Question

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what is the product of 9√5 and 5√10 in simplest radical form?

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√

Explanation:

Step1: Multiply the coefficients and the radicals separately

To find the product of \(9\sqrt{5}\) and \(5\sqrt{10}\), we use the property of radicals \((a\sqrt{b})(c\sqrt{d}) = ac\sqrt{bd}\). So first, multiply the coefficients \(9\) and \(5\), and then multiply the radicals \(\sqrt{5}\) and \(\sqrt{10}\).

The product of the coefficients: \(9\times5 = 45\)

The product of the radicals: \(\sqrt{5}\times\sqrt{10}=\sqrt{5\times10}=\sqrt{50}\)

So now we have \(45\sqrt{50}\)

Step2: Simplify the radical \(\sqrt{50}\)

We can factor \(50\) into \(25\times2\), and since \(\sqrt{ab}=\sqrt{a}\times\sqrt{b}\) (for \(a\geq0,b\geq0\)), we have \(\sqrt{50}=\sqrt{25\times2}=\sqrt{25}\times\sqrt{2}=5\sqrt{2}\)

Step3: Multiply the coefficient with the simplified radical

Now substitute \(\sqrt{50}=5\sqrt{2}\) back into \(45\sqrt{50}\), we get \(45\times5\sqrt{2}\)

Calculate \(45\times5 = 225\), so the expression becomes \(225\sqrt{2}\)

Answer:

\(225\sqrt{2}\)