Problem 26 - Elementary Algebra

Question

Write the following expressions using only positive exponents. Assume all variables are nonzero. $$ y^{-5} $$

Step-by-Step Solution

Answer

Question: Rewrite the expression \(y^{-5}\) using only positive exponents. Answer: \(\frac{1}{y^5}\)

Step 1: Recall laws of exponents

We need to recall the following law of exponents: For any nonzero number a and any integer n, $$ a^{-n} = \frac{1}{a^n} $$

Step 2: Rewrite the expression using positive exponents

Using the above law of exponents, we can rewrite the expression \(y^{-5}\) as: $$ y^{-5} = \frac{1}{y^5} $$ The expression has now been written with only positive exponents.

Key Concepts

Laws of Exponents

Understanding the laws of exponents is crucial for manipulating and simplifying algebraic expressions, and this comprehension begins with knowing the basic rules. One fundamental law is the rule for negative exponents. It states that any nonzero base a raised to a negative exponent n is equivalent to the reciprocal of that base raised to the positive exponent:

For any nonzero number a and any integer n, we have:
\[ a^{-n} = \frac{1}{a^n}\]

• This law helps in converting negative exponents into positive ones, which often simplifies the expression and makes it easier to work with.
• Other laws include handling products and quotients of powers, as well as powers of powers, which are essential for more complex expressions.

Remembering this rule is particularly useful in situations where you're required to express all exponents positively, as illustrated in our exercise solution.

Algebraic Expressions

In the world of algebra, algebraic expressions are combinations of numbers, variables, and arithmetic operations. They represent a wide range of relationships and are a fundamental component of mathematical equations.

When dealing with expressions that include exponents, recognizing the type of expression (monomial, binomial, polynomial, etc.) can guide the simplification process. Our initial problem involved a monomial with a negative exponent. By applying our knowledge of exponents, we rewrote it using only positive exponents to arrive at a standard form that is more familiar and easier to integrate into further calculations or applications.

Using laws of exponents efficiently requires practice, but it empowers students to move through algebraic problems with greater speed and accuracy.

Negative Exponents

A negative exponent signifies the reciprocal of a base raised to a positive exponent. They can seem confusing at first, but they follow a straightforward rule that transforms them into an easier-to-understand form:

\[ a^{-n} = \frac{1}{a^n}\]Where a is a nonzero base and n is a positive integer.

• It's important to note that the base a should not be zero since zero to any power is zero, and a reciprocal of zero is undefined.
• Understanding negative exponents is essential for simplifying expressions and solving equations where such exponents occur.

In our exercise, the negative exponent was simplified to a positive one by placing the variable in the denominator, making the expression easier to read and further manipulate.

Simplifying Expressions

The goal of simplifying expressions is to rewrite them in their most basic or compact form without changing their value. This process can involve a number of steps including:

• Combining like terms.
• Using the distributive property to eliminate parentheses.
• Applying the laws of exponents to simplify powers and roots.

In the context of our exercise, simplifying the expression involved converting a negative exponent to a positive one. This is a common first step in simplification because it often leads to an expression that is easier for most people to understand. Once positive exponents are in place, further simplification may involve combining terms or performing other operations to reduce the expression to its simplest form.

By mastering the art of simplifying expressions, students can make complex problems more approachable and unlock the ability to solve a wider range of algebraic challenges.