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### 3. Finding the Cube Roots by Prime Factorization Method
#### a) Cube root of -13824
1. Recognize the number is negative, so the cube root will be negative.
2. Take the absolute value: [tex]\( 13824 \)[/tex].
3. Prime factorize [tex]\( 13824 \)[/tex]:
- [tex]\( 13824 \div 2 = 6912 \)[/tex]
- [tex]\( 6912 \div 2 = 3456 \)[/tex]
- [tex]\( 3456 \div 2 = 1728 \)[/tex]
- [tex]\( 1728 \div 2 = 864 \)[/tex]
- [tex]\( 864 \div 2 = 432 \)[/tex]
- [tex]\( 432 \div 2 = 216 \)[/tex]
- [tex]\( 216 \div 2 = 108 \)[/tex]
- [tex]\( 108 \div 2 = 54 \)[/tex]
- [tex]\( 54 \div 2 = 27 \)[/tex]
- [tex]\( 27 \div 3 = 9 \)[/tex]
- [tex]\( 9 \div 3 = 3 \)[/tex]
- [tex]\( 3 \div 3 = 1 \)[/tex]
4. Group in triples of prime factors: [tex]\( (2^7 \cdot 3^3) \)[/tex].
5. Cube root: [tex]\( 2^2 \cdot 3 = 4 \cdot 3 = 12 \)[/tex].
6. Account for the negative sign: [tex]\( \boxed{-24} \)[/tex].
#### b) Cube root of 125,000
1. Prime factorize [tex]\( 125000 \)[/tex]:
- [tex]\( 125000 \div 2 = 62500 \)[/tex]
- [tex]\( 62500 \div 2 = 31250 \)[/tex]
- [tex]\( 31250 \div 2 = 15625 \)[/tex]
- [tex]\( 15625 \div 5 = 3125 \)[/tex]
- [tex]\( 3125 \div 5 = 625 \)[/tex]
- [tex]\( 625 \div 5 = 125 \)[/tex]
- [tex]\( 125 \div 5 = 25 \)[/tex]
- [tex]\( 25 \div 5 = 5 \)[/tex]
- [tex]\( 5 \div 5 = 1 \)[/tex]
2. Group in triples of prime factors: [tex]\( (2^3 \cdot 5^6) \)[/tex].
3. Cube root: [tex]\( 2 \cdot 5^2 = 2 \cdot 25 = 50 \)[/tex].
4. [tex]\( \boxed{50} \)[/tex].
#### c) Cube root of 68,921
1. Prime factorize [tex]\( 68921 \)[/tex]:
- [tex]\( 68921 \div 41 = 1681 \)[/tex]
- [tex]\( 1681 \div 41 = 41 \)[/tex]
- [tex]\( 41 \div 41 = 1 \)[/tex]
2. Group in triples of prime factors: [tex]\( (41^3) \)[/tex].
3. Cube root: [tex]\( 41 \)[/tex].
4. [tex]\( \boxed{41} \)[/tex].
#### d) Cube root of [tex]\(\frac{1728}{132651}\)[/tex]
1. Simplify [tex]\( \frac{1728}{132651} \)[/tex]:
- Find cube root of 1728 (factorization: [tex]\( 1728 = 12^3 \)[/tex]):
- Cube root is [tex]\( 12 \)[/tex].
- Find cube root of 132651 (closest simple factorable cube approximation):
- Let's recognize it's practically negligibly small:
- Cube root approaches 0.
2. Hence, [tex]\( \boxed{0} \)[/tex].
#### e) Cube root of [tex]\(216 \times 512\)[/tex]
1. Calculate [tex]\( 216 \times 512 \)[/tex]:
- [tex]\( 216 = 6^3 \)[/tex]
- [tex]\( 512 = 8^3 \)[/tex]
- Product [tex]\( = (6 \times 8)^3 = 48^3 \)[/tex].
2. Cube root: [tex]\( 48 \)[/tex].
3. Hence, [tex]\( \boxed{48} \)[/tex].
### 4. Estimation Method for Cube Roots
#### a) Cube root of -13824
- Recognizing close to [tex]\( (-24)^3 \)[/tex].
- Cube root estimated as: [tex]\( \boxed{-24} \)[/tex].
#### b) Cube root of 125,000
- Close to [tex]\( 50^3 \)[/tex].
- Estimate: [tex]\( \boxed{50} \)[/tex].
#### c) Cube root of 68,921
- Close to [tex]\( 41^3 \)[/tex].
- Estimate: [tex]\( \boxed{41} \)[/tex].
#### d) Cube root of [tex]\(\frac{1728}{132651}\)[/tex]
- Approximate negligible value converges to [tex]\( 0 \)[/tex].
- Estimate: [tex]\( \boxed{0} \)[/tex].
#### e) Cube root of 216 \times 512
- Approximates to [tex]\( 48^3 \)[/tex].
- Estimate: [tex]\( \boxed{48} \)[/tex].
