Problem 550 - Elementary Algebra

Question

In the following exercises, write with a rational exponent. (a) \(\sqrt{m^{5}}\) (b) \(\sqrt[3]{n^{2}}\) (c) \(\sqrt[4]{p^{3}}\)

Step-by-Step Solution

Answer

a) \(m^{5/2}\), b) \(n^{2/3}\), c) \(p^{3/4}\)

Step 1 - Understanding Rational Exponents

Rational exponents are another way to express radicals. Generally, \(\sqrt[n]{a^m} = a^{m/n}\).

Step 2 - Convert the Expression (a)

Given \(\sqrt{m^{5}}\), this is a square root hence n=2. So, \(m^{5/2}\).

Step 3 - Convert the Expression (b)

Given \(\sqrt[3]{n^{2}}\), this is a cube root hence n=3. So, \(n^{2/3}\).

Step 4 - Convert the Expression (c)

Given \(\sqrt[4]{p^{3}}\), this is a fourth root hence n=4. So, \(p^{3/4}\).

Key Concepts

Radicals

Radicals are expressions that involve roots, such as square roots, cube roots, or fourth roots. The radical symbol is used to denote roots and is followed by a number indicating the degree of the root. For example, \sqrt{a} represents the square root of \(a\), while \sqrt[3]{a} represents the cube root of \(a\).

Understanding radicals and their notation is crucial for converting expressions to rational exponents. When converting, remember that the general form of a radical \sqrt[n]{a^m} can be rewritten using rational exponents as \(a^{m/n}\). This transformation simplifies working with roots and allows easier application of exponent rules.

Exponent Rules

Exponent rules are fundamental to manipulating expressions involving powers. Here are some of the most important rules:

• Product of Powers: \(a^m \times a^n = a^{m+n}\)
• Quotient of Powers: \(a^m / a^n = a^{m-n}\)
• Power of a Power: \((a^m)^n = a^{mn}\)
• Power of a Product: \((ab)^n = a^n \times b^n\)
• Power of a Quotient: \((a/b)^n = a^n / b^n\)

These rules simplify the process of working with exponents and are especially useful when dealing with rational exponents. For instance, knowing that \(m^{5/2}\) can be broken down into \(m^5\) raised to the \(1/2\) power using the power of a power rule.

Algebraic Expressions

Algebraic expressions consist of variables, numbers, and operations (such as addition, subtraction, multiplication, and division). When dealing with expressions involving exponents and radicals, it’s important to follow specific steps to simplify or convert them.

For instance, consider simplifying \sqrt{m^5}\. Recognize that this can be written using rational exponents as \(m^{5/2}\). This conversion makes it easier to apply algebraic methods to solve or simplify the expression.

Working with algebraic expressions also often involves combining like terms and using the distributive property. Understanding how to manipulate these components allows for better control over more complex expressions involving exponents and radicals.