Question

what is the following product?
$sqrt3{4} cdot sqrt{3}$
options:
$2\left(\sqrt6{9}\
ight)$
$sqrt6{432}$
$sqrt6{12}$
$2\left(\sqrt6{3,888}\
ight)$

Explanation:

Step1: Rewrite radicals with exponents

Rewrite \(\sqrt[3]{4}\) as \(4^{\frac{1}{3}}\) and \(\sqrt{3}\) as \(3^{\frac{1}{2}}\). So the product is \(4^{\frac{1}{3}}\cdot3^{\frac{1}{2}}\).

Step2: Express 4 as a power of 2

Since \(4 = 2^2\), substitute to get \((2^2)^{\frac{1}{3}}\cdot3^{\frac{1}{2}}=2^{\frac{2}{3}}\cdot3^{\frac{1}{2}}\).

Step3: Find a common denominator for exponents

The least common denominator of 3 and 2 is 6. Rewrite exponents: \(2^{\frac{4}{6}}\cdot3^{\frac{3}{6}}\).

Step4: Rewrite with radical notation

Using \(a^{\frac{m}{n}}=\sqrt[n]{a^m}\), we get \(\sqrt[6]{2^4}\cdot\sqrt[6]{3^3}=\sqrt[6]{16}\cdot\sqrt[6]{27}\).

Step5: Multiply the radicands

Multiply inside the sixth root: \(\sqrt[6]{16\times27}=\sqrt[6]{432}\).

Answer:

\(\sqrt[6]{432}\) (the left - most option in the top row of the given options)