Simplify fifth root of 5a^5b^9c^13

Solution:

Step 1: Decompose the expression

Express $5a^{5}b^{9}c^{13}$ by separating terms that have exponents as multiples of 5 from those that do not:

$\sqrt[5]{(abc^{2})^{5} \cdot 5b^{4}c^{3}}$.

Step 1.1: Extract the perfect fifth power of $b$

Isolate $b^5$ from $b^9$:

$\sqrt[5]{5a^{5}(b^{5}b^{4})c^{13}}$.

Step 1.2: Extract the perfect fifth power of $c$

Isolate $c^{10}$ from $c^{13}$:

$\sqrt[5]{5a^{5}(b^{5}b^{4})(c^{10}c^{3})}$.

Step 1.3: Represent $c^{10}$ as a fifth power

Rewrite $c^{10}$ as $(c^{2})^{5}$:

$\sqrt[5]{5a^{5}(b^{5}b^{4})((c^{2})^{5}c^{3})}$.

Step 1.4: Reorganize $b^{4}$

Move $b^{4}$ to the end of the expression:

$\sqrt[5]{5a^{5}b^{5}(c^{2})^{5}b^{4}c^{3}}$.

Step 1.5: Reorganize the constant 5

Move the constant 5 to the end of the expression:

$\sqrt[5]{a^{5}b^{5}(c^{2})^{5} \cdot 5b^{4}c^{3}}$.

Step 1.6: Combine terms into a fifth power

Combine $a^{5}$, $b^{5}$, and $(c^{2})^{5}$:

$\sqrt[5]{(abc^{2})^{5} \cdot 5b^{4}c^{3}}$.

Step 1.7: Enclose the remaining terms in parentheses

Add parentheses around $5b^{4}c^{3}$:

$\sqrt[5]{(abc^{2})^{5} \cdot (5b^{4}c^{3})}$.

Step 1.8: Ensure proper grouping of terms

Ensure the expression is properly grouped:

$\sqrt[5]{(abc^{2})^{5} \cdot (5b^{4}c^{3})}$.

Step 2: Simplify the radical expression

Extract terms that are perfect fifth powers:

$abc^{2}\sqrt[5]{5b^{4}c^{3}}$.

Knowledge Notes:

To simplify a radical expression involving a root, such as the fifth root, the goal is to express the terms under the radical as powers of the root whenever possible. In this case, we are looking for powers of 5. Here are some relevant knowledge points:

1. Exponent Rules: When simplifying expressions with exponents, remember that $a^{m} \cdot a^{n} = a^{m+n}$ and $(a^{m})^{n} = a^{m \cdot n}$.

2. Radicals and Exponents: The $n$th root of a number can be expressed as an exponent: $\sqrt[n]{a} = a^{\frac{1}{n}}$. For this problem, we are dealing with the fifth root, so we look for terms that can be expressed as a power of 5.

3. Simplifying Radicals: When a term inside a radical is raised to a power that is a multiple of the index of the radical, that term can be taken out of the radical. For example, $\sqrt[n]{a^{n}} = a$.

4. Factoring: Breaking down expressions into factors can help identify terms that can be pulled out of the radical. This involves recognizing that $a^{n} = (a^{\frac{n}{m}})^{m}$ when $m$ divides $n$.

5. Combining Like Terms: When simplifying expressions, we often combine like terms to make the expression easier to understand. In the context of radicals, this means combining terms under the radical that have the same base and can be expressed as a power of the index of the radical.

By applying these principles, we can simplify the given radical expression by extracting terms that are perfect fifth powers and leaving the rest under the radical.