Question

simplify the radical using the product property. assume all variables represent positive real numbers.

\\\sqrt5{-8m^7n^4}\\

\\\sqrt5{-8m^7n^4} = \boxed{\quad}\\
(type an exact answer, using radicals as needed. simplify your answer.)

Explanation:

⚡ Using what you learned: Simplifying radical expressions

Step 1: Factor the radicand into perfect fifth powers

Identify factors inside the fifth root that are perfect fifth powers:

$$ -8 = -1 \cdot 8 $$

$$ m^7 = m^5 \cdot m^2 $$

$$ n^4 \text{ has no factors of fifth powers since } 4 < 5 $$

Group the perfect fifth powers and the remaining factors:

$$ \sqrt[5]{-8m^7n^4} = \sqrt[5]{(-1 \cdot m^5) \cdot (8m^2n^4)} $$

Step 2: Apply the product property of radicals

Split the radical into two parts:

$$ \sqrt[5]{-1 \cdot m^5} \cdot \sqrt[5]{8m^2n^4} $$

Step 3: Simplify the perfect fifth powers

Evaluate the first radical:

$$ \sqrt[5]{-1} = -1 $$

$$ \sqrt[5]{m^5} = m $$

$$ \sqrt[5]{-1 \cdot m^5} = -m $$

Combine the simplified terms:

$$ -m\sqrt[5]{8m^2n^4} $$

Answer:

$$ -m\sqrt[5]{8m^2n^4} $$