what is the cube root of 512
What is the cube root of 512?
Answer:
To find the cube root of 512, you must determine which number, when multiplied by itself three times, equals 512. Let’s explore this process step by step:
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Prime Factorization of 512:
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First, you can break down 512 into its prime factors.
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 -
So, the prime factorization of 512 is (2^9).
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Determining the Cube Root:
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To find the cube root of a number, you take one-third of the exponent of its prime factor when written in powers.
Since (512 = 2^9), the cube root of 512 is:
\sqrt[3]{512} = \sqrt[3]{2^9} = 2^{\frac{9}{3}} = 2^3 = 8
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Therefore, the cube root of 512 is 8. This means that (8^3 = 512).
By using prime factorization and understanding the relationship between exponents and roots, you can easily deduce that the cube root of 512 is \boxed{8}.