Example: $$ f+(g+h)=(f+g) $$ $$ f \times(g \times h)=(f \times g) \times h $$ (multiplication) b. $$ (m+n)+c= $$ a. $$ \begin{array}{l} (a+b)+c= \ a+(b+c) \end{array} $$ d. $$ (c \times d) \times f= $$ c. $$ (k \times z) \times d= $$ c. $$ (g \times h) \times i= $$ $$ \square $$ g. $$ (a \times c) \times d= $$ h. $$ (k \times l) \times m= $$ i. $$ (v+c)+r= $$ $$ \square $$

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UpStudy AI · Accepted Answer

Sure! It looks like you're working on demonstrating the **associative property** for both addition and multiplication. The associative property states that the way in which numbers are grouped does not change their sum or product. Here are the completed expressions based on your examples:

### a. Addition
$$
\begin{align*}
(a + b) + c &= a + (b + c)
\end{align*}
$$
**Explanation:** Whether you add $$a$$ and $$b$$ first and then add $$c$$, or add $$b$$ and $$c$$ first and then add $$a$$, the result is the same.

### b. Addition
$$
(m + n) + c = m + (n + c)
$$
**Explanation:** The grouping of $$m$$, $$n$$, and $$c$$ does not affect the sum.

### c. Multiplication
$$
(k \times z) \times d = k \times (z \times d)
$$
$$
(g \times h) \times i = g \times (h \times i)
$$
**Explanation:** Whether you multiply $$k$$ and $$z$$ first and then multiply by $$d$$, or multiply $$z$$ and $$d$$ first and then multiply by $$k$$, the product remains unchanged. The same applies to the multiplication of $$g$$, $$h$$, and $$i$$.

### d. Multiplication
$$
(c \times d) \times f = c \times (d \times f)
$$
**Explanation:** The product is the same regardless of how you group $$c$$, $$d$$, and $$f$$.

### g. Multiplication
$$
(a \times c) \times d = a \times (c \times d)
$$
**Explanation:** Grouping $$a$$ and $$c$$ first or $$c$$ and $$d$$ first yields the same product.

### h. Multiplication
$$
(k \times l) \times m = k \times (l \times m)
$$
**Explanation:** Whether you multiply $$k$$ and $$l$$ first or $$l$$ and $$m$$ first, the product remains the same.

### i. Addition
$$
(v + c) + r = v + (c + r)
$$
**Explanation:** The sum is unaffected by how you group $$v$$, $$c$$, and $$r$$.

### Summary
The **associative property** allows you to regroup numbers in addition or multiplication without changing the result. This property is fundamental in simplifying expressions and solving equations efficiently.

$$
\square
$$

If you have any more questions or need further clarification, feel free to ask!
The associative property allows you to regroup numbers in addition or multiplication without changing the result. Here are the completed expressions: a. $$ (a + b) + c = a + (b + c) $$ b. $$ (m + n) + c = m + (n + c) $$ c. $$ (k \times z) \times d = k \times (z \times d) $$ and $$ (g \times h) \times i = g \times (h \times i) $$ d. $$ (c \times d) \times f = c \times (d \times f) $$ g. $$ (a \times c) \times d = a \times (c \times d) $$ h. $$ (k \times l) \times m = k \times (l \times m) $$ i. $$ (v + c) + r = v + (c + r) $$ $$ \square $$

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