Question

cd: multiplying radical expressions
possible points: 6
match
√45 × √32
√x × √8x⁴
√10x (3 + √4x³)
12√10; 2x²√2x; 3√10x + 2x²√10; 7√10; 4x²√x

Explanation:

Step1: Simplify \(\boldsymbol{\sqrt{45} \times \sqrt{32}}\)

First, factor the radicands: \(\sqrt{45}=\sqrt{9\times5}=3\sqrt{5}\), \(\sqrt{32}=\sqrt{16\times2}=4\sqrt{2}\). Then multiply: \(3\sqrt{5} \times 4\sqrt{2}=12\sqrt{10}\).

Step2: Simplify \(\boldsymbol{\sqrt{x} \times \sqrt{8x^4}}\)

Combine the radicands: \(\sqrt{x\times8x^4}=\sqrt{8x^5}\). Factor out perfect squares: \(\sqrt{4x^4\times2x}=2x^2\sqrt{2x}\).

Step3: Simplify \(\boldsymbol{\sqrt{10x}(3 + \sqrt{4x^3})}\)

Distribute \(\sqrt{10x}\): \(3\sqrt{10x}+\sqrt{10x}\times\sqrt{4x^3}\). Simplify the second term: \(\sqrt{40x^4}=2x^2\sqrt{10}\). So the result is \(3\sqrt{10x}+2x^2\sqrt{10}\).

Answer:

\(\sqrt{45} \times \sqrt{32}\) matches \(12\sqrt{10}\)
\(\sqrt{x} \times \sqrt{8x^4}\) matches \(2x^2\sqrt{2x}\)
\(\sqrt{10x}(3 + \sqrt{4x^3})\) matches \(3\sqrt{10x}+2x^2\sqrt{10}\)