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Evaluate: $\sqrt[3]{{700 \times 2 \times 49 \times 5}}$.

Last updated date: 06th Sep 2024
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Hint: Here, we will find the cube root of the given number. We will find the factors in the cube root by the method of Prime factorization and by grouping the numbers in thrice. We will multiply the numbers which are in thrice to find the cube root of a given number. The cube root of any number is defined as a number which is multiplied by three times a number gives the original number.

Complete Step by Step Solution:
We are given an expression $\sqrt[3]{{700 \times 2 \times 49 \times 5}}$.
Now, we will find the factors of the numbers in the given expression.
Now, we will find the factors of $700$ by the method of prime factorization.
We can see 700 is an even number, so we will divide it by the least prime number 2. Therefore, we get
$700 \div 2 = 350$
We can see 350 is an even number, so we will divide it by 2. Therefore, we get
$350 \div 2 = 175$
Now dividing 175 by the next least prime number 5, we get
$175 \div 5 = 35$
Again dividing 35 by 5, we get
$35 \div 5 = 7$
Now as we have obtained our quotient as prime number, we will not factorize it further.
So, we can write 700 as $700 = 2 \times 2 \times 5 \times 5 \times 7$.
Now, we will find the factors of $49$ by the method of prime factorization.
We can see 49 is an odd number, so we will divide it by 7. Therefore, we get
$49 \div 7 = 7$
Now as we have obtained our quotient as prime number, we will not factorize it further.
So, we can write 49 as $49 = 7 \times 7$.
Now, by using the factors we will find the cube root of a number.
Rewriting the terms in the form of factors, we get
$\sqrt[3]{{700 \times 2 \times 49 \times 5}} = \sqrt[3]{{2 \times 2 \times 5 \times 5 \times 7 \times 2 \times 7 \times 7 \times 5}}$
Now, we will group the terms in three, so we get
$\Rightarrow \sqrt[3]{{700 \times 2 \times 49 \times 5}} = 2 \times 7 \times 5$
By multiplying the terms, we get
$\Rightarrow \sqrt[3]{{700 \times 2 \times 49 \times 5}} = 70$

Thus, the cube root of $\sqrt[3]{{700 \times 2 \times 49 \times 5}}$ is $70$.

Note:
We will find the cube root of a number by finding the number which is multiplied by itself thrice to give the original number. We should note that the radical sign indicates the root of a number. We know that the radical sign with a small number $n$ is known as ${n^{th}}$ root of a number and the smaller number is called the index. We should also remember that usually the square root of a number is not written with any index and is denoted as $\sqrt x$. However, the cube root of a number is denoted by the radical sign with an index as 3 and is represented as $\sqrt[3]{x}$.