Problem 20 - Algebra 2 and Trigonometry

Question

In \(3-38\) , write each radical in simplest radical form. Variables in the radicand of an even index are non-negative. Variables occurring in the denominator of a fraction are non-zero. $$ \sqrt{\frac{6}{81 y^{6}}} $$

Step-by-Step Solution

Answer

\(\sqrt{\frac{6}{81 y^{6}}} = \frac{\sqrt{6}}{9y^3}\).

Step 1: Simplify the Fraction

Start by simplifying the fraction under the square root. The square root of a fraction is the square root of the numerator divided by the square root of the denominator:\[\sqrt{\frac{6}{81 y^{6}}} = \frac{\sqrt{6}}{\sqrt{81 y^{6}}}.\]

Step 2: Calculate the Square Root of the Denominator

Next, calculate the square root of the denominator. First, simplify the constant and the variable separately. Since \(81 = 9^2\), we have:\[\sqrt{81} = 9.\]For \(y^6\), we use the property \(\sqrt{y^{2n}} = y^n\), so:\[\sqrt{y^6} = y^3.\]Thus, \(\sqrt{81 y^6} = 9y^3.\)

Step 3: Write the Simplified Radical Form

Having the square root of both the numerator and the denominator, the expression simplifies to:\[\frac{\sqrt{6}}{9y^3}.\]

Step 4: Verify the Conditions

Ensure that the conditions are respected: The variable \(y\) is non-negative and non-zero (as clarified in the problem statement), which validates the simplified form. The denominator is non-zero for \(y eq 0\).

Key Concepts

Square Root Simplification

Simplifying square roots can make expressions much easier to work with. When you encounter a square root of a fraction like \(\sqrt{\frac{6}{81 y^{6}}}\), the first step is to apply the property \(\sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}}\). This means you handle the numerator and the denominator separately.
For instance:

• Numerator: \( \sqrt{6} \) can stay as is because 6 is not a perfect square.
• Denominator: \( \sqrt{81} \) simplifies to 9 since 81 is a perfect square \((9^2)\).
• Variable Term: For \( y^6 \), you use the rule \( \sqrt{y^{2n}} = y^n \) to simplify this to \( y^3 \).

This gradual process makes it easier to understand and simplify entire expressions, turning something complex into a more manageable form.

Radicals in Algebra

Radicals, like square roots, are special symbols used in algebra to indicate root values. Radicals can sometimes seem tricky, but by knowing a few basic rules, they become manageable.
When working with radicals:

• A radical like \( \sqrt{x} \) means "the square root of x." It is looking for a number which, when multiplied by itself, gives \( x \).
• Even-index radicals, like square roots, only apply to non-negative numbers when dealing with real numbers. This is crucial, as negative numbers can lead to non-real answers.
• Radicals in denominations require extra care. Ensure the denominator isn't zero to avoid undefined expressions.

In our problem, we ensured the variable \( y \) was non-negative and non-zero, fitting within these algebraic rules. Understanding the properties of radicals helps ensure calculations are both accurate and meaningful.

Simplifying Fractions with Variables

Simplifying fractions that include variables under a square root involves combining radical simplification with fractional simplification. This can be broken down into a few key steps:
1. **Break Apart the Fraction:** Start by rewriting \( \sqrt{\frac{6}{81 y^{6}}} \) as \( \frac{\sqrt{6}}{\sqrt{81 y^{6}}}\), simplifying each part as individual radicals.
2. **Simplify Denominator:** Calculate and simplify the square root of both numerical and variable parts:

• Use \( \sqrt{81} = 9 \)
• For \( y^6 \), remember \( \sqrt{y^{6}} = y^3 \), thanks to the rule \( \sqrt{y^{2n}} = y^n \).

3. **Form a Simple Expression:** These steps lead to \( \frac{\sqrt{6}}{9y^3} \), a much tidier form.
This method is especially useful when dealing with complex algebraic expressions involving both radicals and fractions, ensuring each individual component is simplified and error-free.