Simplify the Radical Expression fourth root of (x^4y^24)/81

Solution:

Step 1:

Express $x^{4}y^{24}$ as $(xy^{6})^{4}$. Thus, the expression becomes $\sqrt[4]{\frac{(xy^{6})^{4}}{81}}$.

Step 2:

Represent $81$ as $3^{4}$. The expression now reads $\sqrt[4]{\frac{(xy^{6})^{4}}{3^{4}}}$.

Step 3:

Transform $\frac{(xy^{6})^{4}}{3^{4}}$ into $(\frac{xy^{6}}{3})^{4}$. The radical expression is now $\sqrt[4]{(\frac{xy^{6}}{3})^{4}}$.

Step 4:

Extract terms from beneath the radical, resulting in $|\frac{xy^{6}}{3}|$.

Step 5:

Eliminate the absolute value for non-negative terms to get $\frac{y^{6}|x|}{3}$.

Knowledge Notes:

1. Radical expressions involve roots, such as square roots, cube roots, and fourth roots. Simplifying radical expressions often requires manipulating the expression to make use of the properties of exponents and radicals.

2. The fourth root of a number $a$, denoted as $\sqrt[4]{a}$, is the number that, when raised to the power of 4, equals $a$.

3. Exponents can be factored out of a radical if the exponent is a multiple of the index of the root. For example, $\sqrt[4]{x^4}$ simplifies to $|x|$ because raising $x$ to the fourth power and then taking the fourth root cancels out.

4. When simplifying expressions under a radical, it's helpful to express numbers as powers that match the root index. For instance, $81$ can be written as $3^4$, which simplifies the process of taking the fourth root.

5. The absolute value, denoted as $|x|$, is the non-negative value of $x$. When simplifying radicals, it's important to consider the absolute value to ensure the result is non-negative, as roots are typically considered to be non-negative in real numbers.

6. When an entire expression under a radical is raised to the power of the index of the root (e.g., $\sqrt[4]{(xy^{6})^{4}}$), the radical and the exponent cancel each other out, leaving the expression without the radical (e.g., $xy^{6}$).