Problem 41 - Algebra for College Students

Question

Write each of the following in radical form. For example, \(3 x^{\frac{2}{3}}=3 \sqrt[3]{x^{2}}\). \(x^{\frac{2}{3}} y^{\frac{1}{3}}\)

Step-by-Step Solution

Answer

\(\sqrt[3]{x^2 y}\)

Step 1: Identify the Expression

We are given the expression with fractional exponents: \(x^{\frac{2}{3}} y^{\frac{1}{3}}\). Our task is to rewrite this in radical form.

Step 2: Rewrite Each Term as a Radical

Start by rewriting each term in the expression using radical notation. Remember the rule: \(a^{\frac{m}{n}} = \sqrt[n]{a^m}\). So, \(x^{\frac{2}{3}}\) can be written as \(\sqrt[3]{x^2}\) and \(y^{\frac{1}{3}}\) as \(\sqrt[3]{y}\).

Step 3: Combine the Radicals

Next, combine the radicals into a single expression to represent the entire term \(x^{\frac{2}{3}} y^{\frac{1}{3}}\). Both terms are cube roots, so they can be written together under the same radical as \(\sqrt[3]{x^2 y}\).

Step 4: Final Expression in Radical Form

The final expression in radical form, by combining and simplifying, is \(\sqrt[3]{x^2 y}\).

Key Concepts

Fractional Exponents

Fractional exponents might seem complex, but they're just another way to express roots and powers together. At first glance, expressions like \(x^{\frac{2}{3}}\) or \(y^{\frac{1}{3}}\) may look daunting. However, understanding them is like unlocking a code. The numerator (top part) of the fraction indicates the power to which the base (e.g., \(x\) or \(y\)) is raised. The denominator (bottom part) shows the root we need to take.
For example, in \(x^{\frac{2}{3}}\), the "2" tells us to square \(x\), and the "3" tells us to take the cube root of \(x^2\). This translates to \(\sqrt[3]{x^2}\) when rewritten with radical notation. Fractional exponents are incredibly useful in algebra for simplifying expressions and solving equations, allowing us to express otherwise cumbersome roots and powers concisely and flexibly.
Here’s a quick reminder on how to interpret fractional exponents:

• Numerator: Power to raise
• Denominator: Root to take

Radical Notation

Radical notation lets us write roots, like square roots or cube roots, in a clear and familiar format. Learning to convert between fractional exponents and radical notation is really useful. It’s like translating between two languages in math. Specifically, when a base raised to a fractional exponent (like \(a^{\frac{m}{n}}\)) is written in radical form, it becomes \(\sqrt[n]{a^m}\).
To visualize this with our example \(x^{\frac{2}{3}}\): it turns into \(\sqrt[3]{x^2}\). Here, the cube root \(\sqrt[3]{\cdot}\) is marked by the index "3," and under the radical sign, \(x^2\) tells us what to take the root of. Similarly, \(y^{\frac{1}{3}}\) becomes \(\sqrt[3]{y}\).
Using radical notation is especially helpful because:

• It makes expressions easier to understand visually.
• Allows for simplification when solving algebraic expressions.
• Helps in recognizing patterns in equations during problem solving.

Algebraic Manipulation

Algebraic manipulation involves transforming mathematical expressions or equations to simplify or solve them. Within the context of radical expressions and fractional exponents, algebraic manipulation is key to changing one form into another.
When we have expressions like \(x^{\frac{2}{3}} y^{\frac{1}{3}}\), we might need to convert them using rules of exponents and radicals. Practicing this skill often involves:

• Recognizing expressions that can be converted using radical notation.
• Applying multiplication rules for radicals, like combining \(\sqrt[3]{x^2}\) and \(\sqrt[3]{y}\) into \(\sqrt[3]{x^2y}\).
• Simplifying complex expressions to make them more manageable.

Whether we are solving equations or simplifying expressions, understanding how to manipulate algebraic forms is essential. It allows us to combine like terms, factor expressions, and discover new ways to approach and solve problems.