Find the cube root of the following using
prime factorization. Leave your answer in
index notation.
(a)
(b)

Ask by Chen Chang. in Singapore

Jan 23,2025

Question

Find the cube root of the following using
prime factorization. Leave your answer in
index notation.
(a)
(b)

Ask by Chen Chang. in Singapore

Jan 23,2025

UpStudy AI · Accepted Answer

To find the cube roots of the given expressions using prime factorization and express the answers in index notation, follow these steps:

### (a) $$ 2^{3} \times 5^{6} $$

1. **Apply the cube root to each factor:**
   $$
   \sqrt[3]{2^{3} \times 5^{6}} = \sqrt[3]{2^{3}} \times \sqrt[3]{5^{6}}
   $$

2. **Simplify each cube root:**
   $$
   \sqrt[3]{2^{3}} = 2^{3/3} = 2^{1} = 2
   $$
   $$
   \sqrt[3]{5^{6}} = 5^{6/3} = 5^{2}
   $$

3. **Combine the simplified terms:**
   $$
   2 \times 5^{2}
   $$

**Final Answer for (a):**
$$
2 \times 5^{2}
$$

---

### (b) $$ 3^{3} \times 5^{9} \times 7^{6} $$

1. **Apply the cube root to each factor:**
   $$
   \sqrt[3]{3^{3} \times 5^{9} \times 7^{6}} = \sqrt[3]{3^{3}} \times \sqrt[3]{5^{9}} \times \sqrt[3]{7^{6}}
   $$

2. **Simplify each cube root:**
   $$
   \sqrt[3]{3^{3}} = 3^{3/3} = 3^{1} = 3
   $$
   $$
   \sqrt[3]{5^{9}} = 5^{9/3} = 5^{3}
   $$
   $$
   \sqrt[3]{7^{6}} = 7^{6/3} = 7^{2}
   $$

3. **Combine the simplified terms:**
   $$
   3 \times 5^{3} \times 7^{2}
   $$

**Final Answer for (b):**
$$
3 \times 5^{3} \times 7^{2}
$$
(a) $$ 2 \times 5^{2} $$ (b) $$ 3 \times 5^{3} \times 7^{2} $$

Find the cube root of the following using
prime factorization. Leave your answer in
index notation.
(a)
(b)

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Find the cube root of the following using prime factorization. Leave your answer in index notation. (a) (b)

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index notation.
(a)
(b)