Question

Find the cube root of the following
\[-729\times 15625\]

Answer 1

Hint: We are asked to find the cube root of the product \[-729\times 15625.\] To find the cube root, we will factor each of the two terms i.e. 729 and 15625. Once we have the prime factor we will triplet the same term and then the term will be taken out of the cube root. \[\sqrt[3]{{}}\] is the sign to show cube root. Once we take the triplet out we product them and calculate the cube root.

Complete step by step answer:
We are given two terms one is – 729 and the other is 15625. We are asked to find the cube root of the product of these two numbers. We know to find the cube root of any number, we will first have to write the number into a prime factor and then we will make a group of 3 same terms and that term will come out of the radical. For example, \[\sqrt[3]{8}\] here \[\sqrt[3]{{}}\] denotes the cube root. We will first find the prime factorization of 8. We know that,
\[8=2\times 2\times 2\]
So, we can write,
\[\sqrt[3]{8}=\sqrt[3]{2\times 2\times 2}\]
Now we make the triplet (pair of 3)
So, \[\sqrt[3]{2\times 2\times 2}\]
So 2 occurs in the triplet. So, we get,
\[\sqrt[3]{8}=\sqrt[3]{2\times 2\times 2}\]
\[\Rightarrow \sqrt[3]{8}=2\]
We will follow these steps to find the cube root of \[-729\times 15625.\]
So we have to find \[\sqrt[3]{-729\times 15625}.\] We will factor each term one by one, we get,
\[729=3\times 3\times 3\times 3\times 3\times 3\]
\[15625=5\times 5\times 5\times 5\times 5\times 5\]
Now, using these, we get,
\[\sqrt[3]{-729\times 15625}=\sqrt[3]{-3\times 3\times 3\times 3\times 3\times 3\times 5\times 5\times 5\times 5\times 5\times 5}\]
In the cube root, we can take the negative sign out. So, we get,
\[\Rightarrow \sqrt[3]{-729\times 15625}=-\sqrt[3]{3\times 3\times 3\times 3\times 3\times 3\times 5\times 5\times 5\times 5\times 5\times 5}\]
Now, we can see,
\[\Rightarrow \sqrt[3]{-729\times 15625}=-\sqrt[3]{\underline{3\times 3\times 3}\times \underline{3\times 3\times 3}\times \underline{5\times 5\times 5}\times \underline{5\times 5\times 5}}\]
3 and 5 are making 2 pairs of triplets. So, we get,
\[\Rightarrow \sqrt[3]{-729\times 15625}=-3\times 3\times 5\times 5\]
Now, simplifying, we get,
\[\Rightarrow \sqrt[3]{-729\times 15625}=-9\times 25\]
So, finally, we get,
\[\Rightarrow \sqrt[3]{-729\times 15625}=-225\]
This means the cube root of \[-729\times 15625\] is – 225.

Note:
 Always remember that as the product of three negative is negative, i.e. \[\left( -1 \right)\times \left( -1 \right)\times \left( -1 \right)=-1\] so if one term of the cube root has a negative sign so that the negative can be taken out, the cube can be negative but the square can never be negative. Also, remember that in the cube root, the pair of 3 will come out as a single term. In the square root, a pair of 2 will come out as one.