Question

How do you find the cube roots of \[ - 125\]?

Answer 1

Hint: Here, we will find the factors of the given number by using the method of Prime factorization. Then by grouping the numbers in thrice, we will 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 that gives the original number.

Complete Step by Step Solution:
We are given with a number \[ - 125\].
Let \[x\] be the given number. So, we get
\[x = - 125\]
Now, we will find the cube root of the given number.
By taking the cube root on both the sides, we get
\[ \Rightarrow \sqrt[3]{x} = \sqrt[3]{{ - 125}}\]
We know that the Cube Root of a negative integer is always a negative integer. So, we get
\[ \Rightarrow \sqrt[3]{x} = - \left( {\sqrt[3]{{125}}} \right)\]……………………………………..\[\left( 1 \right)\]
Now, we will find the factors of the given number can be found using the method of Prime Factorization.
We will divide 125 by the prime number 5. So, we get
\[125 = 25 \div 5\]
Now we will divide 25 by 5.
\[25 = 5 \div 5\]
As the quotient obtained is a prime number, so we will not divide the number further.
So, we can write 125 as \[125 = 5 \times 5 \times 5\].
So writing 125 in terms of factors in equation \[\left( 1 \right)\], we get
\[ \Rightarrow \sqrt[3]{x} = - \left( {\sqrt[3]{{5 \times 5 \times 5}}} \right)\]
As we have to find cube, so we get
\[ \Rightarrow \sqrt[3]{x} = - \left( 5 \right)\]
\[ \Rightarrow \sqrt[3]{x} = - 5\]

Therefore, the cube root of a number \[ - 125\] is \[ - 5\].

Note:
We should note that the radical sign indicates to find 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. 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 \[\sqrt x \] and the cube root of a number is denoted by the radical sign with an index as 3 and is represented as \[\sqrt[3]{x}\]. The method of prime factorization is used to find the factors by using the Prime Numbers.