Problem 1 - Elementary and Intermediate Algebra

Question

In each of Exercises 1–8, match the expression with the equivalent expression from the column on the right. $$ x^{2 / 5} $$ a) \(x^{3 / 5}\) b) \((\sqrt[5]{x})^{4}\) c) \(\sqrt{x^{5}}\) d) \(x^{1 / 2}\) e)\(\frac{1}{(\sqrt{x})^{5}}\) f) \(\sqrt[4]{x^{3}}\) g) \(\sqrt[5]{x^{2}}\) h)\(\frac{1}{(\sqrt[5]{x})^{2}}\)

Step-by-Step Solution

Answer

(g) \(\root{5}\times{x^{2}}\)

Step 1: Understand the Given Expression

First, identify the form of the given expression and the options. The given expression is \(x^{2/5}\).

Step 2: Convert the Expression

Rewrite \(x^{2/5}\) using radicals. Recall that \(x^{a/b}\) can be written as \(\root{b}\times{x^{a}}\). Thus, \(x^{2/5}\) can be written as \(\root{5}\times{x^{2}}\).

Step 3: Match with Given Options

Look at the given options to find an equivalent expression to \(\root{5}\times{x^{2}}\). The correct match is option (g) \(\root{5}\times{x^{2}}\). None of the other options match this form.

Key Concepts

Radicals

Radicals are expressions that include roots, such as square roots or fifth roots. They play a critical role in algebra because they allow us to represent complicated numbers in a simpler form. For example, the square root of 16 is \(\root{2}\times{16}=4\), as 4 times 4 equals 16.

Here are a few key aspects of radicals:

• A radical expression is often written with a root symbol \(\root{} \).
  
• Inside the root symbol, you'll find the radicand, which is the number you want to find the root of. For example, in \(\root{5}\times{32}\), 32 is the radicand.
  
• Different roots solve different powers. A square root solves for a number that, when multiplied by itself, equals the radicand. A fifth root finds a number that, when multiplied by itself five times, equals the radicand.
  

Understanding the basics of radicals will help you more easily grasp other algebraic concepts like fractional exponents and equivalent expressions.

Fractional Exponents

Fractional exponents provide another way to represent roots and simplify expressions. Instead of writing \(\root{5}\times{x^{2}}\), you can write \({x^{2/5}}\). This notation is especially useful in algebra for performing operations on expressions.

Let’s break down how fractional exponents work:

• The numerator (top part) of the fraction indicates the power to which the base is raised.
  
• The denominator (bottom part) signifies the root to be taken.
  
• For example, \({x^{2/5}}\) means that x is raised to the 2nd power, and then the 5th root of that result is taken.
  

By converting between radicals and fractional exponents, you can find equivalent expressions more easily. Both notations are fundamentally the same, just written in different forms. This conversion is visible in the solution where we rewrote \({x^{2/5}}\) as \(\root{5}\times{x^{2}}\). Understanding this helps in simplifying and matching algebraic expressions.

Equivalent Expressions

In algebra, equivalent expressions are different expressions that represent the same value for all values of the variable. Recognizing and transforming expressions to show their equivalence is a key skill.

To identify equivalent expressions, you must understand some core principles:

• Expressions can be manipulated using laws of exponents and radicals.
  
• Two expressions are equivalent if, after simplification, they produce the same result.
  

In the provided exercise, the goal was to match an expression \({x^{2/5}}\) with its equivalent form among given options. By converting \({x^{2/5}}\) into \(\root{5}\times{x^{2}}\), we can easily see that option (g) \(\root{5}\times{x^{2}}\) is equivalent. None of the other options matched this form.

Understanding and recognizing equivalent expressions enables easier problem-solving in algebra, ensuring you can handle complex equations and transformations with confidence.