Prove The Divisibility Of The Following Numbers: 45 10 ⋅ 5 40 45^{10} \cdot 5^{40} 4 5 10 ⋅ 5 40 By 25 20 25^{20} 2 5 20 .Please Write Your Answer In Exponential Form.Answer: □ ⋅ 25 20 \square \cdot 25^{20} □ ⋅ 2 5 20

Prove the Divisibility of $45^{10} \cdot 5^{40}$ by $25^{20}$

In this article, we will explore the concept of divisibility and prove that the number $45^{10} \cdot 5^{40}$ is divisible by $25^{20}$. We will use the properties of exponents and prime factorization to demonstrate this divisibility.

To prove that a number is divisible by another number, we need to show that the first number can be expressed as a product of the second number and another number. In other words, we need to find a number that, when multiplied by the second number, gives us the first number.

The first step in proving divisibility is to find the prime factorization of the numbers involved. The prime factorization of a number is the expression of that number as a product of prime numbers.

• The prime factorization of $45$ is $3^2 \cdot 5$.
• The prime factorization of $5$ is $5$.
• The prime factorization of $25$ is $5^2$.

Using the prime factorization of $45$, we can express $45^{10}$ as $(3^2 \cdot 5)^{10}$.

$(3^2 \cdot 5)^{10} = 3^{20} \cdot 5^{10}$

Similarly, we can express $5^{40}$ as $5^{40}$.

Now, we can combine the exponents of $45^{10}$ and $5^{40}$ to get the final expression.

$45^{10} \cdot 5^{40} = 3^{20} \cdot 5^{10} \cdot 5^{40}$

Using the property of exponents that states $a^m \cdot a^n = a^{m+n}$, we can simplify the expression.

$3^{20} \cdot 5^{10} \cdot 5^{40} = 3^{20} \cdot 5^{50}$

The prime factorization of $25$ is $5^2$. Therefore, we can express $25^{20}$ as $(5^2)^{20}$.

$(5^2)^{20} = 5^{40}$

Now, we can prove that $45^{10} \cdot 5^{40}$ is divisible by $25^{20}$.

$45^{10} \cdot 5^{40} = 3^{20} \cdot 5^{50}$

$25^{20} = 5^{40}$

Since $5^{50}$ can be expressed as $5^{40} \cdot 5^{10}$, we can rewrite the expression as:

$3^{20} \cdot 5^{40} \cdot 5^{10}$

Now, we can see that $45^{10} \cdot 5^{40}$ is divisible by $25^{20}$, since it can be expressed as a product of $25^{20}$ and another number.

In this article, we proved that the number $45^{10} \cdot 5^{40}$ is divisible by $25^{20}$. We used the properties of exponents and prime factorization to demonstrate this divisibility. The final expression is $\boxed{3^{20} \cdot 5^{10}} \cdot 25^{20}$.
Prove the Divisibility of $45^{10} \cdot 5^{40}$ by $25^{20}$: Q&A

In our previous article, we proved that the number $45^{10} \cdot 5^{40}$ is divisible by $25^{20}$. In this article, we will answer some common questions related to this topic.

A: The prime factorization of $45^{10} \cdot 5^{40}$ is $3^{20} \cdot 5^{50}$.

A: To prove that a number is divisible by another number, you need to show that the first number can be expressed as a product of the second number and another number.

A: This property states that when you multiply two numbers with the same base, you can add their exponents.

A: The prime factorization of $25$ is $5^2$. Therefore, we can express $25^{20}$ as $(5^2)^{20}$, which simplifies to $5^{40}$.

A: This is because of the property of exponents that states $a^m \cdot a^n = a^{m+n}$. In this case, $5^{50}$ can be expressed as $5^{40} \cdot 5^{10}$.

A: The final expression for $45^{10} \cdot 5^{40}$ is $\boxed{3^{20} \cdot 5^{10}} \cdot 25^{20}$.

A: This proof demonstrates the concept of divisibility and how to use prime factorization and properties of exponents to prove it.

A: Yes, here are a few more examples:

• Prove that $12^{15} \cdot 2^{30}$ is divisible by $4^{15}$.
• Prove that $9^{20} \cdot 3^{40}$ is divisible by $27^{20}$.
• Prove that $16^{25} \cdot 2^{50}$ is divisible by $64^{25}$.

In this article, we answered some common questions related to the proof that $45^{10} \cdot 5^{40}$ is divisible by $25^{20}$. We hope that this article has provided a better understanding of the concept of divisibility and how to use prime factorization and properties of exponents to prove it.