Question

given x>0 and y>0, match the original radical expression (on the left) with its fully simplified expression (on the right). √6 + √24 -3√2 - 3√18 + 2√3 √8x² + 2x√2 - √20x √32x³ + 2√6x - 8√2x³ + 4√24x 2x√125x²y - x²√80y + 4√20x⁴y :: 3√6 :: 2√6 :: -6√2 + 2√3 :: -12√2 + 2√3 :: 4x√2 - 2√5x :: 4x√2 - 2x√5 :: -4√2x + 10x√6x :: -4x√2x + 10√6x :: 14x²√5y :: 14√5y

Explanation:

Step1: Simplify \(\boldsymbol{\sqrt{6}+\sqrt{24}}\)

Simplify \(\sqrt{24}\): \(\sqrt{24}=\sqrt{4\times6}=2\sqrt{6}\).
Then \(\sqrt{6}+2\sqrt{6}=3\sqrt{6}\).

Step2: Simplify \(\boldsymbol{-3\sqrt{2}-3\sqrt{18}+2\sqrt{3}}\)

Simplify \(\sqrt{18}\): \(\sqrt{18}=\sqrt{9\times2}=3\sqrt{2}\).
Substitute: \(-3\sqrt{2}-3(3\sqrt{2})+2\sqrt{3}=-3\sqrt{2}-9\sqrt{2}+2\sqrt{3}=-12\sqrt{2}+2\sqrt{3}\).

Step3: Simplify \(\boldsymbol{\sqrt{8x^2}+2x\sqrt{2}-\sqrt{20x}}\)

Simplify \(\sqrt{8x^2}\) (since \(x>0\)): \(\sqrt{8x^2}=2x\sqrt{2}\).
Simplify \(\sqrt{20x}\): \(\sqrt{20x}=2\sqrt{5x}\).
Substitute: \(2x\sqrt{2}+2x\sqrt{2}-2\sqrt{5x}=4x\sqrt{2}-2\sqrt{5x}\).

Step4: Simplify \(\boldsymbol{\sqrt{32x^3}+2\sqrt{6x}-8\sqrt{2x^3}+4\sqrt{24x}}\)

Simplify \(\sqrt{32x^3}\) ( \(x>0\) ): \(\sqrt{32x^3}=4x\sqrt{2x}\).
Simplify \(\sqrt{2x^3}\) ( \(x>0\) ): \(\sqrt{2x^3}=x\sqrt{2x}\).
Simplify \(\sqrt{24x}\): \(\sqrt{24x}=2\sqrt{6x}\).
Substitute: \(4x\sqrt{2x}+2\sqrt{6x}-8(x\sqrt{2x})+4(2\sqrt{6x})\)
\(=4x\sqrt{2x}+2\sqrt{6x}-8x\sqrt{2x}+8\sqrt{6x}\)
\(=-4x\sqrt{2x}+10\sqrt{6x}\).

Step5: Simplify \(\boldsymbol{2x\sqrt{125x^2y}-x^2\sqrt{80y}+4\sqrt{20x^4y}}\)

Simplify \(\sqrt{125x^2y}\) ( \(x>0\) ): \(\sqrt{125x^2y}=5x\sqrt{5y}\).
Simplify \(\sqrt{80y}\): \(\sqrt{80y}=4\sqrt{5y}\).
Simplify \(\sqrt{20x^4y}\) ( \(x>0\) ): \(\sqrt{20x^4y}=2x^2\sqrt{5y}\).
Substitute: \(2x(5x\sqrt{5y})-x^2(4\sqrt{5y})+4(2x^2\sqrt{5y})\)
\(=10x^2\sqrt{5y}-4x^2\sqrt{5y}+8x^2\sqrt{5y}\)
\(=14x^2\sqrt{5y}\).

Answer:

• \(\sqrt{6}+\sqrt{24}\) matches \(3\sqrt{6}\)
• \(-3\sqrt{2}-3\sqrt{18}+2\sqrt{3}\) matches \(-12\sqrt{2}+2\sqrt{3}\)
• \(\sqrt{8x^2}+2x\sqrt{2}-\sqrt{20x}\) matches \(4x\sqrt{2}-2\sqrt{5x}\)
• \(\sqrt{32x^3}+2\sqrt{6x}-8\sqrt{2x^3}+4\sqrt{24x}\) matches \(-4x\sqrt{2x}+10\sqrt{6x}\)
• \(2x\sqrt{125x^2y}-x^2\sqrt{80y}+4\sqrt{20x^4y}\) matches \(14x^2\sqrt{5y}\)