Simplify 30 root of 32a^20b^15c^25

Solution:

Step:1 Express $32a^{20}{b}^{15}{c}^{25}$ as $(2a^4{b}^3{c}^5{)}^5$. Then, consider $\sqrt[30]{(2a^4{b}^3{c}^5{)}^5}$.

Step:2 Transform $\sqrt[30]{(2a^4{b}^3{c}^5{)}^5}$ into $\sqrt[6]{\sqrt[5]{(2a^4{b}^3{c}^5{)}^5}}$.

Step:3 Extract terms from under the radical, assuming all variables represent real numbers. Resulting in $\sqrt[6]{2a^4{b}^3{c}^5}$.

Knowledge Notes:

To simplify a radical expression, especially one involving a root of a power, we can use the property that $\sqrt[n]{x^m}=x^{\frac{m}{n}}$ when $x$ is a nonnegative real number and $m$ and $n$ are integers.

In this problem, we are dealing with the 30th root of a power, which can be simplified by recognizing that the exponent of the power and the index of the root have a common factor.

Here are the steps broken down:

1. We first express the given expression in a form that makes it easier to apply the root: $32a^{20}{b}^{15}{c}^{25}$ is rewritten as $(2a^4{b}^3{c}^5{)}^5$ because $32=2^5$ and each of the exponents of $a$, $b$, and $c$ is a multiple of 5.

2. We then recognize that taking the 30th root of a fifth power is the same as taking the sixth root of the fifth root, which simplifies the expression: $\sqrt[30]{(2a^4{b}^3{c}^5{)}^5}=\sqrt[6]{\sqrt[5]{(2a^4{b}^3{c}^5{)}^5}}$.

3. Finally, since the fifth root and the fifth power cancel each other out, we are left with the sixth root of the base terms: $\sqrt[6]{2a^4{b}^3{c}^5}$.

Note that this simplification assumes that all variables represent real numbers to avoid dealing with complex numbers when extracting roots.