and is generally preferred for clarity and simplicity.

Simplification

Simplification in mathematics is the process of rewriting expressions in a simpler, more convenient form. This often involves eliminating complex fractions, combining like terms, or rewriting expressions using basic arithmetic operations. In the exercise provided, simplification involved transforming \(\frac{2}{1/x^5}\) into a form that is more straightforward.

Simplification can include breaking down fractions:

• A complex fraction like \(\frac{a}{1/b}\) can be simplified to \(a \cdot b\).
• You might multiply both the numerator and the denominator by the same quantity to remove a fraction within a fraction.

By applying these techniques, the exercise's expression became \(2x^5\) from the original \(\frac{2}{x^{-5}}\). This is an important step for solving and understanding algebraic problems.

Algebraic Expressions

An algebraic expression is a combination of numbers, variables, and operations (like addition and multiplication) without an equality sign. It represents a mathematical phrase and can be simplified or manipulated using various rules.

Components of algebraic expressions include:

• **Constants**: Fixed numbers like 3, -5, etc.
• **Variables**: Symbols like \(x\) or \(y\) that stand for numbers.
• **Coefficients**: Numbers multiplying the variables.

In algebra, expressions can be expanded, factored, and simplified according to established mathematical conventions.

By writing expressions with positive exponents, interpretation and calculation consistency improves, making it easier to both find solutions and communicate them clearly. In the given exercise, we started with a negative exponent in the denominator. By converting it into a positive exponent, we helped simplify and clarify the algebraic expression for further calculations or usage.