Problem 11 - A Graphical Approach to College Algebra

Question

Write each expression in radical form. Assume that all variables represent positive real numbers. $$(2 m+p)^{2 / 3}$$

Step-by-Step Solution

Answer

The expression in radical form is \(\sqrt[3]{(2m + p)^2}\).

Step 1: Identify the expression

The given expression is \((2m + p)^{2/3}\). We need to convert this expression into radical form.

Step 2: Understand the relationship between rational exponents and radicals

The expression \(a^{m/n}\) can be rewritten in radical form as \(\sqrt[n]{a^m}\), meaning it is the \(n\)-th root of \(a^m\).

Step 3: Apply the relationship to the given expression

Using the rule \(a^{m/n} = \sqrt[n]{a^m}\), we can convert \((2m + p)^{2/3}\) into \(\sqrt[3]{(2m + p)^2}\). This shows the expression in radical form.

Key Concepts

Rational Exponents

Rational exponents are an extension of integer exponents. Unlike integer exponents, which tell us how many times to multiply a number by itself, rational exponents contain fractions and signify both roots and powers. For instance, the exponent \( \frac{m}{n} \) tells us to take the \( n \)-th root of a number and raise it to the \( m \)-th power. Here's a simple breakdown:

• The numerator represents the power.
• The denominator indicates the root.

In our exercise, we see \( (2m + p)^{2/3} \). This means we should take the cube root (since the denominator is 3) of \((2m + p)\), and then square the result (because the numerator is 2). The concept may initially seem tricky, but with practice, it's a powerful tool in algebra.

Radicals

Radicals are mathematical symbols used to denote roots. The most common radical is the square root, represented by the symbol \( \sqrt{} \). However, radicals are versatile and can represent any root with the help of an index.In mathematical notation:

• The symbol \( \sqrt[n]{x} \) denotes the "n-th root" of \( x \).
• The number \( n \) is called the index.
• If the index is omitted, it's usually assumed to be 2, indicating a square root.

Using the properties of radicals, you can convert between radicals and rational exponents effectively. Our exercise transforms the expression \((2m + p)^{2/3}\) into its radical form \( \sqrt[3]{(2m + p)^2} \), capturing the same meaning using a different notation.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and operations that formulate mathematical statements. They can be as simple as a single number, or as complex as a combination of terms with variables and their respective powers.When dealing with expressions involving rational exponents and radicals, it's crucial to simplify or convert them properly for clarity and easier problem-solving.For example, the algebraic expression \((2m + p)^{2/3}\) is changed into \( \sqrt[3]{(2m + p)^2} \). This shows the expression's equivalency in a different form, making it easier to understand and manipulate. Emphasizing the versatility of algebraic expressions helps students learn various ways to solve equations and formulate solutions without being constrained to one form. Understanding these transformations is fundamental in higher mathematics, allowing students to explore deeper algebraic concepts with confidence.