Question

identify the terms and like terms in the expression.
$a^3 + 4 + 2t - 9 + t$
which statements are true? check all that apply.
\\(\square\\) the expression contains seven terms.
\\(\square\\) the terms in the expression are $a^3$, 4, $2t$, 9, and $t$.
\\(\square\\) the constants, 4 and 9, are like terms.
\\(\square\\) like terms have the same variables to the same powers.
\\(\square\\) $a^3$, $2t$, and $t$ are variables, so they are like terms.
\\(\square\\) $2t$ and $t$ are like terms.

Explanation:

Step1: Analyze the expression \(a^3 + 4 + 2t - 9 + t\)

First, identify the terms. Terms are separated by addition or subtraction signs. So the terms are \(a^3\), \(4\), \(2t\), \(-9\), \(t\). That's 5 terms, not 7. So "The expression contains seven terms" is false.

Step2: Identify the terms correctly

The terms in the expression are \(a^3\), \(4\), \(2t\), \(-9\), \(t\). Wait, the option says "The terms in the expression are \(a^3\), 4, \(2t\), 9, and \(t\)" – but -9 is a term, not 9. However, maybe a typo, but let's check constants: 4 and -9 (or 9 if considering absolute, but constants are 4 and -9). Like terms for constants: 4 and -9 are constants, but 4 and 9 (if 9 is -9) – no, 4 and -9 are constants, but are they like terms? Constants are like terms. Now, for the \(t\) terms: \(2t\) and \(t\) have the same variable \(t\) to the same power (1), so they are like terms.

Step3: Check each statement

- "The expression contains seven terms." False, as we have 5 terms (\(a^3\), \(4\), \(2t\), \(-9\), \(t\)).
- "The terms in the expression are \(a^3\), 4, \(2t\), 9, and \(t\)": Close, but -9 is a term, not 9, but maybe a typo. However, the main terms are \(a^3\), 4, \(2t\), -9, \(t\), so this is partially correct (if 9 is -9).
- "The constants, 4 and 9, are like terms": If 9 is -9, then 4 and -9 are constants (like terms). But the statement says 4 and 9 – maybe a sign error, but constants are like terms.
- "Like terms have the same variables to the same powers": This is the definition of like terms, so this is true.
- "\(a^3\), \(2t\), and \(t\) are variables, so they are like terms": \(a^3\) has variable \(a\), \(2t\) and \(t\) have variable \(t\) – different variables, so not like terms. So this is false.
- "\(2t\) and \(t\) are like terms": \(2t\) and \(t\) have the same variable \(t\) with exponent 1, so they are like terms. This is true.

Now, let's re-express:

Wait, the options:

1. The expression contains seven terms. → False (5 terms: \(a^3\), 4, \(2t\), -9, \(t\))
2. The terms in the expression are \(a^3\), 4, \(2t\), 9, and \(t\) → The terms are \(a^3\), 4, \(2t\), -9, \(t\); if 9 is a typo for -9, then the terms are \(a^3\), 4, \(2t\), -9, \(t\), so this is saying the terms are those, so maybe correct (assuming 9 is -9).
3. The constants, 4 and 9, are like terms → Constants are like terms, so if 9 is -9, then 4 and -9 are constants (like terms). So this is true (since constants are like terms regardless of sign, as long as they are constants).
4. Like terms have the same variables to the same powers → This is the definition of like terms, so true.
5. \(a^3\), \(2t\), and \(t\) are variables, so they are like terms → \(a^3\) (variable \(a\)) and \(2t\), \(t\) (variable \(t\)) have different variables, so not like terms → false.
6. \(2t\) and \(t\) are like terms → True, same variable \(t\), same power (1).

So the true statements are:

- The terms in the expression are \(a^3\), 4, \(2t\), 9, and \(t\) (assuming 9 is -9, or maybe the option has a sign error)
- The constants, 4 and 9, are like terms (constants are like terms)
- Like terms have the same variables to the same powers
- \(2t\) and \(t\) are like terms

But let's check again:

Wait, the expression is \(a^3 + 4 + 2t - 9 + t\). So terms: \(a^3\), \(4\), \(2t\), \(-9\), \(t\). So:

1. "The expression contains seven terms." → False (5 terms)
2. "The terms in the expression are \(a^3\), 4, \(2t\), 9, and \(t\)" → The terms are \(a^3\), 4, \(2t\), -9, \(t\). If the option wrote 9 instead of -9, but the terms are those, so maybe this is considered correct (maybe a typo in the option, writing 9 instead of -9). So this is true.
3. "The constants, 4 and 9, are like terms" → Constants are like terms, so 4 and -9 (or 9 if it's a typo) are constants, so like terms. True.
4. "Like terms have the same variables to the same powers" → Definition of like terms (terms with same variable(s) raised to same power). True.
5. "\(a^3\), \(2t\), and \(t\) are variables, so they are like terms" → \(a^3\) has variable \(a\), \(2t\) and \(t\) have variable \(t\) – different variables, so not like terms. False.
6. "\(2t\) and \(t\) are like terms" → Same variable \(t\), same power (1). True.

