Question

question
what is the product of \\(\sqrt{12}\\) and \\(7\sqrt{10}\\) in simplest radical form?

Explanation:

Step1: Recall the product rule of radicals

The product rule of radicals states that \(\sqrt{a}\times\sqrt{b}=\sqrt{ab}\) (for \(a\geq0,b\geq0\)) and also, when multiplying a coefficient with a radical, we multiply the coefficients and the radicals separately. So, for \(a\sqrt{b}\times c\sqrt{d}\), it is \((a\times c)\times\sqrt{b\times d}\).

First, let's write the given expression: \(\sqrt{12}\times7\sqrt{10}\). We can rewrite this as \(7\times\sqrt{12}\times\sqrt{10}\).

Step2: Multiply the radicals using the product rule

Using the product rule \(\sqrt{m}\times\sqrt{n}=\sqrt{m\times n}\), we have \(\sqrt{12}\times\sqrt{10}=\sqrt{12\times10}\). Calculate \(12\times10 = 120\), so now we have \(7\times\sqrt{120}\).

Step3: Simplify \(\sqrt{120}\)

We need to factor 120 into a perfect square and another number. The prime factorization of 120 is \(2\times2\times2\times3\times5\). We can group the perfect square factors: \(120 = 4\times30\), where 4 is a perfect square (\(2^2\)). So, \(\sqrt{120}=\sqrt{4\times30}\). Using the product rule of radicals again, \(\sqrt{4\times30}=\sqrt{4}\times\sqrt{30}\). Since \(\sqrt{4} = 2\), this simplifies to \(2\sqrt{30}\).

Step4: Multiply the coefficient with the simplified radical

Now we have \(7\times2\sqrt{30}\). Multiply the coefficients 7 and 2: \(7\times2 = 14\). So the expression becomes \(14\sqrt{30}\).

Answer:

\(14\sqrt{30}\)