Simplify cube root of x^2* fifth root of x

Solution:

Simplification Process

Step 1: Convert to Common Radical Index

Start by expressing the given radicals with a common index of $15$ for simplification.

Step 1.1: Express $\sqrt[3]{x^2}$ in Exponential Form

Convert $\sqrt[3]{x^2}$ to exponential notation using $a^{\frac{x}{n}}=\sqrt[n]{a^x}$:

$$x^{\frac{2}{3}}\cdot \sqrt[5]{x}$$

Step 1.2: Adjust the Exponent to Match the Common Index

Change $x^{\frac{2}{3}}$ to have a denominator of $15$:

$$x^{\frac{10}{15}}\cdot \sqrt[5]{x}$$

Step 1.3: Rewrite as a Single Radical

Express $x^{\frac{10}{15}}$ as a radical with an index of $15$:

$$\sqrt[15]{x^{10}}\cdot \sqrt[5]{x}$$

Step 1.4: Convert $\sqrt[5]{x}$ to Exponential Form

Rewrite $\sqrt[5]{x}$ using exponential notation:

$$\sqrt[15]{x^{10}}\cdot x^{\frac{1}{5}}$$

Step 1.5: Adjust the Exponent to Match the Common Index

Change $x^{\frac{1}{5}}$ to have a denominator of $15$:

$$\sqrt[15]{x^{10}}\cdot x^{\frac{3}{15}}$$

Step 1.6: Rewrite as a Single Radical

Express $x^{\frac{3}{15}}$ as a radical with an index of $15$:

$$\sqrt[15]{x^{10}}\cdot \sqrt[15]{x^3}$$

Step 2: Combine Radicals

Use the product rule for radicals to combine them into a single radical:

$$\sqrt[15]{x^{10}\cdot x^3}$$

Step 3: Simplify the Exponent

Combine the exponents inside the radical by adding them together.

Step 3.1: Apply the Power Rule

Utilize the power rule $a^m\cdot a^n=a^{m+n}$ to sum the exponents:

$$\sqrt[15]{x^{10+3}}$$

Step 3.2: Perform the Addition

Add the exponents $10$ and $3$:

$$\sqrt[15]{x^{13}}$$

Knowledge Notes:

To simplify expressions involving roots of different orders, it is helpful to understand the following concepts:

1. Radicals and Exponents: The $n$th root of a number $a$ can be expressed as $a^{\frac{1}{n}}$. This is the inverse operation of raising a number to the power of $n$.

2. Common Index: When combining radicals, it's easier to work with a common index. The least common multiple (LCM) of the indices can be used as the common index.

3. Exponential Notation: Radicals can be rewritten in exponential form using the rule $\sqrt[n]{a^x}=a^{\frac{x}{n}}$. This is useful for combining and simplifying expressions with different roots.

4. Product Rule for Radicals: The product rule states that $\sqrt[n]{a}\cdot \sqrt[n]{b}=\sqrt[n]{a\cdot b}$, which allows us to multiply under the same radical.

5. Power Rule: The power rule for exponents states that $a^m\cdot a^n=a^{m+n}$. This is used when multiplying like bases with exponents.

By applying these principles, we can simplify complex radical expressions into a more manageable form.