Problem 44 - Algebra for College Students

Question

Write each of the following in radical form. For example, \(3 x^{\frac{2}{3}}=3 \sqrt[3]{x^{2}}\). \(-4 x^{\frac{3}{4}} y^{\frac{1}{4}}\)

Step-by-Step Solution

Answer

\(-4 \sqrt[4]{x^{3} y}\).

Step 1: Identify the Base and Exponents

In the expression \(-4 x^{\frac{3}{4}} y^{\frac{1}{4}}\), identify the base variables and their respective exponents. Here, \(x\) has an exponent of \(\frac{3}{4}\) and \(y\) has an exponent of \(\frac{1}{4}\).

Step 2: Convert Fractional Exponents to Radical Form

Recall that a fractional exponent \(\frac{a}{b}\) can be written as a radical \(\sqrt[b]{x^{a}}\). Therefore, \(x^{\frac{3}{4}}\) converts to \(\sqrt[4]{x^{3}}\) and \(y^{\frac{1}{4}}\) converts to \(\sqrt[4]{y}\).

Step 3: Rewrite the Complete Expression

Using the converted radical forms, rewrite the entire expression as \[-4 \sqrt[4]{x^{3}} \cdot \sqrt[4]{y}\].This places each base inside their respective root.

Key Concepts

Fractional Exponents

Fractional exponents might look complicated, but they are just another way to represent radicals. Let's simplify this concept. When you see an exponent in the form of a fraction, such as \( \frac{a}{b} \), it represents a root and a power. Here’s how this works: the denominator, \( b \), indicates the root, while the numerator, \( a \), signifies the power.

• The fraction \( \frac{3}{4} \) means that we are taking the fourth root of a number and then raising it to the third power.
• For instance, \( x^{\frac{3}{4}} \) means \( \sqrt[4]{x^3} \).
• In the case of \( y^{\frac{1}{4}} \), it is simply \( \sqrt[4]{y} \).

Remember, the essence of fractional exponents is in recognizing that they combine the concepts of roots and powers in a single expression. Learning this helps greatly in converting expressions back and forth between forms.

Algebraic Transformations

Algebraic transformations involve changing the form of an expression without changing its value. This allows us to simplify or manipulate equations more easily. In the context of fractional exponents, these transformations are very useful. Let's break down how transformations work in practice:

• Identify the component parts, such as coefficients and variables, along with their exponents.
• Use rules for exponents and roots to simplify or rewrite the expression in different forms. For example, \( x^{\frac{3}{4}} \) becomes \( \sqrt[4]{x^{3}} \) in radical form.
• Combine like terms when possible, or distribute products over sums to ease further transformations.

These steps are particularly helpful in exercises involving complex algebraic expressions, as they guide the transition from one form to another cleanly and clearly. Mastering this skill enhances problem-solving abilities in mathematics.

Radical Form

Writing an expression in radical form means expressing it using a root, such as square roots or cube roots, rather than exponents. It's a very useful skill for understanding and simplifying algebraic expressions. Here’s how you can transform fractional exponents to radical form:

• Recognize fractional exponents and their implications. For example, \( x^{\frac{3}{4}} \) is translated to the radical form \( \sqrt[4]{x^{3}} \).
• Pick the root first, indicated by the denominator of the fraction, and then apply the power, represented by the numerator, under the radical symbol.
• The coefficient in front of the variables, such as the \(-4\) in \(-4 x^{\frac{3}{4}} y^{\frac{1}{4}}\), remains unchanged during this transformation. This results in the radical expression \(-4 \sqrt[4]{x^{3}} \cdot \sqrt[4]{y}\).

By understanding and using radical form, you develop a more in-depth comprehension of the nature of roots and powers in mathematics. This knowledge is essential for tackling more advanced algebraic problems with confidence.