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 monomial that can be expressed as the square of another monomial. In other words, it is a monomial that can be written in the form of $(ax)^2$, where $a$ is a constant and $x$ is the variable. Perfect squares are an essential concept in algebra, and identifying them is crucial for simplifying expressions and solving equations.

What are Perfect Squares?

A perfect square is a monomial that can be expressed as the square of another monomial. For example, $x^2$ is a perfect square because it can be written as $(x)^2$. Similarly, $4x^2$ is a perfect square because it can be written as $(2x)^2$. On the other hand, $3x^2$ is not a perfect square because it cannot be written as the square of another monomial.

How to Identify Perfect Squares

To identify perfect squares, we need to look for monomials that can be written in the form of $(ax)^2$. We can do this by checking if the exponent of the variable is even and if the coefficient is a perfect square. If both conditions are met, then the monomial is a perfect square.

Examples of Perfect Squares

Let's consider some examples of perfect squares:

• $x^2$ is a perfect square because it can be written as $(x)^2$.
• $4x^2$ is a perfect square because it can be written as $(2x)^2$.
• $9x^4$ is a perfect square because it can be written as $(3x^2)^2$.

Which Monomials are Perfect Squares?

Now, let's consider the options given in the problem:

A. $6x^2$ B. $9x^8$ C. $17x^9$ D. $25x^{12}$ E. $36x^{16}$

To determine which monomials are perfect squares, we need to check if they can be written in the form of $(ax)^2$. Let's analyze each option:

• Option A: $6x^2$ can be written as $(\sqrt{6}x)^2$, so it is a perfect square.
• Option B: $9x^8$ can be written as $(3x^4)^2$, so it is a perfect square.
• Option C: $17x^9$ cannot be written as the square of another monomial, so it is not a perfect square.
• Option D: $25x^{12}$ can be written as $(5x^6)^2$, so it is a perfect square.
• Option E: $36x^{16}$ can be written as $(6x^8)^2$, so it is a perfect square.

Conclusion

In conclusion, the monomials that are perfect squares are:

• Option A: $6x^2$
• Option B: $9x^8$
• Option D: $25x^{12}$
• Option E: $36x^{16}$

These monomials can be written in the form of $(ax)^2$, where $a$ is a constant and $x$ is the variable. Therefore, they are perfect squares.

Final Answer

The final answer is:

• A. $6x^2$
• B. $9x^8$
• D. $25x^{12}$
• E. $36x^{16}$
  Perfect Squares: A Q&A Guide =============================

Frequently Asked Questions

In this article, we will answer some of the most frequently asked questions about perfect squares.

Q: What is a perfect square?

A: A perfect square is a monomial that can be expressed as the square of another monomial. In other words, it is a monomial that can be written in the form of $(ax)^2$, where $a$ is a constant and $x$ is the variable.

Q: How do I identify a perfect square?

A: To identify a perfect square, you need to check if the exponent of the variable is even and if the coefficient is a perfect square. If both conditions are met, then the monomial is a perfect square.

Q: What are some examples of perfect squares?

A: Some examples of perfect squares include:

• $x^2$ is a perfect square because it can be written as $(x)^2$.
• $4x^2$ is a perfect square because it can be written as $(2x)^2$.
• $9x^4$ is a perfect square because it can be written as $(3x^2)^2$.

Q: Can a perfect square have a negative coefficient?

A: Yes, a perfect square can have a negative coefficient. For example, $(-3x)^2$ is a perfect square because it can be written as $(-3)^2(x)^2$.

Q: Can a perfect square have a fractional coefficient?

A: Yes, a perfect square can have a fractional coefficient. For example, $(\frac{1}{2}x)^2$ is a perfect square because it can be written as $(\frac{1}{2})^2(x)^2$.

Q: Can a perfect square have a coefficient that is a decimal number?

A: Yes, a perfect square can have a coefficient that is a decimal number. For example, $(3.5x)^2$ is a perfect square because it can be written as $(3.5)^2(x)^2$.

Q: Can a perfect square have a variable with a negative exponent?

A: No, a perfect square cannot have a variable with a negative exponent. For example, $x^{-2}$ is not a perfect square because it cannot be written as the square of another monomial.

Q: Can a perfect square have a variable with a fractional exponent?

A: No, a perfect square cannot have a variable with a fractional exponent. For example, $x^{\frac{1}{2}}$ is not a perfect square because it cannot be written as the square of another monomial.

Q: How do I simplify a perfect square?

A: To simplify a perfect square, you need to take the square root of the coefficient and the variable. For example, $(3x)^2$ can be simplified to $9x^2$.

Q: Can a perfect square be a binomial?

A: No, a perfect square cannot be a binomial. A perfect square is a monomial that can be expressed as the square of another monomial.

Q: Can a perfect square have a variable with a coefficient?

A: Yes, a perfect square can have a variable with a coefficient. For example, $2x^2$ is a perfect square because it can be written as $(\sqrt{2}x)^2$.

Conclusion

In conclusion, perfect squares are an essential concept in algebra, and understanding them is crucial for simplifying expressions and solving equations. By following the guidelines and examples provided in this article, you should be able to identify and simplify perfect squares with ease.

Final Tips

• Always check if the exponent of the variable is even and if the coefficient is a perfect square before identifying a monomial as a perfect square.
• Use the square root of the coefficient and the variable to simplify a perfect square.
• Remember that a perfect square cannot have a variable with a negative exponent or a fractional exponent.

Common Mistakes to Avoid

• Don't confuse a perfect square with a binomial.
• Don't forget to check if the exponent of the variable is even and if the coefficient is a perfect square before identifying a monomial as a perfect square.
• Don't simplify a perfect square by taking the square root of the coefficient and the variable without checking if the exponent of the variable is even and if the coefficient is a perfect square.

Final Answer

The final answer is:

• A perfect square is a monomial that can be expressed as the square of another monomial.
• To identify a perfect square, you need to check if the exponent of the variable is even and if the coefficient is a perfect square.
• A perfect square can have a negative coefficient, a fractional coefficient, or a coefficient that is a decimal number.
• A perfect square cannot have a variable with a negative exponent or a fractional exponent.
• To simplify a perfect square, you need to take the square root of the coefficient and the variable.
• A perfect square cannot be a binomial.
• A perfect square can have a variable with a coefficient.