Problem 25 - Elementary Algebra

Question

Write the following expressions using only positive exponents. Assume all variables are nonzero. $$ y^{-1} $$

Step-by-Step Solution

Answer

Question: Rewrite the given expression with only positive exponents: $y^{-1}$. Answer: $\frac{1}{y}$

Step 1: Identify the negative exponent

In this expression, we can see that the exponent is -1. We want to rewrite it with a positive exponent. $$ y^{-1} $$

Step 2: Apply the exponent rule for negative exponents

According to the exponent rule for negative exponents, the expression can be rewritten as the reciprocal of its base with a positive exponent. In other words, we'll change the negative exponent to its reciprocal (positive) form by placing it in the denominator of a fraction. $$ y^{-1} = \frac{1}{y^1} $$

Step 3: Simplify the expression

Since y has an exponent of 1, we can simplify the expression as follows: $$ \frac{1}{y^1} = \frac{1}{y} $$ The given expression with only positive exponents is: $$ y^{-1} = \frac{1}{y} $$

Key Concepts

Negative Exponent Rule

Understanding the negative exponent rule is crucial when dealing with exponents in algebra. Essentially, it tells us that any non-zero number raised to a negative exponent is equal to the reciprocal of that number raised to the corresponding positive exponent.

Let's look at an example. With the negative exponent \(y^{-1}\), the rule states that we can convert this to a positive exponent by finding its reciprocal. This translates the expression \(y^{-1}\) to \(\frac{1}{y^1}\), or simply \(\frac{1}{y}\) since any number raised to the power of one is itself. It's important to remember that the negative exponent rule holds true for any non-zero base, whether it's a number, a variable, or a more complex expression.

Here's a concise bullet-point summary:

• Any non-zero number \(a\) raised to a negative exponent \(b\) is equal to \(1/a^b\).
• \(a^{-b} = \frac{1}{a^b}\), assuming \(a \eq 0\).
• This rule allows us to write expressions with only positive exponents.

Knowing this rule is a stepping stone to mastering more complex algebraic manipulations.

Simplifying Expressions

Simplifying expressions is a fundamental skill in algebra which makes equations easier to understand and solve. The goal of simplification is to break down complicated equations into their most basic form without changing their value.

In our example, we simplify \(\frac{1}{y^1}\) to \(\frac{1}{y}\). This is because any number raised to the first power is the number itself, making the exponent of one unnecessary. Simplifying often involves applying algebraic rules, like the negative exponent rule, combining like terms, and using operations such as addition, subtraction, multiplication, and division.

A few tips for simplification include:

• Combine like terms where applicable.
• Reduce expressions to minimal terms.
• Apply exponent rules correctly.

Effective simplification requires careful attention to the properties of numbers and operations, helping to ensure that the true value of the expression remains constant throughout the process.

Reciprocal of a Number

The reciprocal of a number is essentially a flipped version of the number, usually represented as \(1/x\) for a non-zero number \(x\). When we refer to the reciprocal, we're talking about a special multiplication relationship. Any number multiplied by its reciprocal yields 1, which is the multiplicative identity.

For instance, if we have the number 5, its reciprocal would be \(1/5\). Multiplying these together, \(5 \times \frac{1}{5} = 1\), confirms their reciprocal relationship. The concept plays a vital role when working with negative exponents and dividing fractions. In our example, \(y^{-1}\) is equal to the reciprocal of \(y\), written as \(\frac{1}{y}\).

To keep in mind:

• The reciprocal of \(x\) is \(\frac{1}{x}\).
• Multiplying a number by its reciprocal always results in 1.
• In division, multiplying by the reciprocal is often used to divide fractions.

Understanding reciprocals is not only critical for exponents but also for understanding ratios, rates, and various other mathematical concepts.