Cube root of each of the following by prime factorization: 64000 (full process).
The Correct Answer and Explanation is:
To find the cube root of 64000 using prime factorization, let’s go through the process step by step.
Step 1: Prime Factorization of 64000
Start by breaking down 64000 into prime factors.
- Divide by 2 (since 64000 is even): 64000÷2=3200064000 \div 2 = 3200064000÷2=32000
- Continue dividing by 2: 32000÷2=1600032000 \div 2 = 1600032000÷2=16000 16000÷2=800016000 \div 2 = 800016000÷2=8000 8000÷2=40008000 \div 2 = 40008000÷2=4000 4000÷2=20004000 \div 2 = 20004000÷2=2000 2000÷2=10002000 \div 2 = 10002000÷2=1000 1000÷2=5001000 \div 2 = 5001000÷2=500 500÷2=250500 \div 2 = 250500÷2=250 250÷2=125250 \div 2 = 125250÷2=125
Now, 125 is odd, so we move on to the next prime number, 5.
- Divide by 5: 125÷5=25125 \div 5 = 25125÷5=25 25÷5=525 \div 5 = 525÷5=5 5÷5=15 \div 5 = 15÷5=1
So, the prime factorization of 64000 is: 64000=27×5364000 = 2^7 \times 5^364000=27×53
Step 2: Cube Root Calculation
Now that we have the prime factorization, we can calculate the cube root. The cube root of a number is found by taking the cube root of each prime factor’s exponent.
For 64000=27×5364000 = 2^7 \times 5^364000=27×53, we take the cube root of each factor: Cube root of 27=27/3=22×21/3\text{Cube root of } 2^7 = 2^{7/3} = 2^2 \times 2^{1/3}Cube root of 27=27/3=22×21/3
Since the exponent 7 is not a perfect multiple of 3, we can break it as 26×212^6 \times 2^126×21, where: 26=(23)2=822^6 = (2^3)^2 = 8^226=(23)2=82
So, 27/3=22×21/3=4×21/32^{7/3} = 2^2 \times 2^{1/3} = 4 \times 2^{1/3}27/3=22×21/3=4×21/3.
For 535^353, it’s a perfect cube: Cube root of 53=5\text{Cube root of } 5^3 = 5Cube root of 53=5
Step 3: Simplify the Cube Root Expression
Now, we can combine the results: 640003=27×533=4×5=20\sqrt[3]{64000} = \sqrt[3]{2^7 \times 5^3} = 4 \times 5 = 20364000=327×53=4×5=20
Final Answer:
The cube root of 64000 is 20.
Explanation:
By using prime factorization, we break down the number into its basic prime factors, then apply the rule for cube roots, which allows us to find the root of each factor individually. This makes it much easier to compute the cube root for larger numbers. In this case, 64000=27×5364000 = 2^7 \times 5^364000=27×53, and the cube root of each factor is taken based on the exponents. The result is a simple whole number, 20, which is the cube root of 64000.
