Question

3-6.
match each expression on the left with its equivalent expression on the right.
assume that all variables represent positive values. be sure to justify how you know each pair is equivalent.
a. $sqrt{4x^{2}y^{4}}$ 1. $2x\sqrt{y}$
b. $sqrt{8x^{2}y}$ 2. $2y\sqrt{2x}$
c. $sqrt{4x^{2}y}$ 3. $2xy^{2}$
d. $sqrt{16xy^{2}}$ 4. $2x\sqrt{2y}$
e. $sqrt{8xy^{2}}$ 5. $4y\sqrt{x}$

Explanation:

Step1: Simplify $\sqrt{4x^2y^4}$

Break into perfect squares:
$\sqrt{4x^2y^4} = \sqrt{4} \cdot \sqrt{x^2} \cdot \sqrt{y^4} = 2 \cdot x \cdot y^2 = 2xy^2$

Step2: Simplify $\sqrt{8x^2y}$

Break into perfect squares:
$\sqrt{8x^2y} = \sqrt{4 \cdot 2} \cdot \sqrt{x^2} \cdot \sqrt{y} = \sqrt{4} \cdot \sqrt{2} \cdot x \cdot \sqrt{y} = 2x\sqrt{2y}$

Step3: Simplify $\sqrt{4x^2y}$

Break into perfect squares:
$\sqrt{4x^2y} = \sqrt{4} \cdot \sqrt{x^2} \cdot \sqrt{y} = 2 \cdot x \cdot \sqrt{y} = 2x\sqrt{y}$

Step4: Simplify $\sqrt{16xy^2}$

Break into perfect squares:
$\sqrt{16xy^2} = \sqrt{16} \cdot \sqrt{x} \cdot \sqrt{y^2} = 4 \cdot \sqrt{x} \cdot y = 4y\sqrt{x}$

Step5: Simplify $\sqrt{8xy^2}$

Break into perfect squares:
$\sqrt{8xy^2} = \sqrt{4 \cdot 2} \cdot \sqrt{x} \cdot \sqrt{y^2} = \sqrt{4} \cdot \sqrt{2} \cdot \sqrt{x} \cdot y = 2y\sqrt{2x}$

Answer:

a. $\sqrt{4x^2y^4}$ matches 3. $2xy^2$
b. $\sqrt{8x^2y}$ matches 4. $2x\sqrt{2y}$
c. $\sqrt{4x^2y}$ matches 1. $2x\sqrt{y}$
d. $\sqrt{16xy^2}$ matches 5. $4y\sqrt{x}$
e. $\sqrt{8xy^2}$ matches 2. $2y\sqrt{2x}$