Find the cube root of 512 by the prime factorization method.
Answer
152.7k+ views
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 .
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 .
Recently Updated Pages
If ab and c are unit vectors then left ab2 right+bc2+ca2 class 12 maths JEE_Main

A rod AB of length 4 units moves horizontally when class 11 maths JEE_Main

Evaluate the value of intlimits0pi cos 3xdx A 0 B 1 class 12 maths JEE_Main

Which of the following is correct 1 nleft S cup T right class 10 maths JEE_Main

What is the area of the triangle with vertices Aleft class 11 maths JEE_Main

The coordinates of the points A and B are a0 and a0 class 11 maths JEE_Main

Trending doubts
What was the capital of Kanishka A Mathura B Purushapura class 7 social studies CBSE

Difference Between Plant Cell and Animal Cell

Write an application to the principal requesting five class 10 english CBSE

Ray optics is valid when characteristic dimensions class 12 physics CBSE

Give 10 examples for herbs , shrubs , climbers , creepers

Tropic of Cancer passes through how many states? Name them.

Write the 6 fundamental rights of India and explain in detail

Write a letter to the principal requesting him to grant class 10 english CBSE

Name the Largest and the Smallest Cell in the Human Body ?
