Problem 14 - Algebra and Trigonometry

Question

Write each radical expression using exponents, and each exponential expression $$ \frac{1}{\sqrt{x^{5}}} $$

Step-by-Step Solution

Answer

The expression \( \frac{1}{\sqrt{x^{5}}} \) in exponential form is \( x^{-5/2} \).

Step 1: Understand the Problem

We are given a radical expression \( \frac{1}{\sqrt{x^{5}}} \) and asked to express it using exponents instead of a square root.

Step 2: Express the Radical as an Exponential

The square root \( \sqrt{x^5} \) can be rewritten as \( x^{5/2} \). This is because the square root of a variable raised to a power is the same as the variable raised to the power divided by 2.

Step 3: Find the Reciprocal in Exponential Form

The given expression \( \frac{1}{\sqrt{x^{5}}} \) can now be rewritten as \( x^{-5/2} \). When dividing by a number with an exponent, the exponent becomes negative.

Key Concepts

Radical Expressions

Radical expressions involve roots, such as the square root, of a number or variable. The expression \( \sqrt{x^5} \) is a common example. A radical expression typically includes a radical symbol (\( \sqrt{} \)) and a radicand, which is the number or expression inside the radical. To work with radicals more easily, it is often useful to rewrite them using exponents, as this allows for more straightforward calculations. Root operations are inversely related to exponentiation. This relationship is central to converting between radical and exponential forms, making complex expressions simpler to handle and solve.

Using Exponents

Exponents are a way to represent repeated multiplication of a number by itself. For instance, \( x^5 \) means \( x \times x \times x \times x \times x \). When dealing with radicals in exponential form, exponents can simplify the manipulation of these expressions. In our example, \( \sqrt{x^5} \) translates to an exponent form as \( x^{5/2} \). The exponent \( 5/2 \) indicates that you take \( x \) to the fifth power first, then find the square root, effectively making calculations easier. Learning to convert between these forms helps in simplifying more complicated mathematical problems.

Reciprocals

A reciprocal is simply the inverse of a number or expression when multiplied, resulting in 1. For example, the reciprocal of \( x \) is \( \frac{1}{x} \) because \( x \times \frac{1}{x} = 1 \). Reciprocals are crucial in simplifying expressions and solving equations. In the given expression \( \frac{1}{\sqrt{x^5}} \), we see the reciprocal in action. By writing \( \sqrt{x^5} \) as \( x^{5/2} \), the expression becomes \( x^{-5/2} \). This clearly demonstrates how the reciprocal of a root can be expressed with a negative exponent. The transformation from division by a square root to multiplication by a negative exponent illustrates the power and usefulness of understanding reciprocals.

Mathematical Notation

Mathematical notation is the language used to convey mathematical ideas clearly and concisely. It uses symbols and operators, like those found in exponents and radicals, to express concepts succinctly. In our exercise, mathematical notation helps us transform \( \frac{1}{\sqrt{x^{5}}} \) into \( x^{-5/2} \). The use of notation such as division, exponentiation, and roots standardizes the way we perform and convey mathematical operations. Recognizing and becoming fluent in this notation is vital for solving problems efficiently and communicating mathematical ideas effectively. As you practice, these symbols will become familiar and intuitive, reducing complexity and enabling deeper understanding.