Question

simplify. express your answer a
$r + 3 + \frac{8q^3}{3}$

Explanation:

Step1: Identify like terms

There are no like terms between \( r \), \( 3 \), and \( \frac{8q^{3}}{3} \) since they have different variables or are constants/have different variable expressions.

Step2: Rewrite the expression

We can just present the expression in a combined form. The expression \( r + 3+\frac{8q^{3}}{3} \) can be rewritten as \( r+\frac{8q^{3}}{3}+ 3 \) (by the commutative property of addition, though no simplification of like terms is possible here as the terms have different variables/forms). Wait, actually, maybe there was a typo or missing part? Wait, the original expression is \( r + 3+\frac{8q^{3}}{3} \). Since \( r \), \( 3 \), and \( \frac{8q^{3}}{3} \) are all unlike terms (different variables or constant vs variable term), the simplified form is just combining them as is, but maybe we can write the constant and the fraction? Wait, no, \( 3=\frac{9}{3} \), so:

Step1: Convert 3 to thirds

\( 3=\frac{9}{3} \), so the expression becomes \( r+\frac{9}{3}+\frac{8q^{3}}{3} \)

Step2: Combine the constant fractions

Combine \( \frac{9}{3} \) and \( \frac{8q^{3}}{3} \)? Wait, no, \( \frac{9}{3} \) is a constant term (since it has no variable) and \( \frac{8q^{3}}{3} \) has variable \( q \). Wait, actually, my mistake earlier: \( 3 \) is a constant, \( \frac{8q^{3}}{3} \) is a term with \( q \), and \( r \) is a term with \( r \). So there are three terms: \( r \), \( 3 \), and \( \frac{8q^{3}}{3} \). Since they are all unlike terms, we can't combine them further, but maybe the problem was supposed to have a common denominator or something else? Wait, maybe the original problem was miswritten? Wait, the given expression is \( r + 3+\frac{8q^{3}}{3} \). So the simplified form is just arranging them, but since they are unlike terms, we can write it as \( r+\frac{8q^{3}}{3}+3 \) or factor? Wait, no, unless there's a mistake. Wait, maybe the user made a typo, but assuming the expression is as given, the simplified form is \( r+\frac{8}{3}q^{3}+3 \) (rewriting the fraction as \( \frac{8}{3}q^{3} \)) and then combining the constants? Wait, \( 3 \) is a constant, \( \frac{8}{3}q^{3} \) has \( q \), \( r \) has \( r \). So no, they can't be combined. Wait, maybe the problem was \( r + 3+\frac{8r^{3}}{3} \)? If that were the case, we could combine, but as given, with \( q \) and \( r \) as different variables, we can't. So perhaps the original problem has a typo, but based on the given expression, the simplified form is \( r+\frac{8q^{3}}{3}+3 \), or we can write the constant term as a fraction: \( r+\frac{8q^{3}+9}{3} \) (by getting a common denominator for 3 and \( \frac{8q^{3}}{3} \)? Wait, no, 3 is \( \frac{9}{3} \), so \( r+\frac{9}{3}+\frac{8q^{3}}{3}=r+\frac{8q^{3} + 9}{3} \). Ah! That's a better way. Because 3 is \( \frac{9}{3} \), so we can combine the constant term (3) and the \( \frac{8q^{3}}{3} \) term? Wait, no, 3 is a constant (no variable) and \( \frac{8q^{3}}{3} \) has variable \( q \). Wait, but \( \frac{9}{3} \) is a constant (since \( 9/3 = 3 \)) and \( \frac{8q^{3}}{3} \) has \( q \). So they are still unlike terms. Wait, I'm confused. Wait, no, in algebra, terms are like terms only if they have the same variable(s) raised to the same power(s). So \( r \) (variable \( r \), power 1), \( 3 \) (constant, no variable), \( \frac{8q^{3}}{3} \) (variable \( q \), power 3). So they are all unlike terms. Therefore, the expression is already in a simplified form, or we can write the constant and the \( q \)-term over a common denominator:

Step1: Rewrite 3 as a fraction

\( 3=\frac{9}{3} \)

Step2: Combine the constant and \( q \)-term fractions (but they are still unlike in variables)

\( r+\frac{9}{3}+\frac{8q^{3}}{3}=r+\frac{8q^{3}+9}{3} \) (but note that \( \frac{9}{3} \) is a constant and \( \frac{8q^{3}}{3} \) has \( q \), so this is just combining the two terms with denominator 3, but they are still unlike in variables, so this is just a different way to write the expression)

So the simplified expression is \( r+\frac{8q^{3}+9}{3} \) or \( r+\frac{8}{3}q^{3}+3 \)

Answer:

\( r + 3+\frac{8q^{3}}{3} \) (or \( r+\frac{8q^{3}}{3}+3 \) or \( r+\frac{8q^{3}+9}{3} \))