To find the cube root of 121000 using the prime factorization method, follow these steps:
1. Find the prime factors of 121000:
- We start by finding the prime numbers that divide 121000.
- Firstly, divide the number 121000 by the smallest prime number, which is 2:
[tex]\[
121000 \div 2 = 60500
\][/tex]
[tex]\[
60500 \div 2 = 30250
\][/tex]
[tex]\[
30250 \div 2 = 15125 \quad (\text{15125 is not divisible by 2 anymore})
\][/tex]
- Next, try the next smallest prime number, which is 5:
[tex]\[
15125 \div 5 = 3025
\][/tex]
[tex]\[
3025 \div 5 = 605
\][/tex]
[tex]\[
605 \div 5 = 121 \quad (\text{121 is not divisible by 5 anymore})
\][/tex]
- Next, try the next smallest prime number, which is 11:
[tex]\[
121 \div 11 = 11
\][/tex]
[tex]\[
11 \div 11 = 1
\][/tex]
- Combining these factors, we can write:
[tex]\[
121000 = 2^3 \times 5^3 \times 11^2
\][/tex]
2. Divide the exponents of the prime factors by 3:
- Since we want to find the cube root, we divide each exponent in the prime factorization by 3:
[tex]\[
2^3 \to 2^{3/3} = 2^1 = 2
\][/tex]
[tex]\[
5^3 \to 5^{3/3} = 5^1 = 5
\][/tex]
[tex]\[
11^2 \to 11^{2/3} \quad (\text{which does not form a whole number})
\][/tex]
3. Calculate the cube root:
- The prime factorization whose exponents are whole numbers is:
[tex]\[
2^1 \times 5^1 = 2 \times 5 = 10
\][/tex]
- Therefore, the cube root of 121000, considering only the factors with exponents divisible by 3, is 10.
Hence, the cube root of [tex]\( 121000 \)[/tex] is [tex]\( 10 \)[/tex].
