Simplify sixth root of 256x^8y^4z^10

Solution:

Step 1: Decompose the expression

Express $256 x^{8} y^{4} z^{10}$ as $(2 x z)^{6} \cdot (4 x^{2} y^{4} z^{4})$.

Step 1.1: Extract the perfect sixth power

Identify the perfect sixth power in $256$, which is $64$. Thus, write $\sqrt[6]{64 \cdot (4) x^{8} y^{4} z^{10}}$.

Step 1.2: Represent $64$ as a power of $2$

Express $64$ as $2^{6}$. The expression becomes $\sqrt[6]{2^{6} \cdot 4 x^{8} y^{4} z^{10}}$.

Step 1.3: Separate $x^{6}$ from $x^{8}$

Extract $x^{6}$ from $x^{8}$, yielding $\sqrt[6]{2^{6} \cdot 4 (x^{6} x^{2}) y^{4} z^{10}}$.

Step 1.4: Separate $z^{6}$ from $z^{10}$

Extract $z^{6}$ from $z^{10}$, resulting in $\sqrt[6]{2^{6} \cdot 4 (x^{6} x^{2}) y^{4} (z^{6} z^{4})}$.

Step 1.5: Rearrange $y^{4}$

Position $y^{4}$ next to $z^{6}$, as $\sqrt[6]{2^{6} \cdot 4 (x^{6} x^{2}) z^{6} y^{4} z^{4}}$.

Step 1.6: Rearrange $x^{2}$

Position $x^{2}$ next to $z^{6}$, forming $\sqrt[6]{2^{6} \cdot 4 (x^{6}) z^{6} x^{2} y^{4} z^{4}}$.

Step 1.7: Rearrange $4$

Position $4$ next to $z^{4}$, to get $\sqrt[6]{(2^{6} (x^{6})) z^{6} \cdot 4 x^{2} y^{4} z^{4}}$.

Step 1.8: Rewrite the sixth power

Express $(2^{6} (x^{6})) z^{6}$ as $(2 x z)^{6}$. The expression is now $\sqrt[6]{(2 x z)^{6} \cdot 4 x^{2} y^{4} z^{4}}$.

Step 1.9: Add parentheses around $y^{4} z^{4}$

Enclose $y^{4} z^{4}$ in parentheses, to obtain $\sqrt[6]{(2 x z)^{6} \cdot 4 x^{2} (y^{4} z^{4})}$.

Step 1.10: Add parentheses around $x^{2} (y^{4} z^{4})$

Enclose $x^{2} (y^{4} z^{4})$ in parentheses, resulting in $\sqrt[6]{(2 x z)^{6} \cdot 4 (x^{2} (y^{4} z^{4}))}$.

Step 1.11: Finalize the expression

Confirm the expression $\sqrt[6]{(2 x z)^{6} \cdot (4 x^{2} y^{4} z^{4})}$.

Step 2: Simplify the sixth root

Extract terms from under the sixth root, leading to $2 x z \sqrt[6]{4 x^{2} y^{4} z^{4}}$.

Step 3: Rewrite the remaining expression

Express $4 x^{2} y^{4} z^{4}$ as $(2 x y^{2} z^{2})^{2}$. The expression is now $2 x z \sqrt[6]{(2 x y^{2} z^{2})^{2}}$.

Step 4: Convert the sixth root to a cube root of a square root

Rewrite $\sqrt[6]{(2 x y^{2} z^{2})^{2}}$ as $\sqrt[3]{\sqrt{(2 x y^{2} z^{2})^{2}}}$, resulting in $2 x z \sqrt[3]{\sqrt{(2 x y^{2} z^{2})^{2}}}$.

Step 5: Extract terms under the cube root

Assuming all variables represent positive real numbers, pull terms out from under the cube root to get $2 x z \sqrt[3]{2 x y^{2} z^{2}}$.

Knowledge Notes:

To simplify a radical expression, especially one involving a higher root like the sixth root, the following knowledge points are relevant:

1. Prime Factorization: Breaking down a number into its prime factors can help identify perfect powers within the radical.

2. Properties of Exponents: Understanding how to manipulate exponents, such as factoring them out or combining them, is crucial in simplifying expressions under a radical.

3. Radical Rules: Knowing that $\sqrt[n]{a^n} = a$ when $a$ is a positive real number and $n$ is an integer helps in extracting terms from under the radical.

4. Nested Radicals: Converting between different types of radicals, such as rewriting a sixth root as a cube root of a square root, can simplify the expression further.

5. Algebraic Manipulation: Rearranging terms and grouping them appropriately under the radical can facilitate the simplification process.

In this problem, we used these knowledge points to simplify the given expression by extracting perfect powers and rearranging terms to pull them out from under the sixth root.