Problem 463 - Elementary Algebra

Question

In the following exercises, simplify. (a) \(\sqrt[5]{u^{7}}\) (b) \(\sqrt[6]{v^{11}}\)

Step-by-Step Solution

Answer

(a) \( u^{7/5} \), (b) \( v^{11/6} \)

Step 1 - Use Fractional Exponents

Convert the radicals to expressions with fractional exponents: \ \ (a) \( \sqrt[5]{u^{7}} = u^{7/5} \) \ \ (b) \( \sqrt[6]{v^{11}} = v^{11/6} \)

Step 2 - Simplify the Exponents

Both expressions are already in their simplest form. Fractional exponents cannot be simplified further without additional information. \ \ Therefore, \( \sqrt[5]{u^{7}} \) remains \( u^{7/5} \) and \( \sqrt[6]{v^{11}} \) remains \( v^{11/6} \).

Key Concepts

Fractional Exponents

Fractional exponents, also known as rational exponents, represent another way to express radicals.
This method is useful for simplifying and performing arithmetic operations on root expressions. To convert a radical to a fractional exponent:

• The denominator of the fraction represents the root.
• The numerator represents the power.

For example, \(\root[n]{m^k}\) can be rewritten as \(m^{k/n}\). This is why \(\sqrt[5]{u^{7}}\) converts to \(u^{7/5}\) and \(\sqrt[6]{v^{11}}\) converts to \(v^{11/6}\).
Using fractional exponents can make it easier to differentiate, integrate, or simplify expressions in algebra and calculus.

Radical Expressions

Radical expressions involve roots, such as square roots, cube roots, and so on. Simplifying these involves expressing them in their simplest form.
For example, the expression \(\sqrt[5]{u^{7}}\) means we are looking for the 5th root of \(\text{u raised to the power of 7}\), and \(\sqrt[6]{v^{11}}\) looks for the 6th root of \(\text{v raised to the power of 11}\).
By converting these to their fractional exponent form like \(u^{7/5}\) and \(v^{11/6}\), simplifying becomes more manageable. Note that any radical expression follows the form \root[n]{a}\, where n is the degree of the root.
If the radical cannot be simplified further, it is already in its simplest form.

Exponential Functions

Exponential functions are expressions in which a constant base is raised to a variable exponent. They follow the form \(f(x) = a^x\).
Fractional exponents fit into this category as the exponents can be fractions, not just whole numbers. These functions grow or decay at rates proportional to their values.
Understanding how to manipulate and simplify fractional exponents helps in grasping exponential functions. For example, for \(u^{7/5}\), we interpret this as \(\text{“u raised to the power of } \frac{7}{5}”.\) Utilizing properties of exponents, like \((a^m)^n = a^{m*n}\), can simplify calculations involving these functions.
Hence, exponential functions and fractional exponents are tightly connected concepts that are pivotal in higher-level mathematics.