Question

what is $12\sqrt{2}$ divided by $2\sqrt{6}$ in simplest form?\
\bigcirc a. $3\sqrt{6}$\
\bigcirc b. $3\sqrt{2}$\
\bigcirc c. $6\sqrt{3}$\
\bigcirc d. $2\sqrt{3}$\
reset selection

Explanation:

Step1: Separate coefficients and radicals

We can rewrite the division as the division of the coefficients and the division of the radicals. So, \(\frac{12\sqrt{2}}{2\sqrt{6}}=\frac{12}{2}\times\frac{\sqrt{2}}{\sqrt{6}}\)

Step2: Simplify the coefficient division

Simplify \(\frac{12}{2}\), which equals \(6\). So now we have \(6\times\frac{\sqrt{2}}{\sqrt{6}}\)

Step3: Rationalize the radical fraction

To simplify \(\frac{\sqrt{2}}{\sqrt{6}}\), we rationalize the denominator by multiplying the numerator and denominator by \(\sqrt{6}\): \(\frac{\sqrt{2}\times\sqrt{6}}{\sqrt{6}\times\sqrt{6}}=\frac{\sqrt{12}}{6}\)

Step4: Simplify the new radical

Simplify \(\sqrt{12}\) as \(\sqrt{4\times3}=2\sqrt{3}\). So now we have \(\frac{2\sqrt{3}}{6}\)

Step5: Multiply with the coefficient

Now multiply this by the coefficient \(6\) from Step 2: \(6\times\frac{2\sqrt{3}}{6}\). The \(6\) in the numerator and denominator cancels out, leaving \(2\sqrt{3}\)? Wait, no, wait. Wait, let's go back. Wait, in Step 2, we had \(6\times\frac{\sqrt{2}}{\sqrt{6}}\), then rationalizing: \(\frac{\sqrt{2}}{\sqrt{6}}=\frac{\sqrt{2}\times\sqrt{6}}{6}=\frac{\sqrt{12}}{6}=\frac{2\sqrt{3}}{6}=\frac{\sqrt{3}}{3}\). Then multiply by \(6\): \(6\times\frac{\sqrt{3}}{3}=2\sqrt{3}\)? Wait, no, that's not right. Wait, let's do it again.

Wait, original expression: \(\frac{12\sqrt{2}}{2\sqrt{6}}\). First, divide the coefficients: \(12\div2 = 6\). Then, divide the radicals: \(\sqrt{2}\div\sqrt{6}=\sqrt{\frac{2}{6}}=\sqrt{\frac{1}{3}}=\frac{\sqrt{3}}{3}\) (by simplifying the fraction inside the square root first: \(\frac{2}{6}=\frac{1}{3}\), then taking the square root). Then multiply the coefficient and the radical result: \(6\times\frac{\sqrt{3}}{3}\). The \(6\) and \(3\) simplify: \(6\div3 = 2\), so \(2\sqrt{3}\). Wait, but let's check the options. Option D is \(2\sqrt{3}\). Wait, but let's verify again.

Alternative approach: \(\frac{12\sqrt{2}}{2\sqrt{6}}=\frac{12}{2}\times\frac{\sqrt{2}}{\sqrt{6}} = 6\times\frac{\sqrt{2}}{\sqrt{6}}\). Now, simplify \(\frac{\sqrt{2}}{\sqrt{6}}\) by rationalizing: multiply numerator and denominator by \(\sqrt{6}\): \(\frac{\sqrt{2}\times\sqrt{6}}{\sqrt{6}\times\sqrt{6}}=\frac{\sqrt{12}}{6}=\frac{2\sqrt{3}}{6}=\frac{\sqrt{3}}{3}\). Then \(6\times\frac{\sqrt{3}}{3}=2\sqrt{3}\). So the answer should be D. \(2\sqrt{3}\)

Wait, but let's check the options. Option D is \(2\sqrt{3}\). So that's the correct answer.

Wait, but let's do it step by step again carefully:

1. Start with \(\frac{12\sqrt{2}}{2\sqrt{6}}\)
2. Divide the coefficients: \(12 \div 2 = 6\)
3. Divide the radicals: \(\sqrt{2} \div \sqrt{6} = \sqrt{\frac{2}{6}} = \sqrt{\frac{1}{3}} = \frac{\sqrt{3}}{3}\) (because \(\sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}}\) for \(a,b\geq0\), and then simplify \(\frac{1}{3}\) inside the square root)
4. Multiply the results from step 2 and step 3: \(6 \times \frac{\sqrt{3}}{3}\)
5. Simplify \(6 \div 3 = 2\), so we get \(2\sqrt{3}\)

Yes, that's correct. So the answer is D. \(2\sqrt{3}\)

Answer:

D. \(2\sqrt{3}\)