Question

select the correct answer.

which calculation correctly writes \\(\sqrt{80x^3y^2}\\) in simplest form using prime factorization?

• \\(\sqrt{80x^3y^2} = \sqrt{2 \cdot 2 \cdot 2 \cdot 5 \cdot x \cdot x \cdot x \cdot y \cdot y} = 2xy\sqrt{10x}\\)
• \\(\sqrt{80x^3y^2} = \sqrt{2 \cdot 2 \cdot 2 \cdot 2 \cdot 5 \cdot x \cdot x \cdot x \cdot y \cdot y} = 4xy\sqrt{5x}\\)
• \\(\sqrt{80x^3y^2} = \sqrt{2 \cdot 2 \cdot 2 \cdot 2 \cdot 5 \cdot x \cdot x \cdot x \cdot y \cdot y} = 16x^2y^2\sqrt{5x}\\)
• \\(\sqrt{80x^3y^2} = \sqrt{16 \cdot 5 \cdot 3x \cdot 2y} = 4\sqrt{30xy}\\)

Explanation:

⚡ Using what you learned: simplifying radical expressions

Step 1: Prime factorization of the radicand

Find the prime factorization of the number \(80\) and expand the variables \(x^3\) and \(y^2\):

$$ 80 = 2 \cdot 2 \cdot 2 \cdot 2 \cdot 5 $$

$$ x^3 = x \cdot x \cdot x $$

$$ y^2 = y \cdot y $$

Combine them under the square root:

$$ \sqrt{80x^3y^2} = \sqrt{2 \cdot 2 \cdot 2 \cdot 2 \cdot 5 \cdot x \cdot x \cdot x \cdot y \cdot y} $$

Step 2: Grouping pairs

Group the prime factors and variables into pairs of identical factors to pull them out of the square root:

$$ \sqrt{(2 \cdot 2) \cdot (2 \cdot 2) \cdot 5 \cdot (x \cdot x) \cdot x \cdot (y \cdot y)} $$

Each pair simplifies to a single factor outside the radical:

$$ 2 \cdot 2 \cdot x \cdot y \sqrt{5 \cdot x} = 4xy\sqrt{5x} $$

Comparing this with the given options, the second option correctly shows this complete process.

Answer:

The second option is correct:

$$ \sqrt{80x^3y^2} = \sqrt{2 \cdot 2 \cdot 2 \cdot 2 \cdot 5 \cdot x \cdot x \cdot x \cdot y \cdot y} = 4xy\sqrt{5x} $$