Question

What will be the cube root by the prime factorisation of the number 46656 ?

Answer 1

Hint: Let us find the prime factors by the given number 46656 by dividing the number by the smallest prime factor till we get a number which is not divisible. As we know to find the cube root of the number 46656. First, we had to split this number into its different prime factors.

Complete step-by-step answer:
And after that we had to make pairs of three similar factors and multiply them to get the required cube root of the number.
Now, for finding prime factors we had to divide the number by the smallest prime number by which it is divisible.
So, for finding prime factors.
Now we have to divide the number by 2 till we get a number which is not divisible by 2.
Dividing the number by 2. We get \[\dfrac{{46656}}{2} = 23328\]
Now again dividing the above result by 2. We get \[\dfrac{{23328}}{2} = 11664\]
Now again dividing the above result by 2. We get \[\dfrac{{11664}}{2} = 5832\]
Now again dividing the above result by 2. We get \[\dfrac{{5832}}{2} = 2916\]
Now again dividing the above result by 2. We get \[\dfrac{{2916}}{2} = 1458\]
Now again dividing the above result by 2. We get \[\dfrac{{1458}}{2} = 729\]
Now 729 is not divisible by 2. So, we had to divide it by the next prime number that is 3
Dividing the number by 3. We get \[\dfrac{{729}}{3} = 243\]
Now again dividing the above result by 3. We get \[\dfrac{{243}}{3} = 81\]
Now again dividing the above result by 3. We get \[\dfrac{{81}}{3} = 27\]
Now again dividing the above result by 3. We get \[\dfrac{{27}}{3} = 9\]
Now again dividing the above result by 3. We get \[\dfrac{9}{3} = 3\]
Now again dividing the above result by 3. We get \[\dfrac{3}{3} = 1\]
So, the prime factors of 46656 are \[2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 3 \times 3 \times 3 \times 3 \times 3 \times 3\]
Now we had to find the cube root of the given number.
So, \[\sqrt[3]{{46656}} = \sqrt[3]{{2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 3 \times 3 \times 3 \times 3 \times 3 \times 3}} = \sqrt[3]{{{{\left( 2 \right)}^6}{{\left( 3 \right)}^6}}} = {\left( 2 \right)^2}{\left( 3 \right)^2} = 4 \times 9 = 36\]
Hence, the cube root of 46656 will be 36.

Note: Whenever we come up with this type of problem then first, we have to find all the prime factors of the given number by dividing the number by the smallest prime number by which it is divisible. And then we can also directly find the cube root of the number by directly replacing three pairs of same factors with one factor and then multiplying all. Like if factors of any number is \[{\text{a}} \times {\text{a}} \times {\text{a}} \times {\text{b}} \times {\text{b}} \times {\text{b}}\] then its cube root will be \[{\text{a}} \times {\text{b}}\] because a and b both occurs three times in the factors.