Problem 71 - Elementary Algebra

Question

For the following problems, simplify each of the radical expressions. $$ \sqrt{\frac{6}{x^{3}}} $$

Step-by-Step Solution

Answer

Question: Simplify the radical expression √(6/x³). Answer: The simplified expression is (sqrt(6))/(x^(3/2)).

Step 1: Write the square root as a fractional exponent

Given the square root of a fraction, we can write it as a fractional exponent. Remember that the square root of a number can be written as the exponent 1/2. So, we can rewrite the radical expression as follows: $$ \left(\frac{6}{x^3}\right)^{\frac{1}{2}} $$

Step 2: Apply the power-to-the-fraction rule

The next step to simplify the expression is applying the power to the fraction (a/b)^n = a^n / b^n. Hence, we get: $$ \frac{6^{\frac{1}{2}}}{x^{3\cdot\frac{1}{2}}} $$

Step 3: Simplify the fraction

Now, we can simplify both the numerator and the denominator. In the numerator, the square root of 6 cannot be simplified further as 6 is not a perfect square. So, we have \(6^{\frac{1}{2}} = \sqrt{6}\). In the denominator, we multiply the exponents to get the power of x: \(3\cdot\frac{1}{2} = \frac{3}{2}\). Thus, we have \(x^{\frac{3}{2}}\). Therefore, the simplified expression is: $$ \frac{\sqrt{6}}{x^{\frac{3}{2}}} $$

Key Concepts

Fractional Exponents

When working with square roots or any other type of roots, fractional exponents provide a uniform way to express these roots. For any positive number, the square root can be expressed as raising that number to the power of \( \frac{1}{2} \). This is a fundamental concept which allows us to handle roots using the same rules as other exponents. By expressing the square root of a fraction like \( \sqrt{\frac{6}{x^3}} \) as \( \left(\frac{6}{x^3}\right)^{\frac{1}{2}} \), we can use exponent laws to manipulate the expression further.
Fractional exponents are a bridge between roots and powers. For instance:

• \( x^{\frac{1}{2}} \) represents the square root of \( x \).
• \( x^{\frac{1}{3}} \) represents the cube root of \( x \).
• \( x^{\frac{m}{n}} \) represents \( (\sqrt[n]{x})^m \) or \( \sqrt[n]{x^m} \).

Understanding fractional exponents allows us to tackle root problems with the same techniques we use for polynomial expressions.

Power-to-the-Fraction Rule

The power-to-the-fraction rule is a crucial tool for simplifying expressions involving exponents. When an expression with a fractional power is encountered, such as \( (\frac{a}{b})^n \), it is simplified by raising both the numerator and the denominator to the given power. In the expression \( \left(\frac{6}{x^3}\right)^{\frac{1}{2}} \), this rule is applied to separate the powers:
\[ \left(\frac{6}{x^3}\right)^{\frac{1}{2}} = \frac{6^{\frac{1}{2}}}{x^{3 \cdot \frac{1}{2}}} \]
This results in both components of the fraction being handled independently, making further simplification possible.
Here's how this rule can be further clarified:

• Apply the exponent \( n \) to the numerator: \( a^n \).
• Apply the exponent \( n \) to the denominator: \( b^n \).
• Ensures each part of a fraction is simplified separately.

This rule is effective in disentangling complex expressions into simpler, manageable parts that can easily be interpreted or simplified.

Simplification of Algebraic Expressions

Simplification of algebraic expressions involves reducing them to their simplest form. This often requires the use of basic arithmetic rules, properties of exponents, and an understanding of radical expressions.
In the context of our example, converting \( 6^{\frac{1}{2}} \) into \( \sqrt{6} \) shows how numerical parts of an expression are simplified using known roots. Because 6 isn't a perfect square, \( \sqrt{6} \) remains as it is. The denominator, \( x^{\frac{3}{2}} \), uses multiplication of exponents \( 3 \cdot \frac{1}{2} \)—a direct consequence of the distributive property of exponents—to rewrite it as \( x^{\frac{3}{2}} \).
Key points to remember for simplification:

• Use known root values for numerical simplification when applicable.
• Multiply exponents correctly when removing roots.
• Express the answer in a concise form to avoid overly complex expressions.

The goal of simplification is to make calculations more straightforward and expressions easier to understand and use in future calculations.