Problem 41 - Elementary and Intermediate Algebra

Question

Simplify each radical expression. All variables represent positive real numbers. $$ \sqrt[5]{\frac{3 x^{10}}{32}} $$

Step-by-Step Solution

Answer

The simplified expression is \( \frac{\sqrt[5]{3} \cdot x^2}{2} \).

Step 1: Separate the Radical Fraction

We start by separating the fractional expression under the fifth root: \( \sqrt[5]{\frac{3x^{10}}{32}} = \frac{\sqrt[5]{3x^{10}}}{\sqrt[5]{32}} \).

Step 2: Simplify the Denominator

Find the fifth root of 32. Since \( 32 = 2^5 \), we have \( \sqrt[5]{32} = 2 \). The expression becomes \( \frac{\sqrt[5]{3x^{10}}}{2} \).

Step 3: Simplify the Numerator

The fifth root of \(3x^{10}\) can be rewritten and simplified using the property \( \sqrt[5]{a^m} = a^{m/5} \). Thus, \( \sqrt[5]{3x^{10}} = \sqrt[5]{3} \cdot x^{10/5} = \sqrt[5]{3} \cdot x^2 \).

Step 4: Combine the Simplified Terms

Combine the numerator and denominator from the previous steps: \( \frac{\sqrt[5]{3} \cdot x^2}{2} \). This expression is in its simplest form.

Key Concepts

Understanding the Fifth Root

In this problem, the fifth root is an important concept to grasp. The fifth root of a number essentially means the number that needs to be multiplied by itself five times to get back to the original number. For example, if we consider the operation \( \sqrt[5]{32} \), we are looking for a number that, when multiplied by itself five times, results in 32. This number is 2, because \( 2^5 = 32 \).

Fifth roots can be handled similarly to square roots or cube roots, but instead of multiplying two or three times, you multiply five times. When dealing with fifth roots in algebra, particularly with variables or constants, remember:

• Find any perfect powers of 5 present in the expression.
• Simplify using regular arithmetic rules.

Breaking Down Fractional Expression

A fractional expression like \( \frac{3x^{10}}{32} \) involves both a numerator and a denominator that need treating separately especially when they are under a common radical like a fifth root. First, consider separating the expression so that you can accurately handle each part. Here, we did it by writing \( \sqrt[5]{\frac{3x^{10}}{32}} \) as \( \frac{\sqrt[5]{3x^{10}}}{\sqrt[5]{32}} \).

Separating helps in easing the process of simplification since you can work on the numerator and denominator individually. This breakdown into simpler parts often reveals clearer insights into simplifying each component, especially when combined with our knowledge of roots.

Thus, always remember:

• Separate the fraction into a numerator and a denominator under the same root.
• Simplify each independently.

Applying Exponent Rules

Understanding exponent rules is fundamental when simplifying expressions involving radicals, especially when dealing with variables raised to powers. In this problem, dealing with \( x^{10} \) under a fifth root calls for some specific exponent rules.

According to the property \( \sqrt[n]{a^m} = a^{m/n} \), we simplify \( \sqrt[5]{x^{10}} \) by converting the radical expression into a power with a fractional exponent: \( x^{10/5} \). This simplifies to \( x^2 \) because \( 10/5 = 2 \).

Key points on exponent rules:

• Convert root expressions into fractional exponents for easier handling.
• Divide the original exponent by the root's degree to simplify.

Master these rules to navigate radical expressions efficiently.