Problem 30 - College Algebra with Modeling and Visualization

Question

Use radical notation to rewrite. $$ a^{-2 / 3} b^{3 / 5} $$

Step-by-Step Solution

Answer

\(\frac{\sqrt[5]{b^3}}{\sqrt[3]{a^2}}\)

Step 1: Understand Exponent Rules

Exponent rules tell us that an exponent with a negative sign implies a reciprocal and a fractional exponent specifies a root. For instance, \(a^{-m/n} = \frac{1}{a^{m/n}}\) and \(x^{m/n} = \sqrt[n]{x^m}\). We need to apply these rules to change the expression to radical form.

Step 2: Rewriting Using Radical and Reciprocal

The expression \(a^{-2/3}\) can be rewritten as \(\frac{1}{a^{2/3}}\). This implies the use of the cube root and the reciprocal of the square power. Similarly, \(b^{3/5}\) can be expressed as \(\sqrt[5]{b^3}\). Let's apply these transformations.

Step 3: Applying to Each Part

For \(a^{-2/3}\), it becomes \(\frac{1}{\sqrt[3]{a^2}}\), which denotes the reciprocal of the cube root. For \(b^{3/5}\), it becomes \(\sqrt[5]{b^3}\), denoting the fifth root.Therefore, the entire expression becomes:\(\frac{\sqrt[5]{b^3}}{\sqrt[3]{a^2}}\).

Key Concepts

Exponent Rules

Exponent rules are fundamental to understanding how to manipulate and simplify expressions involving powers. There are several key rules:

• A negative exponent indicates the reciprocal of the base raised to the opposite positive exponent. For example, if you have an expression like \(a^{-1}\), it can be rewritten as \(\frac{1}{a}\).
• A fractional exponent translates into radicals or roots. The denominator of the fraction determines the type of root, while the numerator indicates the power that the base is raised to.

Understanding these rules allows us to transform expressions from one form to another. This makes calculations easier and expressions clearer. Always look out for opportunities to apply these rules especially when dealing with complex expressions.

Reciprocal

The concept of reciprocal is straightforward: it's basically flipping the number. For example, the reciprocal of \(2\) is \(\frac{1}{2}\). In mathematics, if we are dealing with a power, such as \(a^{-m}\), the rule states that this is equal to \(\frac{1}{a^m}\).
Why does this matter? Using a reciprocal can simplify expressions and makes calculations with fractional powers more manageable. When you encounter negative exponents, immediately think of applying the reciprocal rule to simplify the expression. This process aids in converting negative exponents into positive ones, which are easier to interpret and solve.

Fractional Exponents

Fractional exponents might seem intimidating at first, but they are actually quite simple when you break them down. A fractional exponent like \(x^{m/n}\) can be rewritten as \(\sqrt[n]{x^m}\). Here, \(n\) is the root and \(m\) indicates the power.

• The denominator (\(n\)) represents the type of root (e.g., square root, cube root, etc.).
• The numerator (\(m\)) describes the power to which the base is raised.

Converting problems involving fractional exponents into radical form can often make the problem simpler. This understanding bridges the gap between polynomial expressions and radical expressions, making them easier to handle in various mathematical problems.

Cube Root

The cube root is a specific type of root that involves finding an input that when multiplied by itself three times gives the original number. Symbolically, the cube root of a number \(x\) is written as \(\sqrt[3]{x}\).
Cube roots are commonly found in problems involving fractional exponents like \(x^{1/3}\). This particular form means you are looking for the number that times itself three times will equal \(x\). This concept is especially useful when dealing with expressions involving powers of three or similar transformations. Understanding cube root simplifies converting expressions with fractional powers into radical form.

Fifth Root

The fifth root, similar to the cube root, is concerned with finding a number that, when multiplied by itself five times, results in the original number. When you see an expression such as \(\sqrt[5]{x}\), it is asking for the fifth root of \(x\).
Fifth roots often appear in conjunction with fractional exponents like \(x^{1/5}\). This represents the fifth root form and is essential for simplifying expressions in equations and radical expressions. Recognizing and being able to convert fifth roots from and to fractional exponents allows for a deeper understanding and flexibility in solving algebraic equations.