Which Monomials Are Perfect Squares? Select Three Options.☐ 6x² ☐ 9x⁸ ☐ 17x⁹ ☐ 25x¹² ☐ 36x¹⁶

In mathematics, a perfect square is a monomial that can be expressed as the square of another monomial. This means that the exponent of the variable in the monomial must be an even number. In this article, we will explore three options that are perfect squares and identify the correct ones.

What are Perfect Squares?

A perfect square is a monomial that can be expressed as the square of another monomial. For example, x² is a perfect square because it can be expressed as (x)². Similarly, x⁴ is a perfect square because it can be expressed as (x²)². On the other hand, x³ is not a perfect square because it cannot be expressed as the square of another monomial.

Properties of Perfect Squares

Perfect squares have several properties that make them easy to identify. One of the most important properties is that the exponent of the variable in a perfect square must be an even number. This means that if a monomial has an odd exponent, it cannot be a perfect square.

Another property of perfect squares is that they can be expressed as the square of another monomial. For example, x² can be expressed as (x)², and x⁴ can be expressed as (x²)². This property makes it easy to identify perfect squares by looking for monomials that can be expressed as the square of another monomial.

Option 1: 6x²

The first option is 6x². To determine if this monomial is a perfect square, we need to check if the exponent of the variable (x) is an even number. In this case, the exponent is 2, which is an even number. Therefore, 6x² is a perfect square.

Option 2: 9x⁸

The second option is 9x⁸. To determine if this monomial is a perfect square, we need to check if the exponent of the variable (x) is an even number. In this case, the exponent is 8, which is an even number. Therefore, 9x⁸ is a perfect square.

Option 3: 17x⁹

The third option is 17x⁹. To determine if this monomial is a perfect square, we need to check if the exponent of the variable (x) is an even number. In this case, the exponent is 9, which is an odd number. Therefore, 17x⁹ is not a perfect square.

Option 4: 25x¹²

The fourth option is 25x¹². To determine if this monomial is a perfect square, we need to check if the exponent of the variable (x) is an even number. In this case, the exponent is 12, which is an even number. Therefore, 25x¹² is a perfect square.

Option 5: 36x¹⁶

The fifth option is 36x¹⁶. To determine if this monomial is a perfect square, we need to check if the exponent of the variable (x) is an even number. In this case, the exponent is 16, which is an even number. Therefore, 36x¹⁶ is a perfect square.

Conclusion

In conclusion, the three options that are perfect squares are:

• 6x²
• 9x⁸
• 25x¹²
• 36x¹⁶

These monomials can be expressed as the square of another monomial, and the exponent of the variable in each monomial is an even number. Therefore, they meet the criteria for perfect squares.

Final Answer

The correct options are:

In the previous article, we discussed the concept of perfect squares and identified three options that are perfect squares. In this article, we will answer some frequently asked questions about perfect squares.

Q: What is the difference between a perfect square and a monomial?

A: A perfect square is a monomial that can be expressed as the square of another monomial. For example, x² is a perfect square because it can be expressed as (x)². On the other hand, x³ is not a perfect square because it cannot be expressed as the square of another monomial.

Q: How do I identify a perfect square?

A: To identify a perfect square, you need to check if the exponent of the variable in the monomial is an even number. If the exponent is an even number, then the monomial is a perfect square.

Q: Can a perfect square have a coefficient?

A: Yes, a perfect square can have a coefficient. For example, 2x² is a perfect square because it can be expressed as (2x)².

Q: Can a perfect square have a negative exponent?

A: No, a perfect square cannot have a negative exponent. The exponent of the variable in a perfect square must be an even number, and negative numbers are not even.

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

A: Yes, a perfect square can have a variable with a negative sign. For example, (-x)² is a perfect square because it can be expressed as ((-x)²)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 fractional exponent?

A: No, a perfect square cannot have a variable with a fractional exponent. The exponent of the variable in a perfect square must be an even number, and fractional numbers are not even.

Q: Can a perfect square be a polynomial?

A: No, a perfect square cannot be a polynomial. 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 negative fractional exponent?

A: No, a perfect square cannot have a variable with a negative fractional exponent. The exponent of the variable in a perfect square must be an even number, and negative fractional numbers are not even.

Q: Can a perfect square be a rational expression?

A: No, a perfect square cannot be a rational expression. 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 negative rational exponent?

A: No, a perfect square cannot have a variable with a negative rational exponent. The exponent of the variable in a perfect square must be an even number, and negative rational numbers are not even.

Conclusion

In conclusion, perfect squares are monomials that can be expressed as the square of another monomial. They have several properties that make them easy to identify, including the fact that the exponent of the variable in a perfect square must be an even number. We hope that this article has helped to answer some of the frequently asked questions about perfect squares.

Final Answer

• A perfect square is a monomial that can be expressed as the square of another monomial.
• The exponent of the variable in a perfect square must be an even number.
• A perfect square can have a coefficient, but it cannot have a negative exponent.
• A perfect square can have a variable with a negative sign, but it cannot have a variable with a negative fractional exponent.
• A perfect square cannot be a binomial, a polynomial, a rational expression, or a variable with a negative rational exponent.