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Multiply and write the answer in simplest form \[3\sqrt {12} \times 7 \sqrt {15} \].

Last updated date: 21st Jul 2024
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Hint: We are given with two terms in product. First we will multiply the numbers outside the roots and then the numbers in the root. If the numbers in the root are in complete square form then we will take the root and if they are not then we will take only the perfect square number outside and we will keep the remaining number inside the root only. Then multiply the numbers outside the roots.

Step by step solution:
Given that,
Now multiply the numbers outside the root first,
\[ \Rightarrow 3 \times 7\sqrt {12} \sqrt {15} \]
\[ \Rightarrow 21\sqrt {12} \sqrt {15} \]
Now numbers inside the roots,
\[ \Rightarrow 21\sqrt {12 \times 15} \]
Now we will write the numbers inside the roots in factored form
\[ \Rightarrow 21\sqrt {2 \times 2 \times 3 \times 5 \times 3} \]
Here we will take the numbers together whose product will give us a perfect square. Like product of 2 will give 4 as product and that of 3 will give 9.
\[ \Rightarrow 21\sqrt {4 \times 9 \times 5} \]
Now the root of 4 is 2 and the root of 9 is 3.
\[ \Rightarrow 21 \times 2 \times 3\sqrt 5 \]
Now multiply the numbers outside the root now.
\[ \Rightarrow 126\sqrt 5 \]

This is our answer. \[ \Rightarrow 126\sqrt 5 \]

Simplifying the given question is nothing but simplifying the terms only. Moreover if we are given with the value of root of 5 we will multiply it with 126 and that will be a more simplified answer. But note that the numbers which we are multiplying should be under the same root. Like numbers in square root are only multiplied with the numbers in square root only and not with those in cube root.