Simplify 30 root of 32a^20b^15c^25
The question is asking for a simplification of a mathematical expression which involves a radical (root). Specifically, you are given the 30th root of the algebraic term 32a^20b^15c^25. The task is to simplify this expression by reducing it to its simplest form, which might involve factoring out perfect powers from under the radical as appropriate for a 30th root and simplifying any numerical coefficients.
$\sqrt[30]{32 a^{20} b^{15} c^{25}}$
Answer
Solution:
Step:1 Express $32a^{20}b^{15}c^{25}$ as $(2a^4b^3c^5)^5$. Then, consider $\sqrt[30]{(2a^4b^3c^5)^5}$.
Step:2 Transform $\sqrt[30]{(2a^4b^3c^5)^5}$ into $\sqrt[6]{\sqrt[5]{(2a^4b^3c^5)^5}}$.
Step:3 Extract terms from under the radical, assuming all variables represent real numbers. Resulting in $\sqrt[6]{2a^4b^3c^5}$.
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:
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 $(2a^4b^3c^5)^5$ because $32 = 2^5$ and each of the exponents of $a$, $b$, and $c$ is a multiple of 5.
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: $\sqrt[30]{(2a^4b^3c^5)^5} = \sqrt[6]{\sqrt[5]{(2a^4b^3c^5)^5}}$.
Finally, since the fifth root and the fifth power cancel each other out, we are left with the sixth root of the base terms: $\sqrt[6]{2a^4b^3c^5}$.
Note that this simplification assumes that all variables represent real numbers to avoid dealing with complex numbers when extracting roots.
