Use prime factorization, to find the value of 3√4096

To find the value of \( \sqrt[3]{4096} \) using prime factorization, let's break it down step by step.

### Step 1: Prime factorization of 4096
We begin by factoring 4096 into its prime factors:

\[
4096 \div 2 = 2048
\]
\[
2048 \div 2 = 1024
\]
\[
1024 \div 2 = 512
\]
\[
512 \div 2 = 256
\]
\[
256 \div 2 = 128
\]
\[
128 \div 2 = 64
\]
\[
64 \div 2 = 32
\]
\[
32 \div 2 = 16
\]
\[
16 \div 2 = 8
\]
\[
8 \div 2 = 4
\]
\[
4 \div 2 = 2
\]
\[
2 \div 2 = 1
\]

Thus, the prime factorization of 4096 is:

\[
4096 = 2^{12}
\]

### Step 2: Taking the cube root
Now we need to find the cube root of \( 4096 = 2^{12} \). The cube root of \( 2^{12} \) is:

\[
\sqrt[3]{2^{12}} = 2^{12/3} = 2^4 = 16
\]

### Final answer:
Thus, \( \sqrt[3]{4096} = 16 \).

Would you like further details or have any questions?

Here are 5 related questions you can explore:
1. What is the cube root of other powers of 2, such as \( \sqrt[3]{512} \)?
2. How do you find the cube root of non-perfect cubes using approximation?
3. Can you find the fourth root of 4096 using a similar method?
4. How do prime factorizations help in simplifying higher-order roots?
5. What is the connection between prime factorization and the Greatest Common Divisor (GCD)?

**Tip:** When calculating roots using prime factorization, simplify by dividing exponents based on the type of root (square root, cube root, etc.).

Find the cube root of 4096 using prime factorization with this step-by-step solution. Learn how to break down 4096 into its prime factors and apply the cube root to get 16. This problem demonstrates the use of prime factorization and is ideal for students in Grades 7-9.

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