Simplify fourth root of (x^2)/(9x)

Solution:

Simplify the expression: $\sqrt[4]{\frac{x^2}{9x}}$

Step 1: Simplify the fraction by removing common factors.

• Step 1.1: Extract $x$ from $x^2$. $\sqrt[4]{\frac{x \cdot x}{9x}}$
• Step 1.2: Extract $x$ from $9x$. $\sqrt[4]{\frac{x \cdot x}{x \cdot 9}}$
• Step 1.3: Eliminate the common $x$. $\sqrt[4]{\frac{\cancel{x} \cdot x}{\cancel{x} \cdot 9}}$
• Step 1.4: Present the simplified fraction. $\sqrt[4]{\frac{x}{9}}$

Step 2: Convert the fourth root of a fraction into a fraction of fourth roots. $\frac{\sqrt[4]{x}}{\sqrt[4]{9}}$

Step 3: Simplify the fourth root of the denominator.

• Step 3.1: Express 9 as $3^2$. $\frac{\sqrt[4]{x}}{\sqrt[4]{3^2}}$
• Step 3.2: Represent $\sqrt[4]{3^2}$ as the square root of a square root. $\frac{\sqrt[4]{x}}{\sqrt{\sqrt{3^2}}}$
• Step 3.3: Extract terms from under the radical, assuming x is positive. $\frac{\sqrt[4]{x}}{\sqrt{3}}$

Step 4: Rationalize the denominator by multiplying by $\frac{\sqrt{3}}{\sqrt{3}}$. $\frac{\sqrt[4]{x}}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}}$

Step 5: Combine and simplify the terms in the denominator.

• Step 5.1: Multiply the numerator and denominator by $\sqrt{3}$. $\frac{\sqrt[4]{x} \cdot \sqrt{3}}{\sqrt{3} \cdot \sqrt{3}}$
• Step 5.2: Rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$. $\frac{\sqrt[4]{x} \cdot \sqrt{3}}{(3^{\frac{1}{2}})^2}$
• Step 5.3: Apply the power rule to combine exponents. $\frac{\sqrt[4]{x} \cdot \sqrt{3}}{3^{\frac{1}{2} \cdot 2}}$
• Step 5.4: Simplify the exponent. $\frac{\sqrt[4]{x} \cdot \sqrt{3}}{3^{\frac{2}{2}}}$
• Step 5.5: Cancel out the common factor of 2. $\frac{\sqrt[4]{x} \cdot \sqrt{3}}{3^{\cancel{2}/\cancel{2}}}$
• Step 5.6: Present the simplified denominator. $\frac{\sqrt[4]{x} \cdot \sqrt{3}}{3}$

Step 6: Simplify the numerator.

• Step 6.1: Use the least common index of 4 to combine the radicals.

• Step 6.2: Apply the product rule for radicals. $\frac{\sqrt[4]{x \cdot 3^2}}{3}$

• Step 6.3: Simplify the expression. $\frac{\sqrt[4]{x \cdot 9}}{3}$

Step 7: Rearrange the factors. $\frac{\sqrt[4]{9x}}{3}$

Knowledge Notes:

To solve the given problem, several mathematical concepts and rules are applied:

1. Simplifying Fractions: Fractions are simplified by canceling out common factors in the numerator and denominator.

2. Radicals: A radical expression involves roots, such as square roots or fourth roots. The fourth root of a number $a$ is written as $\sqrt[4]{a}$.

3. Rationalizing the Denominator: This process involves removing radicals from the denominator of a fraction by multiplying the numerator and denominator by an appropriate form of 1, such as $\frac{\sqrt{3}}{\sqrt{3}}$.

4. Exponent Rules: These rules include the power rule ($a^m \cdot a^n = a^{m+n}$) and the rule for raising a power to a power ($(a^m)^n = a^{m \cdot n}$).

5. Radical Rules: The product rule for radicals allows us to combine radicals by multiplying the terms inside the radical if the index is the same ($\sqrt[n]{a} \cdot \sqrt[n]{b} = \sqrt[n]{a \cdot b}$).

6. Least Common Index: When dealing with multiple radicals, it's often useful to rewrite them with the least common index to combine them more easily.

By applying these concepts and rules systematically, the original radical expression is simplified to a more elementary form.