Problem 531 - Elementary Algebra

Question

In the following exercises, write with a rational exponent. (a) \(\sqrt[5]{u}\) (b) \(\sqrt{v}\) (c) \(\sqrt[16]{w}\)

Step-by-Step Solution

Answer

(a) \( u^{1/5} \), (b) \( v^{1/2} \), (c) \( w^{1/16} \)

Step 1: Understand Rational Exponents

Rational exponents can be written in the form of fractional powers. The general form is \( \sqrt[n]{a} = a^{1/n} \), where \( n \) is the root and \( a \) is the base.

Step 2: Convert \( \sqrt[5]{u} \)

Using the formula for rational exponents, rewrite \( \sqrt[5]{u} \) as \( u^{1/5} \).

Step 3: Convert \( \sqrt{v} \)

Similarly, rewrite \( \sqrt{v} \) as \( v^{1/2} \).

Step 4: Convert \( \sqrt[16]{w} \)

Finally, rewrite \( \sqrt[16]{w} \) as \( w^{1/16} \).

Key Concepts

Fractional Powers

When we talk about fractional powers, we are essentially discussing rational exponents. These exponents are fractions instead of whole numbers. For example, writing a number like 8 raised to the 1/3 power \(8^{1/3}\). This is the same as taking the cube root of 8 \(\sqrt[3]{8}\). The idea is straightforward: the numerator of the fraction is the power, and the denominator is the root.

Radicals

Radicals are another way of expressing roots in mathematics. The square root of a number is the most common radical and is represented by the symbol \(\sqrt{a}\). For instance, \(\sqrt{9} = 3\). There are also cube roots \(\sqrt[3]{a}\), fourth roots \(\sqrt[4]{a}\), and so on.

Radicals can always be turned into fractional powers. Take the fifth root of \(u\) which is written as \(\sqrt[5]{u}\) and convert it into fractional exponent form \(u^{1/5}\). This method makes calculations easier, especially when dealing with higher-level algebraic concepts.

Algebra Basics

Algebra is the branch of mathematics dealing with symbols and the rules for manipulating those symbols. It is a unifying thread of almost all of mathematics and includes everything from solving elementary equations to the study of abstractions like groups, rings, and fields.

One fundamental concept in algebra involves understanding how to work with exponents and roots. For example, when we change a root, such as \(\sqrt[16]{w}\), into a rational exponent, we write \(w^{1/16}\). This conversion is part of simplifying expressions and solving equations more efficiently.

Exponent Rules

Exponent rules are crucial in simplifying expressions and solving mathematical problems. These rules include:

• Multiplying Exponents: \(a^m \cdot a^n = a^{m+n}\)
• Dividing Exponents: \(a^m / a^n = a^{m-n}\)
• Power of a Power: \( (a^m)^n = a^{m \cdot n}\)
• Negative Exponents: \(a^{-n} = 1/a^n\)

Understanding these rules helps in converting roots into fractional exponents, making it simpler to manage and manipulate algebraic expressions.

For example, converting \(\sqrt{v}\) into fractional exponent form makes it \(v^{1/2}\), making subsequent operations straightforward and manageable.