To determine if $3^4 \times 5^{10}$ is a square number, let's break it down step by step.

What is a Square Number?

A square number is a number that can be written as the square of an integer. In terms of prime factors, a number is a square if all the exponents in its prime factorization are even.

Prime Factorization of $3^4 \times 5^{10}$

1. $3^4$ means 3 is raised to the power of 4.
2. $5^{10}$ means 5 is raised to the power of 10.

So, the expression $3^4 \times 5^{10}$ is already in its prime factorized form:

• The base 3 has an exponent of 4.
• The base 5 has an exponent of 10.

Are the Exponents Even?

For a number to be a square, all exponents in the prime factorization must be even.

• The exponent of 3 is 4, which is even.
• The exponent of 5 is 10, which is also even.

Since both exponents (4 and 10) are even, $3^4 \times 5^{10}$ is indeed a square number.

Conclusion

The number $3^4 \times 5^{10}$ is a square number because both exponents in its prime factorization are even, meaning it can be written as the square of another number.

Would you like further details or examples to clarify this concept?

Related Questions:

1. What is the square root of $3^4 \times 5^{10}$?
2. Is $2^5 \times 7^4$ a square number? Why or why not?
3. How do you check if a product of multiple numbers is a square number?
4. What is the general method for finding the prime factorization of a large number?
5. How does $3^4 \times 5^{10}$ compare to other square numbers?

Tip:

To quickly check if a number is a square, look at its prime factorization and ensure that all exponents are even.