Question

How do you multiply radical expressions with different indices?

Answer 1

Hint: We will solve this problem with the help of an example. If the radicals are with different indices we take the LCM of the power and then proceed to find the answer. When the powers are the same we can multiply the number inside the radical. And if we know the root of the number inside the radical we can write the simplified answer. We also need to use the rules of indices here if required.

Complete step-by-step answer:
Let’s start the solution.
Let the two numbers with different radicals be \[\sqrt 2 \& \sqrt[3] {5}\]
Here the numbers are one having square root and other is having cube root.
Now we can write these roots as \[{5^{\dfrac{1}{3}}}\] and \[{2^{\dfrac{1}{2}}}\] .
Now the LCM of 2 and 3 is 6. So we need to make the powers of root equal to 6.
 \[{5^{\dfrac{1}{2} \times \dfrac{3}{3}}}\] and \[{2^{\dfrac{1}{3} \times \dfrac{2}{2}}}\]
 \[ = {5^{\dfrac{1}{6} \times 3}}\]and \[ {2^{\dfrac{1}{6} \times 2}}\]
Now we can write the powers as,
 \[ = {\left( {{5^3}} \right)^{\dfrac{1}{6}}}\] and \[{\left( {{2^2}} \right)^{\dfrac{1}{6}}}\]
Now writing this in radical form we get,
 \[ = \sqrt[6] {{{5^3}}}\] and \[\sqrt[6] {{{2^2}}}\]
Now the numbers are with the same radicals. So we can multiply the numbers in the root.
 \[ = \sqrt[6] {{{5^3} \times {2^2}}}\]
Taking the respective powers,
 \[ = \sqrt[6] {{125 \times 4}}\]
Taking the product,
 \[ = \sqrt[6] {{500}}\]
This is the answer.

Note: Here note that until the power of radical is same we cannot multiply the numbers under the radical. Also note that if after multiplication we know the number inside the root is a power of a number then we can simplify it. Radical is nothing but the root.
When we found the LCM we multiplied and divided the power with the number that completes the LCM. But note that the number remaining is included in the power of the number inside the root.