Question

Evaluate the value of the cube root of 343.

Answer 1

Hint- Here, we will proceed by representing the number whose cube root is required (i.e., number 343) as the product of the prime factors and then applying the formula ${\left( {{a^b}} \right)^c} = {a^{bc}}$ in order to obtain the required value.

Complete step-by-step answer:
We have to find the value of the cube root of the number 343.
Let us suppose the value of the cube root of the number 343 be x
i.e., \[x = \sqrt[3]{{343}} = {\left( {343} \right)^{\dfrac{1}{3}}}\]
Now, we will represent the number whose cube root is required i.e., 343 as the product of its prime factors.
The number 343 can be written as 343 = 7$ \times $7$ \times $7 = ${7^3}$
i.e., the cube of number 7 will obtain the number 343.
\[ \Rightarrow x = {\left( {{7^3}} \right)^{\dfrac{1}{3}}}\]
Using the formula ${\left( {{a^b}} \right)^c} = {a^{bc}}$ in the above equation, we get
\[ \Rightarrow x = {\left( {{7^3}} \right)^{\dfrac{1}{3}}} = {\left( 7 \right)^{3 \times \dfrac{1}{3}}} = 7\]
Therefore, \[x = \sqrt[3]{{343}} = {\left( {343} \right)^{\dfrac{1}{3}}} = 7\]
Hence, the cube root of the number 343 (i.e., \[\sqrt[3]{{343}}\]) is equal to 7.

Note- In this particular problem, there are three equal prime factors of the number 343 which comes out to be 7. A prime factor is simply a prime number by which the number whose prime factors are identified is divisible. The square root of any number a is given by $\sqrt a = {\left( a \right)^{\dfrac{1}{2}}}$ whereas the cube root of the same number a is given by $\sqrt[3]{a} = {\left( a \right)^{\dfrac{1}{3}}}$.