#### a) Cube root of -13824
1. Recognize the number is negative, so the cube root will be negative.
2. Take the absolute value: [tex]\( 13824 \)[/tex].
3. Prime factorize [tex]\( 13824 \)[/tex]:
- [tex]\( 13824 \div 2 = 6912 \)[/tex]
- [tex]\( 6912 \div 2 = 3456 \)[/tex]
- [tex]\( 3456 \div 2 = 1728 \)[/tex]
- [tex]\( 1728 \div 2 = 864 \)[/tex]
- [tex]\( 864 \div 2 = 432 \)[/tex]
- [tex]\( 432 \div 2 = 216 \)[/tex]
- [tex]\( 216 \div 2 = 108 \)[/tex]
- [tex]\( 108 \div 2 = 54 \)[/tex]
- [tex]\( 54 \div 2 = 27 \)[/tex]
- [tex]\( 27 \div 3 = 9 \)[/tex]
- [tex]\( 9 \div 3 = 3 \)[/tex]
- [tex]\( 3 \div 3 = 1 \)[/tex]
4. Group in triples of prime factors: [tex]\( (2^7 \cdot 3^3) \)[/tex].
5. Cube root: [tex]\( 2^2 \cdot 3 = 4 \cdot 3 = 12 \)[/tex].
6. Account for the negative sign: [tex]\( \boxed{-24} \)[/tex].
#### b) Cube root of 125,000
1. Prime factorize [tex]\( 125000 \)[/tex]:
- [tex]\( 125000 \div 2 = 62500 \)[/tex]
- [tex]\( 62500 \div 2 = 31250 \)[/tex]
- [tex]\( 31250 \div 2 = 15625 \)[/tex]
- [tex]\( 15625 \div 5 = 3125 \)[/tex]
- [tex]\( 3125 \div 5 = 625 \)[/tex]
- [tex]\( 625 \div 5 = 125 \)[/tex]
- [tex]\( 125 \div 5 = 25 \)[/tex]
- [tex]\( 25 \div 5 = 5 \)[/tex]
- [tex]\( 5 \div 5 = 1 \)[/tex]
2. Group in triples of prime factors: [tex]\( (2^3 \cdot 5^6) \)[/tex].
3. Cube root: [tex]\( 2 \cdot 5^2 = 2 \cdot 25 = 50 \)[/tex].
4. [tex]\( \boxed{50} \)[/tex].
#### c) Cube root of 68,921
1. Prime factorize [tex]\( 68921 \)[/tex]:
- [tex]\( 68921 \div 41 = 1681 \)[/tex]
- [tex]\( 1681 \div 41 = 41 \)[/tex]
- [tex]\( 41 \div 41 = 1 \)[/tex]
2. Group in triples of prime factors: [tex]\( (41^3) \)[/tex].
3. Cube root: [tex]\( 41 \)[/tex].
4. [tex]\( \boxed{41} \)[/tex].
#### d) Cube root of [tex]\(\frac{1728}{132651}\)[/tex]
1. Simplify [tex]\( \frac{1728}{132651} \)[/tex]:
- Find cube root of 1728 (factorization: [tex]\( 1728 = 12^3 \)[/tex]):
- Cube root is [tex]\( 12 \)[/tex].
- Find cube root of 132651 (closest simple factorable cube approximation):
- Let's recognize it's practically negligibly small:
- Cube root approaches 0.
2. Hence, [tex]\( \boxed{0} \)[/tex].
#### e) Cube root of [tex]\(216 \times 512\)[/tex]
1. Calculate [tex]\( 216 \times 512 \)[/tex]:
- [tex]\( 216 = 6^3 \)[/tex]
- [tex]\( 512 = 8^3 \)[/tex]
- Product [tex]\( = (6 \times 8)^3 = 48^3 \)[/tex].
2. Cube root: [tex]\( 48 \)[/tex].
3. Hence, [tex]\( \boxed{48} \)[/tex].
### 4. Estimation Method for Cube Roots
#### a) Cube root of -13824
- Recognizing close to [tex]\( (-24)^3 \)[/tex].
- Cube root estimated as: [tex]\( \boxed{-24} \)[/tex].
#### b) Cube root of 125,000
- Close to [tex]\( 50^3 \)[/tex].
- Estimate: [tex]\( \boxed{50} \)[/tex].
#### c) Cube root of 68,921
- Close to [tex]\( 41^3 \)[/tex].
- Estimate: [tex]\( \boxed{41} \)[/tex].
#### d) Cube root of [tex]\(\frac{1728}{132651}\)[/tex]
- Approximate negligible value converges to [tex]\( 0 \)[/tex].
- Estimate: [tex]\( \boxed{0} \)[/tex].
#### e) Cube root of 216 \times 512
- Approximates to [tex]\( 48^3 \)[/tex].
- Estimate: [tex]\( \boxed{48} \)[/tex].
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