Question

Find the cube root of -17576 using factorization:

Answer 1

Hint: Here we need to find the cube root of the given negative number. For that, we will first find the factors of the given number using the factorization method. Then we will find the cube root of the given negative number. The cube root of any negative number gives a negative number.

Complete step by step solution:
Here we need to find the cube root of the given number using the prime factorization method.
The given number is -17576.
So we will first find the prime factors of the given number using the prime factorization method.
We will first divide the -17576 by the least prime factor 2.
\[{\text{ - 17576}} \div 2 = - 8788\]
Again we will divide the obtained number by the least prime factor 2.
\[ \Rightarrow - 8788 \div 2 = - 4394\]
Again we will divide the obtained number by the next least prime factor 2.
\[ \Rightarrow - 4394 \div 2 = - 2197\]
Again we will divide the obtained number by the next least prime factor 13.
\[ \Rightarrow - 2197 \div 13 = - 169\]
Again we will divide the obtained number by the next least prime factor 13.
\[ \Rightarrow - 169 \div 13 = - 13\]
As we can see that the 13 is a prime number. So we can’t factorize that number further.
Therefore, the prime factors of the given number i.e. -17576 is equal to $2 \times 2 \times 2 \times 13 \times 13 \times - 13$
But we can also write the factors as $2 \times 2 \times 2 \times - 13 \times - 13 \times - 13$
Now, we will find the cube of the given number i.e. -17576
We can write it as
$ \Rightarrow \sqrt[3]{{ - 17576}} = \sqrt[3]{{2 \times 2 \times 2 \times - 13 \times - 13 \times - 13}}$
On further simplification, we get
 $ \Rightarrow \sqrt[3]{{ - 17576}} = \sqrt[3]{{{2^3} \times {{\left( { - 13} \right)}^3}}}$
Now, we will write the cube root in the power form.
$ \Rightarrow \sqrt[3]{{ - 17576}} = {\left( {{2^3} \times {{\left( { - 13} \right)}^3}} \right)^{\frac{1}{3}}}$
We know from the basics properties of the exponential function that
$ \Rightarrow \sqrt[3]{{ - 17576}} = {2^{3 \times \frac{1}{3}}} \times {\left( { - 13} \right)^{3 \times \frac{1}{3}}}$
On further simplifying the terms, we get
$ \Rightarrow \sqrt[3]{{ - 17576}} = 2 \times \left( { - 13} \right)$
On multiplying the terms, we get
$ \Rightarrow \sqrt[3]{{ - 17576}} = - 26$

Hence, the cube root of the given number i.e. -17576 is equal to -26.

Note: Here we have obtained the cube root of the given negative number. Here the cube root of any number is defined as the factor which when multiplied by it gives the original number. We have seen that when we multiply the factor -26 three times we get the original number -17576.