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, we need to break down the number into its prime factors and then identify the cube root.
Step 1: Prime Factorization of 64000
We start by dividing 64000 by the smallest prime number, which is 2: 64000÷2=3200064000 \div 2 = 3200064000÷2=32000 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 not divisible by 2, so we move to the next smallest prime number, which is 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=28×5364000 = 2^8 \times 5^364000=28×53
Step 2: Taking the Cube Root
The cube root of a number can be found by taking the cube root of each factor in the prime factorization. In this case, we have 282^828 and 535^353.
For 282^828, we can rewrite it as: 28=(23)2×222^8 = (2^3)^2 \times 2^228=(23)2×22
Thus, the cube root of 282^828 is 222^222, because the cube root of 232^323 is just 2. The remaining factor 222^222 will be left as a square.
For 535^353, the cube root of 535^353 is simply 5, because the cube root of 535^353 is 5.
Thus, we now have: 640003=28×533=22×5=4×5=20\sqrt[3]{64000} = \sqrt[3]{2^8 \times 5^3} = 2^2 \times 5 = 4 \times 5 = 20364000=328×53=22×5=4×5=20
Step 3: Final Answer
The cube root of 64000 is 202020.
Explanation:
Prime factorization involves breaking down a number into its prime factors. In this case, 64000 is factored as 28×532^8 \times 5^328×53. When finding the cube root, we take the cube root of each factor. The cube root of 282^828 is 222^222, and the cube root of 535^353 is 5. The final result is the product of these values, which gives 20.
Thus, the cube root of 64000 is 202020.
