Problem 8 - Algebra and Trigonometry

Question

Write each radical expression using exponents, and each exponential expression using radicals. Radical expression \(\quad\) Exponential expression \(\frac{1}{\sqrt{x^{5}}}\) \(\quad\) ______

Step-by-Step Solution

Answer

Exponential expression: \( x^{-\frac{5}{2}} \)

Step 1: Understanding Radicals and Exponents

To convert a radical expression to an exponent, recall that any expression of the form \(\sqrt[n]{a}\) can be rewritten using exponents as \(a^{1/n}\). Similarly, \(\frac{1}{\sqrt[n]{a^m}}\) translates to \(a^{-m/n}\).

Step 2: Identifying the Radical

Start with the given radical expression \(\frac{1}{\sqrt{x^5}}\). Recognize that \(\sqrt{x^5}\) is \((x^5)^{1/2}\) because a square root corresponds to raising to the power of \(\frac{1}{2}\).

Step 3: Converting the Expression

Convert \(\frac{1}{\sqrt{x^5}}\) to its exponential form. First express \(\sqrt{x^5}\) as \((x^5)^{1/2}\). This becomes \((x^5)^{1/2}\). Since this is in the denominator, we need to take its reciprocal: \((x^5)^{-1/2}\).

Step 4: Solution and Simplification

The exponential form of \(\frac{1}{\sqrt{x^5}}\) is \(x^{-5/2}\), which represents the same expression using exponents instead of a radical.

Key Concepts

Radical Expressions

Radical expressions involve the use of roots, such as square roots, cube roots, and more. A radical sign, represented by \( \sqrt{} \), indicates that we want the root of a number or variable. For example, \( \sqrt{x} \) means "the square root of \( x \)." It's important to remember that taking the square root is equivalent to raising that number or expression to the power of \( \frac{1}{2} \).
Radicals are common in various mathematical disciplines and are used to simplify expressions and solve equations where variables are under a root sign. Here's a useful tip: higher roots like cube roots can be expressed as \( \sqrt[3]{x} = x^{1/3} \). This leads us into exponential expressions, which are closely related to radical expressions.

Exponential Expressions

Exponential expressions use exponents to show powers of numbers or variables. An exponent tells us how many times to multiply a number by itself. For instance, \( x^3 \) or "\( x \) to the power of three" means \( x \times x \times x \). This kind of expression is crucial when handling large numbers or values that grow rapidly, such as in scientific calculations.
Typically, an expression like \( x^{1/n} \) means taking the \( n \)-th root of \( x \), reflecting the broader relationship between radicals and exponents. Thus, a square root, which might initially appear as \( \sqrt{x} \), is actually \( x^{1/2} \). Exponents can also be negative, indicating an inverse operation, such as \( x^{-n} = \frac{1}{x^n} \).
Understanding exponential notations is key to manipulating algebraic expressions, solving equations, and simplifying complex problems efficiently.

Conversion Between Radicals and Exponents

Converting between radicals and exponents is all about changing how we represent expressions while maintaining their value. This is particularly useful when simplifying expressions or solving equations.
To convert a radical expression to an exponential form, remember the rule: \( \sqrt[n]{a} = a^{1/n} \). For example, to express \( \sqrt{x^5} \) as an exponent, rewrite it as \((x^5)^{1/2}\). Therefore, the expression \( \frac{1}{\sqrt{x^5}} \) becomes \( (x^5)^{-1/2} \), simplifying to \( x^{-5/2} \).
Similarly, if you have \( a^m \) with a fractional exponent such as \( a^{m/n} \), this is equivalent to the n-th root of \( a \) raised to the m-th power: \((\sqrt[n]{a})^m \). Mastering this conversion allows you to switch easily between forms, aiding in understanding and tackling complex problems.