Problem 24 - Contemporary Precalculus

Question

Write the given expression without using radicals. $$\sqrt[5]{x^{2}}$$

Step-by-Step Solution

Answer

Question: Rewrite the expression $\sqrt[5]{x^{2}}$ without radicals. Answer: $x^{\frac{2}{5}}$

Step 1: Rewrite the radical expression as a fractional exponent

Rewrite the given expression using fractional exponents. Recall that a radical can be expressed as a fractional exponent, where the denominator of the fractional exponent is equal to the root of the radical. In this case, the radical is a fifth (5) root, so the denominator of the fractional exponent will be 5. $$\sqrt[5]{x^{2}} = x^{\frac{2}{5}}$$

Step 2: Final expression without radicals

Now, our expression is written without any radicals: $$x^{\frac{2}{5}}$$

Key Concepts

Radicals to Fractional Exponents

Transitioning from radical notation to fractional exponents is a vital skill in algebra that simplifies the operation of roots. The general principle is that a radical can be rewritten as a power with a fractional exponent. For example, the fifth root of x squared, denoted as \(\sqrt[5]{x^{2}}\), is equivalent to raising x to the power of 2/5, written as \(x^{\frac{2}{5}}\). The numerator of the fraction represents the power to which the base is raised, while the denominator indicates the root being taken. Embracing this conversion allows for more streamlined manipulation of expressions when performing operations such as multiplication, division, or even finding derivatives in calculus.

Here's how this works in practice: taking any radical \(\sqrt[n]{x^m}\) can be expressed as \(x^{\frac{m}{n}}\), where m is the exponent and n is the radical's index, the number indicating the degree of the root.

Simplifying Radical Expressions

Simplifying radical expressions, especially when variables are involved, requires understanding the properties of exponents. To simplify a radical expression, one should look for factors that are perfect powers of the radical index. For example, if you are dealing with a square root, you would search for factors that are perfect squares.

In situations where variables are raised to powers within a radical, similar principles apply. By rewriting radicals as fractional exponents, not only is the expression often simplified, but it also becomes easier to apply further algebraic rules. Always remember that simplifying involves reducing the expression to its simplest form while keeping its value unchanged.

Radical Notation

Radical notation is a mathematical symbol that signifies the root of a number or expression. It is characterized by the radical symbol (also known as the root symbol) and an index, which indicates the degree of the root. For example, the square root of 9 is written as \(\sqrt{9}\) whereas the cube root of 27 is \(\sqrt[3]{27}\). When no index is shown, as in \(\sqrt{}\), it's assumed to be 2, indicating a square root.

It's important to be comfortable with reading and writing radical notation since it is commonly used in mathematics. When dealing with higher-level math, radical functions and their transformations are essential concepts that rely heavily on a strong grasp of radical notation.

Fractional Exponents

Fractional exponents express powers and roots in an alternate, streamlined manner. A fractional exponent such as \(x^{m/n}\) conveys two operations: the nth root of x followed by raising the result to the mth power. Understanding fractional exponents is crucial when solving algebraic equations, simplifying expressions, and in more advanced topics like functions and calculus.

One of the essential properties of fractional exponents is their conformity with exponentiation rules. This familiarity eases the learning curve as students can apply what they already know about exponents to solve problems involving fractional exponents. This includes operations such as multiplying and dividing exponents, raising powers to powers, and converting between radical and exponential forms.