Find the cube root of 512 by the prime factorization method.

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Hint: Use the prime factorization method to obtain the prime factors of 512 and write 512 as the product of primes. To find the cube root, convert the triplet into a single number.

Complete step-by-step answer:
Cube root of a number is the number that multiplies by itself three times to produce the first number. For example, the cube root of 8 is 2 because \[2 \times 2 \times 2\] is equal to 8.
We can find the cube root of a number by using the prime factorization method.
In this method, we start dividing the number by the first prime number and continue dividing by 2 until we get a remainder. Then we proceed with the next prime number 3 and so on. Then, we represent the number as the product of prime numbers so obtained. Then, we form triplets by grouping together three of the same factors and write only one from each triplet. We finally multiply the resulting factors to obtain the cube root.
Now, let us use the prime factorization method to write 512 as the product of prime numbers.
\[\begin{gathered}
  2\left| \!{\underline {\,
  {512} \,}} \right. \\
  2\left| \!{\underline {\,
  {256} \,}} \right. \\
  2\left| \!{\underline {\,
  {128} \,}} \right. \\
  2\left| \!{\underline {\,
  {64{\text{ }}} \,}} \right. \\
  2\left| \!{\underline {\,
  {32{\text{ }}} \,}} \right. \\
  2\left| \!{\underline {\,
  {{\text{16 }}} \,}} \right. \\
  2\left| \!{\underline {\,
  {{\text{8 }}} \,}} \right. \\
  2\left| \!{\underline {\,
  {{\text{4 }}} \,}} \right. \\
  2\left| \!{\underline {\,
  {{\text{2 }}} \,}} \right. \\
  {\text{ 1}} \\
\end{gathered} \]
Hence, we have:
\[512 = 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2\]
We now form triplets as follows:
\[512 = \mathop {2 \times 2 \times 2}\limits_{\_\_\_\_\_\_\_\_\_\_\_} \times \mathop {2 \times 2 \times 2}\limits_{\_\_\_\_\_\_\_\_\_\_\_} \times \mathop {2 \times 2 \times 2}\limits_{\_\_\_\_\_\_\_\_\_\_\_} \]
Next, we just take one among each triplet and multiply to get the cube root.
\[2 \times 2 \times 2 = 8\]
Hence, the cube root of 512 is 8.

Note: You can cross-check your answer by multiplying the answer three times with itself to get 512. This method can only be used to find cube roots of perfect cube numbers. For finding the solution we make the cube pairs of prime numbers .