Simplify square root of (x^4)/3* square root of (x^5)/3

Solution:

Step 1:

Combine the expressions $\frac{x^{4}}{3}$ and $\sqrt{\frac{x^{5}}{3}}$ to form $\sqrt{\frac{x^{4} \sqrt{\frac{x^{5}}{3}}}{3}}$.

Step 2:

Begin simplification of the numerator.

Step 2.1:

Express $\sqrt{\frac{x^{5}}{3}}$ as $\frac{\sqrt{x^{5}}}{\sqrt{3}}$ to get $\sqrt{\frac{x^{4} \frac{\sqrt{x^{5}}}{\sqrt{3}}}{3}}$.

Step 2.2:

Further simplify the numerator.

Step 2.2.1:

Decompose $x^{5}$ into $\left(x^{2}\right)^{2} x$.

Step 2.2.1.1:

Extract $x^{4}$ from the radical to obtain $\sqrt{\frac{x^{4} \frac{\sqrt{x^{4} x}}{\sqrt{3}}}{3}}$.

Step 2.2.1.2:

Represent $x^{4}$ as $\left(x^{2}\right)^{2}$, yielding $\sqrt{\frac{x^{4} \frac{\sqrt{\left(x^{2}\right)^{2} x}}{\sqrt{3}}}{3}}$.

Step 2.2.2:

Extract terms from under the radical, resulting in $\sqrt{\frac{x^{4} \frac{x^{2} \sqrt{x}}{\sqrt{3}}}{3}}$.

Step 2.3:

Multiply $\frac{x^{2} \sqrt{x}}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ to rationalize the denominator.

Step 2.4:

Simplify the denominator.

Step 2.4.1:

Multiply $\frac{x^{2} \sqrt{x}}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ to get $\sqrt{\frac{x^{4} \frac{x^{2} \sqrt{x} \sqrt{3}}{\sqrt{3} \sqrt{3}}}{3}}$.

Step 2.4.2:

Express $\sqrt{3}$ as $3^{\frac{1}{2}}$.

Step 2.4.3:

Apply the power rule to combine the exponents of $\sqrt{3}$, resulting in $\sqrt{\frac{x^{4} \frac{x^{2} \sqrt{x} \sqrt{3}}{3}}{3}}$.

Step 2.4.4:

Combine the product of $\sqrt{3}$ terms using the power rule.

Step 2.4.5:

Add the exponents of $\sqrt{3}$.

Step 2.4.6:

Simplify $\left(\sqrt{3}\right)^{2}$ to $3$.

Step 2.5:

Apply the product rule for radicals to combine the terms under the radical.

Step 3:

Merge $x^{4}$ and $\frac{x^{2} \sqrt{3 x}}{3}$ to form $\sqrt{\frac{\frac{x^{4} \left(x^{2} \sqrt{3 x}\right)}{3}}{3}}$.

Step 4:

Add the exponents of $x$ when multiplying $x^{4}$ and $x^{2}$.

Step 4.1:

Rearrange $x^{2}$ to precede $x^{4}$.

Step 4.2:

Combine exponents using the power rule.

Step 4.3:

Add the exponents $2$ and $4$ to get $\sqrt{\frac{\frac{x^{6} \sqrt{3 x}}{3}}{3}}$.

Step 5:

Multiply the numerator by the reciprocal of the denominator.

Step 6:

Combine the terms under the radical.

Step 7:

Perform the multiplication.

Step 7.1:

Multiply $x^{6}$ by $1$.

Step 7.2:

Multiply $3$ by $3$ to get $\sqrt{\frac{x^{6} \sqrt{3 x}}{9}}$.

Step 8:

Rewrite $\frac{x^{6} \sqrt{3 x}}{9}$ as $\left(\frac{x^{3}}{3}\right)^{2} \sqrt{3 x}$.

Step 8.1:

Factor out the perfect square $\left(x^{3}\right)^{2}$ from $x^{6} \sqrt{3 x}$.

Step 8.2:

Factor out the perfect square $3^{2}$ from $9$.

Step 8.3:

Rearrange the fraction to $\sqrt{\left(\frac{x^{3}}{3}\right)^{2} \sqrt{3 x}}$.

Step 9:

Extract terms from under the radical.

Step 10:

Express $\sqrt{\sqrt{3 x}}$ as $\sqrt[4]{3 x}$.

Step 11:

Combine $\frac{x^{3}}{3}$ and $\sqrt[4]{3 x}$ to get $\frac{x^{3} \sqrt[4]{3 x}}{3}$.

Knowledge Notes:

The problem involves simplifying a radical expression that contains variables with exponents. The key knowledge points used in the solution include:

1. Radical Simplification: Simplifying expressions under a square root involves factoring out perfect squares and rationalizing the denominator if necessary.

2. Exponent Rules: The power rule $a^{m} a^{n} = a^{m + n}$ is used to combine like bases with exponents. Additionally, $\left(a^{m}\right)^{n} = a^{m \cdot n}$ is used to raise a power to another power.

3. Rationalizing the Denominator: Multiplying by a conjugate or an appropriate form of 1 (such as $\frac{\sqrt{3}}{\sqrt{3}}$) to eliminate radicals from the denominator.

4. Combining Radicals: Using the product rule for radicals, $\sqrt{a} \cdot \sqrt{b} = \sqrt{a \cdot b}$, to combine radicals into a single radical.

5. Higher-Order Roots: Understanding that $\sqrt{\sqrt{a}}$ is equivalent to $\sqrt[4]{a}$, which represents the fourth root of $a$.

6. Fraction Multiplication: Multiplying the numerator by the reciprocal of the denominator, which is a basic operation with fractions.

These concepts are fundamental in algebra and are often used when working with polynomial expressions, radical expressions, and rational expressions.