Question

1. which two expressions are equivalent?

a. $9(6 + x)$ and $9 \cdot 6 + 9 \cdot x$
b. $x + (8 \cdot 9)$ and $(x + 8) \cdot 9$
c. $8 \cdot 6 \div x$ and $8 \cdot x \div 6$
d. $6 \cdot x + 3$ and $6 \cdot (x + 3)$

Explanation:

Step1: Analyze Option a

Apply distributive property: $9(6+x) = 9\cdot6 + 9\cdot x$

Step2: Analyze Option b

Simplify both expressions:
Left: $x+(8\cdot9)=x+72$
Right: $(x+8)\cdot9=9x+72$
These are not equivalent.

Step3: Analyze Option c

Simplify both expressions:
Left: $8\cdot6\div x=\frac{48}{x}$
Right: $8\cdot x\div6=\frac{8x}{6}=\frac{4x}{3}$
These are not equivalent.

Step4: Analyze Option d

Simplify the right expression:
$6\cdot(x+3)=6x+18$
Left: $6\cdot x+3=6x+3$
These are not equivalent.

Answer:

a. $9(6 + x)$ and $9 \cdot 6+ 9 \cdot x$