2. In each expression, what cancels, if anything? a. $$ \frac{8}{5 * 4} * 4 $$ b. $$ \frac{8}{5 * 4} * 8 $$ c. $$ \frac{8}{5 * 4} * 5 * 4 $$ d. $$ \frac{8}{5 * 4} *(4+5) $$

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1. For $$ \displaystyle \frac{8}{5 \cdot 4} \cdot 4 $$:

- Write the expression as  
     $$
     \frac{8 \cdot 4}{5 \cdot 4}.
     $$
   - The factor $$4$$ appears in both the numerator and the denominator, so it cancels:
     $$
     \frac{8 \cdot \cancel{4}}{5 \cdot \cancel{4}} = \frac{8}{5}.
     $$
   - **Cancellation:** The common factor $$4$$ cancels.

2. For $$ \displaystyle \frac{8}{5 \cdot 4} \cdot 8 $$:

- Write the expression as  
     $$
     \frac{8 \cdot 8}{5 \cdot 4} = \frac{64}{20}.
     $$
   - Although $$8$$ itself does not appear directly in the denominator, notice that one way to simplify is to factor one of the $$8$$s as $$4 \cdot 2$$:
     $$
     \frac{8 \cdot (4 \cdot 2)}{5 \cdot 4}.
     $$
   - Now, the factor $$4$$ in the numerator cancels with the $$4$$ in the denominator:
     $$
     \frac{8 \cdot 2}{5} = \frac{16}{5}.
     $$
   - **Cancellation:** A factor of $$4$$ cancels.

3. For $$ \displaystyle \frac{8}{5 \cdot 4} \cdot 5 \cdot 4 $$:

- Write the expression as  
     $$
     \frac{8 \cdot 5 \cdot 4}{5 \cdot 4}.
     $$
   - The factors $$5$$ and $$4$$ in the numerator cancel with the identical factors in the denominator:
     $$
     \frac{8\cancel{ \cdot 5 \cdot 4}}{\cancel{5 \cdot 4}} = 8.
     $$
   - **Cancellation:** Both factors $$5$$ and $$4$$ cancel.

4. For $$ \displaystyle \frac{8}{5 \cdot 4} \cdot (4+5) $$:

- Write the expression as  
     $$
     \frac{8}{5 \cdot 4} \cdot 9 = \frac{8 \cdot 9}{20} = \frac{72}{20}.
     $$
   - Here, the multiplier $$(4+5)=9$$ does not share a common factor with the $$5$$ or $$4$$ in the denominator, so nothing cancels directly before performing the multiplication.
   - **Cancellation:** Nothing cancels.

- In $$ \frac{8}{5 \cdot 4} \cdot 4 $$, the $$4$$ cancels out. - In $$ \frac{8}{5 \cdot 4} \cdot 8 $$, the $$4$$ cancels out. - In $$ \frac{8}{5 \cdot 4} \cdot 5 \cdot 4 $$, both $$5$$ and $$4$$ cancel out. - In $$ \frac{8}{5 \cdot 4} \cdot (4+5) $$, nothing cancels out.

In each expression, what cancels, if
anything?
a.
b.
c.
d.

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