To find the cube root of a number using prime factorization, you break down the number into its prime factors, group these factors into triplets, and then take one factor from each triplet to multiply together. The product is the cube root.
Here's a step-by-step guide:
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Prime Factorization: Find the prime factors of the given number. This means expressing the number as a product of prime numbers (numbers only divisible by 1 and themselves). You can do this by repeatedly dividing the number by the smallest prime number that divides it evenly, and continuing until you're left with only prime numbers.
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Grouping into Triplets: Arrange the prime factors and group them into sets of three identical factors (triplets). If you're trying to find the cube root and you can't form perfect triplets, then the number is not a perfect cube, and its cube root will be an irrational number.
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Extracting One Factor from Each Triplet: From each group of three identical prime factors, take only one factor.
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Multiplying Extracted Factors: Multiply all the factors you extracted in the previous step. The result is the cube root of the original number.
Example: Find the cube root of 216
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Prime Factorization of 216:
- 216 ÷ 2 = 108
- 108 ÷ 2 = 54
- 54 ÷ 2 = 27
- 27 ÷ 3 = 9
- 9 ÷ 3 = 3
- 3 ÷ 3 = 1
Therefore, 216 = 2 × 2 × 2 × 3 × 3 × 3
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Grouping into Triplets:
We can group the prime factors as (2 × 2 × 2) × (3 × 3 × 3). This gives us one triplet of 2s and one triplet of 3s.
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Extracting One Factor from Each Triplet:
From the triplet (2 × 2 × 2), we take one 2. From the triplet (3 × 3 × 3), we take one 3.
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Multiplying Extracted Factors:
Multiply the extracted factors: 2 × 3 = 6.
Therefore, the cube root of 216 is 6.
