Simplify y sixth root of 27x^3

Solution:

Step 1:

Express $27x^3$ in the form of $(3x{)}^3$. Thus, we have $y\sqrt[6]{(3x{)}^3}$.

Step 2:

Transform $\sqrt[6]{(3x{)}^3}$ into $\sqrt{\sqrt[3]{(3x{)}^3}}$. Now the expression is $y\sqrt{\sqrt[3]{(3x{)}^3}}$.

Step 3:

Extract terms from under the radical, assuming all numbers are real. The final simplified form is $y\sqrt{3x}$.

Knowledge Notes:

To simplify an expression like $y\sqrt[6]{27x^3}$, we need to understand several mathematical concepts:

1. Radical Notation: The notation $\sqrt[n]{a}$ represents the nth root of a number 'a'. In this case, $\sqrt[6]{a}$ means the sixth root of 'a'.

2. Exponent Rules: The expression $a^b$ where 'a' is the base and 'b' is the exponent, means 'a' multiplied by itself 'b' times. For example, $(3x{)}^3$ means $3x$ multiplied by itself three times.

3. Simplifying Radicals: When simplifying radicals, we look for factors that are perfect powers of the radical index. For example, since $27$ is $3^3$, it can be taken out of a cube root as a single 3.

4. Nested Radicals: A nested radical is a radical within another radical. The expression $\sqrt{\sqrt[3]{a}}$ is a nested radical. In this case, it's a square root outside of a cube root.

5. Real Numbers: The assumption that all numbers are real is important because it affects how we can manipulate and simplify expressions involving roots. Some operations are not valid with complex numbers.

By applying these concepts, we can simplify the given expression step by step, eventually extracting terms from under the radical to find the simplest form.