Question

Find the product of \[\sqrt[3]{2}\] and \[\sqrt[4]{3}\].

Answer 1

Hint: Use the exponent product rule to find the product of the 2 given expressions. Make the base of both expressions equal by taking LCM of their power.

Complete step-by-step answer:
We have been given the expression \[\sqrt[3]{2}\] and \[\sqrt[4]{3}\]. We need to find the product of these 2 expressions.
\[\sqrt[3]{2}\times \sqrt[4]{3}\]
We can also write the above expression as \[{{\left( 2 \right)}^{\dfrac{1}{3}}}.{{\left( 3 \right)}^{\dfrac{1}{4}}}\].
The exponent “product rule” can be used to solve this expression. According to exponent product rule, when multiplied by 2 powers that have the same base, you can add the exponents.
i.e. \[{{x}^{m}}.{{x}^{n}}={{x}^{m+n}}\]
But \[{{\left( 2 \right)}^{\dfrac{1}{3}}}.{{\left( 3 \right)}^{\dfrac{1}{4}}}\] doesn’t have same base, so to use exponent product rule, we need to make the base same.
So to make the base the same, take LCM of 3 and 4 is LCM of power LCM (3, 4) = 12.
Thus \[\dfrac{1}{3}=\dfrac{4}{12}\] and \[\dfrac{1}{4}=\dfrac{3}{12}\].
So we can write \[{{\left( 2 \right)}^{\dfrac{1}{3}}}\] as \[{{\left( 2 \right)}^{\dfrac{4}{12}}}\].
Similarly, you can write \[{{\left( 3 \right)}^{\dfrac{1}{4}}}\] as \[{{\left( 3 \right)}^{\dfrac{3}{12}}}\].
\[\therefore {{\left( 2 \right)}^{\dfrac{1}{3}}}.{{\left( 3 \right)}^{\dfrac{3}{4}}}={{\left( 2 \right)}^{\dfrac{4}{12}}}.{{\left( 3 \right)}^{\dfrac{3}{12}}}\]
Thus we can write, \[{{\left( {{2}^{4}} \right)}^{\dfrac{1}{12}}}.{{\left( {{3}^{3}} \right)}^{\dfrac{1}{12}}}\].
Now this is equal to, \[\sqrt[12]{{{2}^{4}}{{.3}^{3}}}={{\left( {{2}^{4}}{{.3}^{3}} \right)}^{\dfrac{1}{12}}}\].
\[\begin{align}
  & ={{\left( 16\times 27 \right)}^{\dfrac{1}{12}}} \\
 & =\sqrt[12]{432} \\
\end{align}\]
Thus we got the product of \[\sqrt[3]{2}.\sqrt[4]{3}=\sqrt[12]{432}\].

Note: In a question like this the base is not the same, you can’t use the exponent product rule directly. Thus make the power the same before solving the product.