Simplify cube root of x^2* fifth root of x
The question is asking to combine and simplify two different types of roots—a cube root and a fifth root—both of which have the same base, x. Specifically, you are asked to simplify the expression which involves taking the cube root of x squared (written as ∛(x^2)) and multiplying it by the fifth root of x (written as ⁵√x). The task is to apply the properties of exponents and roots to express this as a single, simplified radical expression, if possible.
$\sqrt[3]{x^{2}} \cdot \sqrt[5]{x}$
Answer
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:
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 $n$.
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.
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.
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.
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.
