Question

If $a,b,c$ are positive real numbers, then $\sqrt[5]{3125a^{10}{b}^5{c}^{10}}$ is equals to:

• A. a) $5a^{2}b{c}^2$
• B. b) $5a^{3}b{c}^3$
• C. c) $25a{b}^{2}c$
• D. d) $125a^{2}b{c}^2$

Answer 1

The correct option is A

a) $5a^{2}b{c}^2$

Explanation for correct options

Option (a): $5a^{2}b{c}^2$

Given an expression $\sqrt[5]{3125a^{10}{b}^5{c}^{10}}$ where $a,b,c$ are positive real numbers.

We proceed by simplifying the expression.

$$\begin{gathered} \sqrt[5]{3125a^{10}{b}^5{c}^{10}}=\sqrt[5]{5^5{\left(a^2\right)}^5{\left(b\right)}^5{\left(c^2\right)}^5}[\because 3125=5\times 5\times 5\times 5\times 5=5^5\mathbf{;}\mathbf{}{a}^{mn}={\left(a^m\right)}^n] \\ =\sqrt[5]{{\left(5a^{2}b{c}^2\right)}^5} \\ =5a^{2}b{c}^2 \end{gathered}$$

Therefore, the simplified form of the expression $\sqrt[5]{3125a^{10}{b}^5{c}^{10}}$ is $5a^{2}b{c}^2$.

Thus, the correct answer is option (a).