Which Monomials Are Perfect Squares? Select Three Options.A. $6x^2$ B. $9x^8$ C. $17x^9$ D. $25x^{12}$ E. $36x^{16}$

Understanding Perfect Squares

In mathematics, a perfect square is a number or expression that can be expressed as the square of an integer or a variable. For example, 4 is a perfect square because it can be expressed as 2^2, and x^2 is a perfect square because it can be expressed as (x)^2. In the context of monomials, a perfect square is a monomial that can be expressed as the square of another monomial.

What are Monomials?

A monomial is an algebraic expression that consists of only one term. It can be a number, a variable, or a product of numbers and variables. For example, 3x, 2y^2, and 5z are all monomials.

Which Monomials are Perfect Squares?

To determine which monomials are perfect squares, we need to look for monomials that can be expressed as the square of another monomial. Let's examine the options:

Option A: $6x^2$

To determine if $6x^2$ is a perfect square, we need to find a monomial that, when squared, gives us $6x^2$. We can start by factoring out the coefficient 6:

$$6x^2 = (2x)^2$$

Since $6x^2$ can be expressed as the square of another monomial, it is a perfect square.

Option B: $9x^8$

To determine if $9x^8$ is a perfect square, we need to find a monomial that, when squared, gives us $9x^8$. We can start by factoring out the coefficient 9:

$$9x^8 = (3x^4)^2$$

Since $9x^8$ can be expressed as the square of another monomial, it is a perfect square.

Option C: $17x^9$

To determine if $17x^9$ is a perfect square, we need to find a monomial that, when squared, gives us $17x^9$. However, we cannot factor out the coefficient 17 in a way that gives us a perfect square. Therefore, $17x^9$ is not a perfect square.

Option D: $25x^{12}$

To determine if $25x^{12}$ is a perfect square, we need to find a monomial that, when squared, gives us $25x^{12}$. We can start by factoring out the coefficient 25:

$$25x^{12} = (5x^6)^2$$

Since $25x^{12}$ can be expressed as the square of another monomial, it is a perfect square.

Option E: $36x^{16}$

To determine if $36x^{16}$ is a perfect square, we need to find a monomial that, when squared, gives us $36x^{16}$. We can start by factoring out the coefficient 36:

$$36x^{16} = (6x^8)^2$$

Since $36x^{16}$ can be expressed as the square of another monomial, it is a perfect square.

Conclusion

In conclusion, the monomials that are perfect squares are:

• $6x^2$
• $9x^8$
• $25x^{12}$
• $36x^{16}$

Q: What is a perfect square in mathematics?

A: A perfect square is a number or expression that can be expressed as the square of an integer or a variable. For example, 4 is a perfect square because it can be expressed as 2^2, and x^2 is a perfect square because it can be expressed as (x)^2.

Q: What is a monomial?

A: A monomial is an algebraic expression that consists of only one term. It can be a number, a variable, or a product of numbers and variables. For example, 3x, 2y^2, and 5z are all monomials.

Q: How do I determine if a monomial is a perfect square?

A: To determine if a monomial is a perfect square, you need to find a monomial that, when squared, gives you the original monomial. You can start by factoring out the coefficient and the variable(s) in the monomial.

Q: What are some examples of perfect squares in monomials?

A: Some examples of perfect squares in monomials include:

• $6x^2 = (2x)^2$
• $9x^8 = (3x^4)^2$
• $25x^{12} = (5x^6)^2$
• $36x^{16} = (6x^8)^2$

Q: How do I identify a perfect square in a monomial with a negative exponent?

A: To identify a perfect square in a monomial with a negative exponent, you need to rewrite the monomial with a positive exponent. For example, $x^{-2}$ can be rewritten as $\frac{1}{x^2}$, which is a perfect square.

Q: Can a monomial with a variable and a coefficient be a perfect square?

A: Yes, a monomial with a variable and a coefficient can be a perfect square. For example, $4x^2$ is a perfect square because it can be expressed as $(2x)^2$.

Q: Can a monomial with a variable and a coefficient and a negative exponent be a perfect square?

A: Yes, a monomial with a variable and a coefficient and a negative exponent can be a perfect square. For example, $-4x^{-2}$ is a perfect square because it can be expressed as $-(2x^{-1})^2$.

Q: What are some common mistakes to avoid when identifying perfect squares in monomials?

A: Some common mistakes to avoid when identifying perfect squares in monomials include:

• Not factoring out the coefficient and the variable(s) in the monomial.
• Not rewriting the monomial with a positive exponent when it has a negative exponent.
• Not checking if the monomial can be expressed as the square of another monomial.

Q: How do I apply the concept of perfect squares in monomials to real-world problems?

A: The concept of perfect squares in monomials can be applied to real-world problems in various fields, such as physics, engineering, and computer science. For example, in physics, the concept of perfect squares is used to describe the relationship between energy and momentum. In engineering, the concept of perfect squares is used to design and optimize systems. In computer science, the concept of perfect squares is used to develop algorithms and data structures.

Conclusion

In conclusion, perfect squares in monomials are an important concept in mathematics that can be applied to real-world problems. By understanding how to identify perfect squares in monomials, you can solve problems and develop new ideas in various fields.