Question

why are the following expressions not monomials?
\\(3a^x\\)
\\(x+2w\\)
\\(\frac{3}{h}\\)
\\(ab^{-1}\\)

dropdown options:

• the variable is in the denominator.
• the exponent is not a non-negative integer.
• the expression is not a product.

which expressions are monomials?
\\(-4+6\\)
\\(b+2b+2\\)
\\((x-2x)^2\\)
\\(\frac{rs}{t}\\)
\\(36x^2yz^3\\)
\\(a^x\\)
\\(x^{-1}\\)

Explanation:

🆕 New Concept Discovered: Identifying Monomials
A single term with non-negative integer exponents.

Step 1: Define a Monomial

A monomial is a number, a variable, or a product of numbers and variables with non-negative integer exponents. This means:

• No variables in denominators (which would mean a negative exponent).
• No fractional or negative exponents on variables.
• No addition or subtraction of terms (it must be a single product).

---

Step 2: Analyze "Why are the following expressions not monomials?"

We evaluate each given expression against the definition of a monomial:

1. \( 3a^{x} \)

• The exponent is a variable \( x \), not a fixed non-negative integer.
• Reason: The exponent is not a non-negative integer.

1. \( x + 2w \)

• This expression is a sum of two separate terms, not a single product.
• Reason: The expression is not a product.

1. \( \frac{3}{h} \)

• The variable \( h \) is in the denominator (which is equivalent to \( 3h^{-1} \)).
• Reason: The variable is in the denominator.

1. \( ab^{-1} \)

• The exponent on the variable \( b \) is \(-1\), which is negative.
• Reason: The exponent is not a non-negative integer.

---

Step 3: Analyze "Which expressions are monomials?"

We check each option in the checklist:

• \( -4 + 6 \)
• This simplifies to \( 2 \), which is a constant (a single number). A constant is a monomial.
• Monomial: Yes.

• \( b + 2b + 2 \)
• This contains addition of multiple terms that cannot be simplified into a single product term (it simplifies to \( 3b + 2 \)).
• Monomial: No.

• \( (x - 2x)^2 \)
• This simplifies to \( (-x)^2 = x^2 \), which is a single variable raised to a positive integer power.
• Monomial: Yes.

• \( \frac{rs}{t} \)
• The variable \( t \) is in the denominator.
• Monomial: No.

• \( 36x^2yz^3 \)
• This is a product of a number and variables with positive integer exponents (\( 2 \), \( 1 \), and \( 3 \)).
• Monomial: Yes.

• \( a^x \)
• The exponent is a variable \( x \), not a fixed integer.
• Monomial: No.

• \( x^{\frac{1}{3}} \)
• The exponent is a fraction, not an integer.
• Monomial: No.

Answer:

Part 1: Why are the following expressions not monomials?

• \( 3a^x \): The exponent is not a non-negative integer.
• \( x + 2w \): The expression is not a product.
• \( \frac{3}{h} \): The variable is in the denominator.
• \( ab^{-1} \): The exponent is not a non-negative integer.

Part 2: Which expressions are monomials? (Select these options)

• [x] \( -4 + 6 \)
• [x] \( (x - 2x)^2 \)
• [x] \( 36x^2yz^3 \)