Problem 24 - A Graphical Approach to Precalculus with Limits

Question

Use positive rational exponents to rewrite each expression. Assume variables represent positive numbers. $$\sqrt[5]{x^{2}}$$

Step-by-Step Solution

Answer

The expression is rewritten as \( x^{2/5} \).

Step 1: Understand the Expression

The expression given is \( \sqrt[5]{x^{2}} \). This is a radical expression where \( x^2 \) is under the fifth root.

Step 2: Convert Radical to Exponent Form

Radical expressions can be converted to exponents by using the rule that \( \sqrt[n]{a^m} = a^{m/n} \). In this expression, we have \( \sqrt[5]{x^{2}} = x^{2/5} \).

Step 3: Review the Exponent

We converted the expression to use a rational exponent. Here the exponent \( \frac{2}{5} \) is positive, signifying a fifth root followed by a square.

Key Concepts

Understanding Radical Expressions

Radical expressions involve roots, such as square roots, cube roots, and even beyond. In simple terms, they represent a number or variable raised to some fraction. For example, the square root of a number is when it is raised to the power of one-half. This is because multiplying a number by itself gives you the original number. The cube root, on the other hand, is when the number is raised to a third. It works similarly for other roots like the fourth, fifth, and so forth.
When you see an expression like \( \sqrt[5]{x^{2}} \), it means you're looking for a value that, when raised to the fifth power, returns something related to \( x^{2} \). Understanding this is crucial to navigating between radical expressions and exponent forms effectively. Radical expressions can often seem complex at first glance, but breaking down the concept shows that they are just another way to represent exponents. Always remember, the number outside the root sign tells you the degree of the root you are dealing with.

Exploring Positive Exponents

Positive exponents are straightforward and simply represent how many times to multiply the base by itself. They indicate repeated multiplication. A positive rational exponent, like \( x^{2/5} \), denotes a root. Here, 2 is the power and 5 is the root.
To break it down:

• The numerator of the fraction (2 in \( \frac{2}{5} \)) is the power applied to the base \( x \).
• The denominator (5 in \( \frac{2}{5} \)) signifies the root.

Positive exponents help simplify operations as you don’t have to carry around radical symbols, making calculations more efficient. They also make it easier to apply further operations, like multiplication and division, when dealing with algebraic expressions.

Conversion to Exponent Form

Converting radical expressions to their equivalent exponent form is a key skill in algebra and helps in simplifying expressions further. The conversion follows a basic rule:

• \( \sqrt[n]{a^m} = a^{m/n} \)

Here, \( a \) is the base, \( m \) is the power of the base, and \( n \) is the degree of the root. In the given expression \( \sqrt[5]{x^{2}} \), conversion involves shifting from the radical to the exponential representation: \( x^{2/5} \). This transformation helps in various ways, particularly for solving equations and performing algebraic manipulations, saving you from dealing with radical signs.The practice strengthens your understanding of how roots and powers interplay and lets you handle more complex equations with ease. Over time, converting between these forms will become second nature, enhancing your mathematical fluency.