Problem 32 - Algebra for College Students

Question

Write each of the following in radical form. For example, \(3 x^{\frac{2}{3}}=3 \sqrt[3]{x^{2}}\). \(x^{\frac{2}{5}}\)

Step-by-Step Solution

Answer

\(x^{\frac{2}{5}} = \sqrt[5]{x^2}\)

Step 1: Identify the Given Expression

The exercise asks us to convert the fractional exponent expression into its equivalent radical form. The given expression is \(x^{\frac{2}{5}}\).

Step 2: Understand the Fractional Exponent

The expression \(x^{\frac{2}{5}}\) has a fractional exponent of \(\frac{2}{5}\), where \(2\) is the numerator and \(5\) is the denominator.

Step 3: Recall the Radical Form

In radical form, a fractional exponent \(\frac{m}{n}\) can be expressed as \(\sqrt[n]{x^m}\). Here \(n\) is the index of the root, and \(m\) is the power of the radicand \(x\).

Step 4: Convert to Radical Form

Using the formula \(x^{\frac{m}{n}} = \sqrt[n]{x^m}\), the expression \(x^{\frac{2}{5}}\) can be rewritten in radical form as \(\sqrt[5]{x^2}\).

Step 5: Review the Result

Ensure that \(x^{\frac{2}{5}} = \sqrt[5]{x^2}\) accurately reflects the transformation from fractional exponent to radical form. The fifth root represents the denominator, and the square represents the numerator of the original fraction.

Key Concepts

Fractional Exponents

Fractional exponents are a unique way to express powers and roots in a compact form. When we see something like \(x^{\frac{2}{5}}\), it means we are dealing with both an exponent and a root. Here's a basic breakdown to help you understand this:

• The numerator (which is 2 in this case) indicates the power we raise the base to, so \(x^2\).
• The denominator (that's 5 here) tells you the root you need to take, specifically the fifth root in this example.

Consider fractional exponents as a form of shorthand. They can transform expressions with roots and powers into a single term. This can simplify calculations and make algebraic manipulation easier. Remember, fractional exponents follow the same rules as other exponents:

• Adding exponents when multiplying like bases.
• Subtracting exponents when dividing like bases.
• Multiplying exponents when raising a power to a power.

Radical Form

The radical form is how we visually and mathematically represent roots. When you convert a fractional exponent to radical form, you maintain the mathematical integrity while changing the expression's appearance. For \(x^{\frac{2}{5}}\), converting it to radical form involves understanding a simple concept:

• \(\frac{2}{5}\) becomes \(\sqrt[5]{x^2}\).
• The 5 is the root (or index), meaning it is the fifth root.
• The 2 stays as an exponent on \(x\) inside the radical sign, denoting \(x^2\).

This conversion is crucial because radical form often provides a clearer visual perspective of what calculation is taking place. It can help when solving equations or simplifying expressions by making the orders of operations more apparent. Radical notation mirrors this as \(\sqrt[n]{x^m}\), showing the root \(n\) and the power \(m\) without re-writing with a fraction.

Algebraic Expressions

Algebraic expressions in mathematics are essentially combinations of numbers, variables, and operations such as addition, subtraction, multiplication, and division. When you mix these with fractional exponents, you open up a wider range of expressions that can be both seen in symbolic form and evaluated numerically.

For example, dealing with an expression like \(3x^{\frac{2}{3}}\), you would rewrite it using radical notation as \(3 \sqrt[3]{x^2}\). This highlights a couple of important aspects:

• Constants (like 3 in this instance) can be carried through the transformation without being affected by exponent conversion.
• Understanding the structure helps when simplifying, factoring, or solving algebraic equations involving exponents and radicals.

In algebra, recognizing and using different forms—whether fractional exponents or radical form—makes solving equations more intuitive, allowing for diverse approaches depending on the context or the specific problem you face.