Problem 3 - Algebra 2 and Trigonometry

Question

In \(3-10,\) write each expression as a rational number without an exponent. $$ 5^{-1} $$

Step-by-Step Solution

Answer

\(5^{-1}\) as a rational number is \(\frac{1}{5}\).

Step 1: Understanding Negative Exponents

A negative exponent means that the base of the exponent is on the wrong side of the fraction line, so you need to flip it to the other side. For example, if you have \(a^{-n}\), it is equivalent to \(\frac{1}{a^n}\).

Step 2: Applying the Negative Exponent Rule

Given the expression \(5^{-1}\), apply the negative exponent rule. Rewrite it as \(\frac{1}{5^1}\).

Step 3: Simplifying Exponent

Next, simplify \(5^1\). Since any number raised to the power of 1 is the number itself, \(5^1\) is simply 5.

Step 4: Final Expression

Substitute back the simplified expression into the fraction from the previous steps: \(\frac{1}{5}\). Therefore, \(5^{-1}\) written without an exponent is \(\frac{1}{5}\).

Key Concepts

Understanding Rational Numbers

Rational numbers include any number that can be expressed as the quotient or fraction of two integers, where the numerator is an integer and the denominator is a non-zero integer. Examples include fractions like \(\frac{1}{2}\), whole numbers such as 3 (which can be expressed as \(\frac{3}{1}\)), and decimals that either terminate or repeat like 0.75 or 0.333\.

In the case of negative exponents, transforming them to rational numbers makes the concept clearer. A negative exponent like \(5^{-1}\) can be rewritten as \(\frac{1}{5}\), clearly displaying its rational number form. Transposing negative exponents into rational numbers helps in simplifying expressions and solving equations.

Exploring Exponent Rules

Exponent rules are essential tools in mathematics, providing the framework to manipulate expressions involving powers. Here are some key rules relevant to this concept:

• Negative Exponent Rule: The expression \(a^{-n}\) translates to \(\frac{1}{a^n}\). This rule moves the base from the numerator of a fraction to the denominator, effectively 'flipping' its position.
• Zero Exponent Rule: Any base with an exponent of zero equals one. For example, \(5^0 = 1\).
• Product of Powers Rule: When multiplying two powers with the same base, add the exponents. For instance, \(a^m \times a^n = a^{m+n}\).

Understanding these rules allows for seamless manipulation of exponential expressions, aiding in their simplification and transformation.

Mastering Fraction Simplification

Simplifying fractions involves reducing them to their simplest form. This means making the numerator and the denominator as small as possible while retaining the same ratio.

For instance, let's break down the simplification of \( \frac{4}{8} \). Both numbers can be divided by their greatest common divisor, which is 4 here. Thus, \( \frac{4}{8} \) simplifies to \( \frac{1}{2} \).

• Divide by the Greatest Common Divisor: Identify the largest number that can evenly divide both the numerator and denominator.
• Ensure Irreducibility: If no further common factors exist other than 1, the fraction is in its simplest form.

In terms of the original expression \(5^{-1}\), converting it to \(\frac{1}{5}\) already depicts it in its simplest possible form as a fraction, since both 1 and 5 are coprime.