Problem 80 - Algebra A Combined Function

Question

Use rational expressions to write as a single radical expression. $$ \frac{\sqrt[5]{b^{2}}}{\sqrt[10]{b^{3}}} $$

Step-by-Step Solution

Answer

\( \frac{\sqrt[5]{b^{2}}}{\sqrt[10]{b^{3}}} = \sqrt[10]{b} \)

Step 1: Express the radicals as fractional exponents

The expression \( \frac{\sqrt[5]{b^{2}}}{\sqrt[10]{b^{3}}} \) can be rewritten using fractional exponents. The expression \( \sqrt[5]{b^{2}} \) is equivalent to \( b^{\frac{2}{5}} \) and \( \sqrt[10]{b^{3}} \) is equivalent to \( b^{\frac{3}{10}} \).

Step 2: Apply the quotient rule for exponents

The quotient rule for exponents states that \( \frac{a^m}{a^n} = a^{m-n} \). Apply this rule to \( \frac{b^{\frac{2}{5}}}{b^{\frac{3}{10}}} \) to get \( b^{\frac{2}{5} - \frac{3}{10}} \).

Step 3: Subtract the exponents

To subtract these fractions, find a common denominator. The common denominator for \( \frac{2}{5} \) and \( \frac{3}{10} \) is 10. Convert \( \frac{2}{5} \) to \( \frac{4}{10} \). Now \( \frac{4}{10} - \frac{3}{10} = \frac{1}{10} \). Thus, the new expression is \( b^{\frac{1}{10}} \).

Step 4: Convert back to radical expression

The expression \( b^{\frac{1}{10}} \) can be written as a radical \( \sqrt[10]{b} \), which is already a single radical expression.

Key Concepts

Fractional Exponents

Fractional exponents might seem intimidating at first, but they are quite straightforward once you get the hang of it. They offer a handy way to express roots and powers in a single format.
For example, the expression \(root5{b^2}\) is equivalent to \(b^{\frac{2}{5}}\). Here, 5 is the root, and 2 is the exponent of the base, b. These fractional exponents allow for a more uniform way of handling powers and roots, especially when performing operations such as multiplication and division.
This format is beneficial when simplifying expressions like \(root[10]{b^3}\), which becomes \(b^{\frac{3}{10}}\). When both parts of a complex expression are converted to this form, it becomes easier to apply algebraic rules, as shown in this exercise.

Quotient Rule for Exponents

The quotient rule is a handy tool when you need to simplify fractions that involve powers with the same base. It states \(\frac{a^m}{a^n} = a^{m-n}\).
Essentially, you keep the base unchanged and subtract the exponent of the denominator from the exponent of the numerator. In the exercise, we have \(\frac{b^{\frac{2}{5}}}{b^{\frac{3}{10}}}\).
To simplify, we subtract the exponents: \(\frac{2}{5} - \frac{3}{10}\).
It's important to have a common denominator when subtracting fractions, which was solved by converting \(\frac{2}{5}\) into \(\frac{4}{10}\), making it easy to perform the subtraction and simplifying the expression to \(b^{\frac{1}{10}}\). The correct application of this rule can significantly simplify complex rational expressions.

Rational Expressions

Rational expressions are fractions in which both the numerator and denominator are polynomials. They aren't restricted to integers or whole numbers but can include variable expressions as well. In our context, they are used as tools to simplify and manipulate expressions involving roots.
In the given exercise, we deal with an expression involving exponents in a radical form: \(\frac{root5{b^2}}{root[10]{b^3}}\). When rewritten using fractional exponents, the problem becomes more manageable, allowing us to apply exponent rules with ease.
Finally, converting the simplified fractional exponent form \(b^{\frac{1}{10}}\) back into a radical results in \(root[10]{b}\). These manipulations exemplify how rational expressions can streamline solving and simplifying problems involving radicals and exponents.