Question

Express each of the following as a product of the powers of their prime factors:

1. $648$
2. $405$
3. $540$
4. $3600$

Answer 1

Step $\mathbf{1}$: Write the prime factors of $\mathbf{648}$.

So the prime factorization of $648$ is

$648=2\times 2\times 2\times 3\times 3\times 3\times 3$

Next, we will write this as a product of the powers of their prime factors. That is,

$648=2^3\times 3^4$

Step $\mathbf{2}$: Write the prime factors of $\mathbf{405}$.

So the prime factorization of $405$ is

$405=3\times 3\times 3\times 3\times 5$

Next, we will write this as a product of the powers of their prime factors. That is,

$405=3^4\times 5$

Step $\mathbf{3}$: Write the prime factors of $\mathbf{540}$.

So the prime factorization of $540$ is

$540=2\times 2\times 3\times 3\times 3\times 5$

Next, we will write this as a product of the powers of their prime factors. That is,

$540=2^2\times 3^3\times 5$

Step $\mathbf{4}$: Write the prime factors of $\mathbf{3600}$.

So the prime factorization of $3600$ is

$3600=2\times 2\times 2\times 2\times 3\times 3\times 5\times 5$

Next, we will write this as a product of the powers of their prime factors. That is,

$3600=2^4\times 3^2\times 5^2$

Hence the required answers are

1. $\mathbf{648}\mathbf{=}{\mathbf{2}}^{\mathbf{3}}\mathbf{\times }{\mathbf{3}}^{\mathbf{4}}$
2. $\mathbf{405}\mathbf{=}{\mathbf{3}}^{\mathbf{4}}\mathbf{\times }\mathbf{5}$
3. $\mathbf{540}\mathbf{=}\mathbf{}{\mathbf{2}}^{\mathbf{2}}\mathbf{\times }\mathbf{}{\mathbf{3}}^{\mathbf{3}}\mathbf{}\mathbf{\times }\mathbf{}\mathbf{5}$
4. $\mathbf{3600}\mathbf{=}{\mathbf{2}}^{\mathbf{4}}\mathbf{\times }{\mathbf{3}}^{\mathbf{2}}\mathbf{\times }{\mathbf{5}}^{\mathbf{2}}$