Question

simplify: 2(6) ^ (1/2) * (12) ^ (1/2)

Explanation:

Step1: Use the property of exponents \(a^{m} \cdot b^{m}=(ab)^{m}\)

We have \(2\times6^{\frac{1}{2}}\times12^{\frac{1}{2}} = 2\times(6\times12)^{\frac{1}{2}}\)

Step2: Calculate the product inside the exponent

First, calculate \(6\times12 = 72\), so the expression becomes \(2\times72^{\frac{1}{2}}\)

Step3: Simplify the square root

We know that \(72^{\frac{1}{2}}=\sqrt{72}=\sqrt{36\times2} = 6\sqrt{2}\)

Step4: Multiply with the coefficient

Now, \(2\times6\sqrt{2}=12\sqrt{2}\)? Wait, no, wait. Wait, actually, let's re - check. Wait, \(72^{\frac{1}{2}}=\sqrt{72}=\sqrt{36\times2}=6\sqrt{2}\), then \(2\times6\sqrt{2} = 12\sqrt{2}\)? Wait, no, that's wrong. Wait, no, wait, \(6\times12 = 72\), \(72^{\frac{1}{2}}=\sqrt{72}=\sqrt{36\times2}=6\sqrt{2}\), then \(2\times6\sqrt{2}=12\sqrt{2}\)? Wait, no, wait, maybe I made a mistake. Wait, no, let's do it again.

Wait, the property \(a^{m}\times b^{m}=(ab)^{m}\), so \(6^{\frac{1}{2}}\times12^{\frac{1}{2}}=(6\times12)^{\frac{1}{2}} = 72^{\frac{1}{2}}=\sqrt{72}\)

\(\sqrt{72}=\sqrt{36\times2}=6\sqrt{2}\)

Then \(2\times\sqrt{72}=2\times6\sqrt{2}=12\sqrt{2}\)? Wait, no, wait, no, that's incorrect. Wait, wait, \(6\times12 = 72\), \(\sqrt{72}=\sqrt{36\times2}=6\sqrt{2}\), then \(2\times6\sqrt{2}=12\sqrt{2}\)? But wait, let's check with another approach.

Alternative approach:

\(6^{\frac{1}{2}}=\sqrt{6}\), \(12^{\frac{1}{2}}=\sqrt{12}=\sqrt{4\times3}=2\sqrt{3}\)

Then \(2\times\sqrt{6}\times2\sqrt{3}=2\times2\times\sqrt{6}\times\sqrt{3}=4\times\sqrt{18}=4\times\sqrt{9\times2}=4\times3\sqrt{2}=12\sqrt{2}\)

Wait, but let's check the value. \(\sqrt{6}\approx2.45\), \(\sqrt{12}\approx3.46\), \(2\times2.45\times3.46\approx2\times8.477 = 16.954\)

\(12\sqrt{2}\approx12\times1.414 = 16.968\), which is close (due to rounding). But wait, maybe there is a simpler way. Wait, \(6\times12 = 72\), \(2\times\sqrt{72}=2\times\sqrt{36\times2}=2\times6\sqrt{2}=12\sqrt{2}\). But wait, is there a mistake? Wait, no, let's see:

Wait, \(6^{\frac{1}{2}}\times12^{\frac{1}{2}}=(6\times12)^{\frac{1}{2}}=72^{\frac{1}{2}}=\sqrt{72}\)

\(\sqrt{72}=\sqrt{36\times2}=6\sqrt{2}\)

Then \(2\times6\sqrt{2}=12\sqrt{2}\approx16.97\)

But let's check with actual multiplication: \(\sqrt{6}\approx2.449\), \(\sqrt{12}\approx3.464\)

\(2\times2.449\times3.464\approx2\times8.485 = 16.97\), which matches \(12\sqrt{2}\approx12\times1.4142 = 16.9704\)

Wait, but maybe we can simplify it more? Wait, no, \(12\sqrt{2}\) is the simplified radical form. But wait, wait, maybe I made a mistake in the property. Wait, the original expression is \(2\times6^{\frac{1}{2}}\times12^{\frac{1}{2}}\)

Using \(a^{m}\times b^{m}=(ab)^{m}\), so \(6^{\frac{1}{2}}\times12^{\frac{1}{2}}=(6\times12)^{\frac{1}{2}} = 72^{\frac{1}{2}}\)

\(72^{\frac{1}{2}}=\sqrt{72}=\sqrt{36\times2}=6\sqrt{2}\)

Then \(2\times6\sqrt{2}=12\sqrt{2}\)

Wait, but let's check if \(12\sqrt{2}\) is correct. Alternatively, \(6\times12 = 72\), \(2\times\sqrt{72}=2\times\sqrt{36\times2}=2\times6\sqrt{2}=12\sqrt{2}\)

Yes, that's correct.

Wait, but another way: \(6 = 2\times3\), \(12=2^{2}\times3\)

So \(6^{\frac{1}{2}}=(2\times3)^{\frac{1}{2}}=2^{\frac{1}{2}}\times3^{\frac{1}{2}}\)

\(12^{\frac{1}{2}}=(2^{2}\times3)^{\frac{1}{2}}=2^{1}\times3^{\frac{1}{2}}\)

Then \(2\times6^{\frac{1}{2}}\times12^{\frac{1}{2}}=2\times(2^{\frac{1}{2}}\times3^{\frac{1}{2}})\times(2\times3^{\frac{1}{2}})\)

\(=2\times2^{\frac{1}{2}}\times3^{\frac{1}{2}}\times2\times3^{\frac{1}{2}}\)

\(=2\times2\times2^{\frac{1}{2}}\times3^{\frac{1}{2}}\times3^{\frac{1}{2}}\)

\(=4\times2^{\frac{1}{2}}\times3^{1}\)

\(=12\…

Answer:

\(12\sqrt{2}\)