Problem 42 - Algebra for College Students

Question

Write each of the following in radical form. For example, \(3 x^{\frac{2}{3}}=3 \sqrt[3]{x^{2}}\). \(x^{\frac{3}{7}} y^{\frac{5}{7}}\)

Step-by-Step Solution

Answer

\(x^{\frac{3}{7}} y^{\frac{5}{7}} = \sqrt[7]{x^3} \cdot \sqrt[7]{y^5}\)

Step 1: Identify the Base and Exponent

The expression is given as \(x^{\frac{3}{7}} y^{\frac{5}{7}}\). Here, \(x^{\frac{3}{7}}\) and \(y^{\frac{5}{7}}\) have fractional exponents. In this form, the base is \(x\) and \(y\), and the exponents are \(\frac{3}{7}\) and \(\frac{5}{7}\), respectively.

Step 2: Identify the Root from the Denominator

In fractional exponents, the denominator indicates the root. For the expression \(x^{\frac{3}{7}}\), the root is the \(7\)-th root since the denominator is \(7\). Similarly, for \(y^{\frac{5}{7}}\), the \(7\)-th root is applied as well.

Step 3: Adjust the Numerator as Power Inside the Radical

In fractional exponents \(x^{\frac{a}{b}}\), \(a\) is the power to which the base is raised. Therefore, \(x^{\frac{3}{7}}\) becomes \(\sqrt[7]{x^3}\) and \(y^{\frac{5}{7}}\) becomes \(\sqrt[7]{y^5}\). Here, \(3\) and \(5\) are the powers inside the radicals.

Step 4: Write the Complete Radical Form

Combine the radical expressions obtained from the previous steps. Therefore, \(x^{\frac{3}{7}} y^{\frac{5}{7}}\) is written in radical form as \(\sqrt[7]{x^3} \cdot \sqrt[7]{y^5}\).

Key Concepts

Fractional Exponents

Fractional exponents might seem intimidating at first, but they offer a simple way to represent roots and powers in a single expression. Here's how to break them down:

• The top number (numerator) represents the power that you raise the base to.
• The bottom number (denominator) indicates the root that you take of the base.

For example, let's consider the expression \(x^{\frac{3}{7}}\). Here, \(3\) is the power, and \(7\) is the root. This means you are taking the 7th root of \(x\) and then raising it to the 3rd power.
Understanding fractional exponents is essential because they simplify expressions. Instead of dealing with complex radicals directly, you can use fractional exponents for easy manipulation and calculation.
Whether adding, subtracting, or multiplying numbers with fractional exponents, the key is to pay attention to both parts of the fraction to perform calculations accurately.

Nth Roots

Nth roots are a generalization of square roots and cube roots. They express the operation of finding a number that, when raised to the nth power, gives the original number. In mathematical terms:

• The "n" in the nth root represents the degree of the root.
• For instance, the square root is the 2nd root, and the cube root is the 3rd root.

If we consider the term \(\sqrt[n]{a}\), it essentially means the number that, when raised to the power of \(n\), results in \(a\).
Using our example from above, \(x^{\frac{3}{7}}\) converts to \(\sqrt[7]{x^3}\). It asks us to find the 7th root of \(x^3\). This is the same as finding the number that, when raised to the 7th power, equals \(x^3\).
Understanding nth roots provides a deeper insight into simplifying complex radical expressions and solving equations that involve roots.

Exponential Notation

Exponential notation provides a compact way to express repeated multiplication. It's particularly powerful when dealing with very large or very small numbers.
Here's a quick breakdown:

• The base is the number that gets multiplied by itself.
• The exponent (or power) tells us how many times to multiply the base.

For example, \(x^3\) means multiply \(x\) by itself three times: \(x \cdot x \cdot x\).
In the context of fractional exponents, exponential notation allows us to handle calculations involving both roots and powers succinctly. Instead of writing out long equations, you can use exponential notation to quickly demonstrate the idea of repeated multiplication intertwined with root taking.
This notation is also foundational in scientific calculations and helps communicate mathematical ideas clearly and efficiently across various fields.