So the true statements are the second, third, fourth, and sixth? Wait, no, let's check the options again as per the image:

Options:

- The expression contains seven terms. → False
- The terms in the expression are \(a^3\), 4, \(2t\), 9, and \(t\) → Let's count terms: \(a^3\) (1), 4 (2), \(2t\) (3), -9 (4), \(t\) (5). So 5 terms. The option says 5 terms? Wait, \(a^3\), 4, \(2t\), 9, \(t\) – that's 5 terms (if 9 is -9). So maybe the option has a sign error, but the terms are those five. So this is true.
- The constants, 4 and 9, are like terms → Constants are like terms, so 4 and -9 (or 9) are constants, so like terms. True.
- Like terms have the same variables to the same powers → True (definition)
- \(a^3\), \(2t\), and \(t\) are variables, so they are like terms → False (different variables)
- \(2t\) and \(t\) are like terms → True (same variable, same power)

So the correct statements are:

- The terms in the expression are \(a^3\), 4, \(2t\), 9, and \(t\) (assuming 9 is -9)
- The constants, 4 and 9, are like terms
- Like terms have the same variables to the same powers
- \(2t\) and \(t\) are like terms

But let's confirm each:

1. "The expression contains seven terms." → False (5 terms)
2. "The terms in the expression are \(a^3\), 4, \(2t\), 9, and \(t\)" → The terms are \(a^3\), 4, \(2t\), -9, \(t\). If the option wrote 9 instead of -9, but the terms are those five, so this is true (maybe a typo in the option, writing 9 instead of -9). So this is correct.
3. "The constants, 4 and 9, are like terms" → Constants are like terms, so yes, 4 and -9 (or 9) are constants, so like terms. True.
4. "Like terms have the same variables to the same powers" → True (by definition of like terms: terms with the same variable(s) raised to the same power)
5. "\(a^3\), \(2t\), and \(t\) are variables, so they are like terms" → \(a^3\) has variable \(a\), \(2t\) and \(t\) have variable \(t\) – different variables, so not like terms. False.
6. "\(2t\) and \(t\) are like terms" → True (same variable \(t\), same exponent 1)

So the true statements are the second, third, fourth, and sixth? Wait, no, let's check the original options again as per the image:

Looking at the image:

Options:

1. The expression contains seven terms. → False (5 terms: \(a^3\), 4, \(2t\), -9, \(t\))
2. The terms in the expression are \(a^3\), 4, \(2t\), 9, and \(t\) → The terms are \(a^3\), 4, \(2t\), -9, \(t\). If 9 is a typo for -9, then this is correct (5 terms). So this is true.
3. The constants, 4 and 9, are like terms → Constants are like terms, so 4 and -9 (or 9) are constants, so like terms. True.
4. Like terms have the same variables to the same powers → True (definition)
5. \(a^3\), \(2t\), and \(t\) are variables, so they are like terms → False (different variables)
6. \(2t\) and \(t\) are like terms → True (same variable, same power)

So the correct options (assuming the 9 is a typo for -9) are:

- The terms in the expression are \(a^3\), 4, \(2t\), 9, and \(t\)
- The constants, 4 and 9, are like terms
- Like terms have the same variables to the same powers
- \(2t\) and \(t\) are like terms

But maybe the 9 is a mistake, and the correct terms are \(a^3\), 4, \(2t\), -9, \(t\), so the second option is correct (with 9 as -9). The third option: constants 4 and -9 (or 9) are like terms, so true. The fourth is the definition, true. The sixth is true.

So the true statements are:

- The terms in the expression are \(a^3\), 4, \(2t\), 9, and \(t\) (assuming 9 is -9)
- The constants, 4 and 9, are like terms (constants are like terms)
- Like terms have the same variables to the same powers (definition)
- \(2t\) and \(t\) are like terms (same variable, same power)

Answer:

• The terms in the expression are \(a^3\), 4, \(2t\), 9, and \(t\)
• The constants, 4 and 9, are like terms
• Like terms have the same variables to the same powers
• \(2t\) and \(t\) are like terms

(Note: The first option "The expression contains seven terms" is false. The fifth option "\(a^3\), \(2t\), and \(t\) are variables, so they are like terms" is false.)