Question

Evaluate the following \[\sqrt[3]{{ - 2197}}\].

Answer 1

Hint: First we will use the prime factorization method to find the prime factors of the number given in the cube root term.
Using those factors as we will simply the factors in the form of a cube of the factors of the given number so that we can apply the formula for \[\sqrt[3]{N} = \sqrt[3]{{{a^3}{b^3}{c^3}}}\] which results in the answer as \[\sqrt[3]{N} = abc\], therefore using this on the given expression we will get the required answer.

Complete step by step solution: Given data: \[\sqrt[3]{{ - 2197}}\]

Using the prime factorization method we can say that

\[2197 = 13 \times 13 \times 13\]

Therefore we can write \[\sqrt[3]{{ - 2197}}\] as

\[ \Rightarrow \sqrt[3]{{ - 2197}} = \sqrt[3]{{\left( { - 13} \right)\left( { - 13} \right)\left( { - 13} \right)}}\]

Using \[a \times a \times a \times ......(n\,\,times) = {a^n}\]

\[ \Rightarrow \sqrt[3]{{ - 2197}} = \sqrt[3]{{{{\left( { - 13} \right)}^3}}}\]

Now we know that if \[\sqrt[3]{N} = \sqrt[3]{{{{\left( n \right)}^3}}}\]

Then \[\sqrt[3]{N} = n\]

Therefore, \[\sqrt[3]{{ - 2197}} = - 13\]

Therefore the required value is -13.

Note: An alternative method for the above solution can be

Using the prime factorization method we can say that

\[2197 = 13 \times 13 \times 13\]

Therefore we can write \[\sqrt[3]{{ - 2197}}\] as

\[ \Rightarrow \sqrt[3]{{ - 2197}} = \sqrt[3]{{ - 1\left( {13} \right)\left( {13} \right)\left( {13} \right)}}\]

Using \[a \times a \times a \times ......(n\,\,times) = {a^n}\]

\[ \Rightarrow \sqrt[3]{{ - 2197}} = \sqrt[3]{{ - 1{{\left( {13} \right)}^3}}}\]

Now we know that (-1) to the odd integer exponent also results in the same value that is (-1)

Mathematically, ${\left( { - 1} \right)^{2n + 1}} = - 1$ , where n is a natural number

\[ \Rightarrow \sqrt[3]{{ - 2197}} = \sqrt[3]{{{{\left( { - 1} \right)}^3}{{\left( {13} \right)}^3}}}\]

Now we know that if \[\sqrt[3]{N} = \sqrt[3]{{{{\left( a \right)}^3}{{\left( b \right)}^3}}}\]

Then \[\sqrt[3]{N} = ab\]

\[ \Rightarrow \sqrt[3]{{ - 2197}} = ( - 1)(13)\]

Therefore, \[\sqrt[3]{{ - 2197}} = - 13\]

Therefore the required value is -13, which a similar result as of the above solution result.