Question

Find the cube root of$\dfrac{{ - 125}}{{68921}}$
(A)$\dfrac{5}{{41}}$
(B) $-\dfrac{5}{{41}}$
(C) $\dfrac{5}{{61}}$
(D) $-\dfrac{5}{{61}}$

Answer 1

Hint: Keep the negative sign aside and factorize the numerator and the denominator to obtain
their respective cube roots.
Use the facts:
1) the cube root of a fraction is the fraction formed by the cube roots of the numerator and the
denominator.
2) the cube root of a negative number is negative.
This will give us the required answer.

Complete step by step solution:
We are given a fraction$\dfrac{{ - 125}}{{68921}}$
We need to find the cube root of this fraction.
Here, we will use a property of cube root:
If$a$and$b$are two real numbers such that$b \ne 0$, then the cube root of the
fraction$\dfrac{a}{b}$can be calculated as follows:\[\sqrt[3]{{\dfrac{a}{b}}} =
\dfrac{{\sqrt[3]{a}}}{{\sqrt[3]{b}}}\]where\[\sqrt[3]{{}}\]denotes the cube root of a number.
This means that we can find the cube root of $\dfrac{{ - 125}}{{68921}}$by computing the cube roots of
the numerator and the denominator separately.
Now, here are the steps to compute the cube root of a number.
1) Choose the number for which you wish to compute the cube root.
In our case, it will be the numerator of the given fraction first and then we will repeat the same process
for the denominator.
2) Factorize the number. That is, write the number as a product of prime factors.
3) Now, group the primes in such a way that each group contains only 3 numbers and the 3 numbers in a group are the same.
4) Consider one prime factor from each group.

5) Multiply the prime factors thus obtained in the previous step, i.e. step 4.
The product we get will be the required cube root.

So, we are going to apply steps 1 to 5 on both the numerator and the denominator.
Consider the numerator -125.
We know that if a number is negative, then its cube root will also be negative.
So, we can factorize the number ignoring its sign and then add the sign to the final answer.
We will do the same for -125.
Let’s factorize 125.
Since 125 has 5 in its unit place, we know that 5 will be one of its prime factors.
Thus, we get the following:
$125 = 5 \times 25 = 5 \times 5 \times 5$
Therefore, cube root of 125 is 5 and hence the cube root of -125 is -5.
Now, consider the denominator 68921.
Upon factorization, we get the following:
$68921 = 41 \times 1681 = 41 \times 41 \times 41$
Therefore, the cube root of 68921 is 41.
Thus, going back to the fraction, we have the cube root of$\dfrac{{ - 125}}{{68921}}$is$\dfrac{{ -
5}}{{41}}$.
Note: It is not necessary to compute the cube roots of the numerator and the denominator of a fraction
separately. However, it is advisable for numbers with a large number of prime factors. This is because
students often make mistakes while calculating the cube roots simultaneously.