Problem 128 - Intermediate Algebra

Question

In electronics, the angular frequency of oscillations in a certain type of circuit is given by the expression \((L C)^{-1 / 2} .\) Use radical notation to write this expression.

Step-by-Step Solution

Answer

The expression \((LC)^{-1/2}\) in radical notation is \(\frac{1}{\sqrt{LC}}\).

Step 1: Understand the Expression

The given expression for angular frequency is \((LC)^{-1/2}\). This expression is written in terms of exponents and involves the variables \(L\) and \(C\).

Step 2: Convert Exponential Notation to Radical Notation

In order to convert an expression from exponential notation to radical notation, we need to remember that a negative exponent represents the reciprocal and a fractional exponent represents a root. Specifically, an exponent of \(1/n\) corresponds to the \(n\)th root. So, the expression \((LC)^{-1/2}\) can be rewritten as \((LC)^{-1/2} = \frac{1}{\sqrt{LC}}\).

Step 3: Write the Final Radical Expression

Following the conversion, the angular frequency expression in radical notation is rewritten as \(\frac{1}{\sqrt{LC}}\). This expression now clearly represents the reciprocal of the square root of the product \(L\) and \(C\).

Key Concepts

Exponential Notation

Exponential notation is a way to express numbers that are multiplied by themselves a number of times. For example, instead of writing 2 multiplied by itself three times (2 x 2 x 2), we use exponential notation as 2^3. Here, 2 is the base, and 3 is the exponent which shows how many times the base is used as a factor.
In our exercise, the expression \[(LC)^{-1/2}\] uses exponential notation to convey more complex mathematical ideas efficiently. The negative exponent highlights a reciprocal relationship, while the fractional part indicates a root.
This structure helps simplify and solve equations, especially when dealing with large or very small numbers as often encountered in electronics and physics.

Angular Frequency

Angular frequency is a concept commonly used in electronics and physics to describe the rate of oscillation of an object or wave. It tells us how fast an angle changes over time.
For example, in a circuit, angular frequency is used to predict how quickly a current might change directions. It is measured in radians per second.
In the formula \[(LC)^{-1/2}\], angular frequency is represented in a mathematical form that shows how it depends on the components of the circuit, specifically the inductance \(L\) and capacitance \(C\).
This dependency is crucial because it helps engineers adjust these components to achieve the desired frequency, allowing devices to function properly on different settings.

Reciprocal

The reciprocal of a number is what you multiply that number by to get the result 1. For example, the reciprocal of 5 is 1/5 because 5 x 1/5 = 1.
In exponent terms, a negative exponent indicates a reciprocal. For instance, \(a^{-n}\) is equivalent to \(1/a^n\).
In the context of our exercise, the expression \[(LC)^{-1/2}\] uses a negative exponent, which is then converted to a reciprocal in radical notation: \[rac{1}{ ext{something}}\]. This indicates that we are dealing with the reciprocal of the square root of \(LC\).
Understanding reciprocals is vital in manipulating expressions and solving equations, effectively simplifying numerical relationships in mathematics and the sciences.

Fractional Exponents

Fractional exponents are used to express roots in exponential notation. Essentially, an expression like \(a^{1/n}\) means the \(n\)th root of \(a\).
In this notation, the fraction in the exponent tells us which root to take. For example, \(a^{1/2}\) is the square root of \(a\), and \(a^{1/3}\) is the cube root of \(a\).
In the exercise given, the exponent \(-1/2\) indicates two things: the negative sign shows a reciprocal, and the \(1/2\) signals a square root.
By combining these interpretations, \[(LC)^{-1/2}\] translates to the reciprocal of the square root of \(LC\) when simplified into radical notation.
Learning to understand and manipulate fractional exponents allows for more flexible and comprehensive mathematical solutions, especially when simplifying complex formulas.