Problem 35 - Algebra 2 and Trigonometry

Question

In \(35-63,\) write each expression with only positive exponents and express the answer in simplest form. The variables are not equal to zero. $$ x^{-4} $$

Step-by-Step Solution

Answer

The expression with positive exponents is \(\frac{1}{x^{4}}\).

Step 1: Understand the Problem

We need to rewrite the expression with only positive exponents. The original expression is \(x^{-4}\). A negative exponent indicates that the term should be taken as the reciprocal, moving it from the numerator to the denominator or vice versa.

Step 2: Apply the Negative Exponent Rule

The rule for negative exponents is \(a^{-n} = \frac{1}{a^{n}}\). So, for \(x^{-4}\), we apply this rule: \(x^{-4} = \frac{1}{x^{4}}\).

Step 3: Simplify the Expression

Since the expression \(\frac{1}{x^{4}}\) is already in its simplest form with no negative exponents, there is no further simplification needed.

Key Concepts

Exponent Rules

Understanding exponent rules is fundamental to working with exponents. These rules help us manipulate expressions involving powers, especially when simplifying them, or converting to different forms. The approach for handling exponents depends mainly on their sign and the operation performed:

• Product of Powers Rule: When multiplying like bases, add the exponents, as in \(a^m \cdot a^n = a^{m+n}\).
• Quotient of Powers Rule: When dividing like bases, subtract the exponents, shown as \(\frac{a^m}{a^n} = a^{m-n}\).
• Power of a Power Rule: When raising a power to another power, multiply the exponents: \((a^m)^n = a^{mn}\).
• Negative Exponent Rule: For an expression with a negative exponent, such as \(a^{-n}\), convert it to \( \frac{1}{a^n} \), effectively flipping it to its reciprocal.

These rules simplify complex expressions into more manageable forms. They are the building blocks for advanced algebraic operations and calculus. Understanding them is crucial for tackling a variety of mathematical problems.

Reciprocal

The concept of a reciprocal is pivotal when dealing with negative exponents. A reciprocal of a number is essentially "flipped" on its head. For any non-zero number \(a\), its reciprocal is \(\frac{1}{a}\).When expressions with negative exponents are encountered, the reciprocal is used to convert them into positive exponents.This is a technique for simplifying expressions involving negative powers.

• A simple example is \(x^{-2}\), which is the reciprocal of \(x^2\), leading to \(\frac{1}{x^2}\).
• When the negative exponent is resolved into the reciprocal of a higher power, it becomes more digestible and follows standard procedures for further simplification.

Reciprocals ensure that we only handle positive exponents in simplified mathematical expressions.They play a significant role when transforming equations, especially in calculus, to more understandable forms.

Simplifying Expressions

Simplifying expressions is an important skill in algebra and beyond. It involves rewriting expressions in a simpler form, making them easier to work with. This often means using positive exponents instead of negative ones.

• Start by identifying terms with negative exponents and apply the reciprocal rule to convert them.
• Use other exponent rules to combine like terms, reducing the expression further.
• Ensure no terms can be reduced or cancelled further, confirming that the simplest form is reached.

Understanding how to simplify expressions not only aids in algebraic calculations but also sets the foundation for advanced mathematics. It helps prevent errors in computations and provides clearer insights into the structure of the mathematical relationships being studied.