Question

what is the following product?
$sqrt3{5} cdot sqrt{2}$
options:
$sqrt6{1000000}$
$sqrt6{10}$
$sqrt6{500}$
$sqrt6{200}$

Explanation:

Step1: Rewrite radicals with exponents

Recall that \(\sqrt[n]{a}=a^{\frac{1}{n}}\). So, \(\sqrt[3]{5}=5^{\frac{1}{3}}\) and \(\sqrt{2}=2^{\frac{1}{2}}\).

Step2: Find a common denominator for exponents

To multiply these, we need a common denominator for the exponents. The least common multiple of 3 and 2 is 6. Rewrite the exponents: \(5^{\frac{1}{3}} = 5^{\frac{2}{6}}\) and \(2^{\frac{1}{2}}=2^{\frac{3}{6}}\).

Step3: Use exponent rule \(a^m\cdot b^m=(ab)^m\)

Now, \(5^{\frac{2}{6}}\cdot2^{\frac{3}{6}}=(5^2)^{\frac{1}{6}}\cdot(2^3)^{\frac{1}{6}}\) (since \(a^{\frac{m}{n}}=(a^m)^{\frac{1}{n}}\)). Calculate \(5^2 = 25\) and \(2^3=8\). Then, \((25)^{\frac{1}{6}}\cdot(8)^{\frac{1}{6}}=(25\times8)^{\frac{1}{6}}\) (by \(a^m\cdot b^m=(ab)^m\)).

Step4: Simplify the product inside the radical

\(25\times8 = 200\), so \((200)^{\frac{1}{6}}=\sqrt[6]{200}\). Wait, no, wait: Wait, \(5^{\frac{2}{6}}\cdot2^{\frac{3}{6}}=(5^2\times2^3)^{\frac{1}{6}}=(25\times8)^{\frac{1}{6}}=(200)^{\frac{1}{6}}\)? Wait, no, wait, maybe I made a mistake. Wait, let's check again. Wait, \(\sqrt[3]{5}\cdot\sqrt{2}=5^{\frac{1}{3}}\cdot2^{\frac{1}{2}}\). Let's convert to sixth roots: \(\sqrt[6]{5^2}\cdot\sqrt[6]{2^3}=\sqrt[6]{25}\cdot\sqrt[6]{8}=\sqrt[6]{25\times8}=\sqrt[6]{200}\)? Wait, no, wait, 25*8 is 200? Wait, 25*8=200? Yes. Wait, but let's check the options. Wait, maybe I messed up. Wait, alternatively, let's compute \(\sqrt[3]{5}\cdot\sqrt{2}\) and then raise to the 6th power to see which option matches. Let's compute \((\sqrt[3]{5}\cdot\sqrt{2})^6\). Using \((ab)^n=a^n b^n\), we get \((\sqrt[3]{5})^6\cdot(\sqrt{2})^6\). \((\sqrt[3]{5})^6 = 5^2 = 25\), \((\sqrt{2})^6=2^3 = 8\), so \(25\times8 = 200\). So \((\sqrt[3]{5}\cdot\sqrt{2})^6 = 200\), which means \(\sqrt[3]{5}\cdot\sqrt{2}=\sqrt[6]{200}\)? Wait, no, \((\sqrt[6]{200})^6 = 200\), so that's correct. Wait, but let's check the options. Wait, the options are \(\sqrt[6]{1000000}\), \(\sqrt[6]{500}\), \(\sqrt[6]{10}\), \(\sqrt[6]{200}\)? Wait, no, the options in the image: Wait, the user's image shows options: \(\sqrt[6]{1000000}\), \(\sqrt[6]{500}\), \(\sqrt[6]{10}\), \(\sqrt[6]{200}\)? Wait, no, let's re - evaluate. Wait, maybe I made a mistake in the exponent conversion. Wait, \(\sqrt[3]{5}=5^{\frac{1}{3}}\), \(\sqrt{2}=2^{\frac{1}{2}}\). The product is \(5^{\frac{1}{3}}\times2^{\frac{1}{2}}\). Let's find a common index. The least common multiple of 3 and 2 is 6. So, \(5^{\frac{1}{3}}=5^{\frac{2}{6}}=\sqrt[6]{5^2}=\sqrt[6]{25}\), \(2^{\frac{1}{2}}=2^{\frac{3}{6}}=\sqrt[6]{2^3}=\sqrt[6]{8}\). Then, multiplying these two sixth roots: \(\sqrt[6]{25}\times\sqrt[6]{8}=\sqrt[6]{25\times8}=\sqrt[6]{200}\). Wait, but let's check \(\sqrt[6]{1000000}\): \(\sqrt[6]{1000000}=\sqrt[6]{10^6}=10\). \(\sqrt[6]{500}\) is different. \(\sqrt[6]{10}\) is different. \(\sqrt[6]{200}\) matches our calculation. Wait, but let's verify with another approach. Let's compute \(\sqrt[3]{5}\approx1.71\), \(\sqrt{2}\approx1.414\), their product is approximately \(1.71\times1.414\approx2.419\). Now, compute \(\sqrt[6]{200}\): \(200^{\frac{1}{6}}\). \(2^6 = 64\), \(3^6=729\), so \(200^{\frac{1}{6}}\) is between 2 and 3. \(2.419^6\approx(2.419)^2 = 5.85\), \((5.85)^3\approx5.85\times5.85 = 34.22\), \(34.22\times5.85\approx200\). Yes, that works. So the correct option is \(\sqrt[6]{200}\). Wait, but let's check the options again. Wait, the options in the image: the first option is \(\sqrt[6]{1000000}\), second \(\sqrt[6]{500}\), third \(\sqrt[6]{10}\), fourth \(\sqrt[6]{200}\)? Wait, no, the user's image: let's see, the options are:

1. \(\sqrt[6]{1000000}\)

2. \(\sqrt[6]{500}\)

3. \(\sqrt[6]{10}\)

4. \(\sqrt[6]{200}\)

Wait, our calculation shows that \(\sqrt[3]{5}\cdot\sqrt{2}=\sqrt[6]{200}\), so the correct option is the fourth one, \(\sqrt[6]{200}\).

Answer:

\(\sqrt[6]{200}\) (the last option among the given choices)