Question

1. use the information from above to fill in the blanks in the proof. proof: show that $(-1) \cdot (-1) = 1$. if $(-1) \cdot 0 = 0$ by the then $(-1) \cdot 1 + (-1) = 0$ by the additive inverse property by the distributive property $(-1) + (-1)(-1) = 0$ by the by the

Explanation:

Step1: Identify zero multiplication property

$\text{Zero multiplication property: } a \cdot 0 = 0$

Step2: Substitute 0 with additive inverse

$0 = 1 + (-1)$

Step3: Substitute into the equation

$(-1) \cdot [1 + (-1)] = 0$

Step4: Apply distributive property

$(-1) \cdot 1 + (-1) \cdot (-1) = 0$

Step5: Simplify $(-1) \cdot 1$

$(-1) \cdot 1 = -1$ (multiplicative identity property)

Step6: Solve for $(-1)\cdot(-1)$

Add 1 to both sides: $(-1)\cdot(-1) = 1$ (additive inverse property)

Answer:

1. First blank (right, for $(-1)\cdot0=0$): $\text{Zero Multiplication Property}$
2. Second blank (left, for additive inverse): $0 = 1 + (-1)$
3. Third blank (right, for $(-1)\cdot[1+(-1)]=0$): $\text{Substitution Property}$
4. Fourth blank (left, for distributive step): $(-1)\cdot1 + (-1)\cdot(-1) = 0$
5. Fifth blank (right, for $(-1)+(-1)(-1)=0$): $\text{Multiplicative Identity Property}$
6. Sixth blank (left, final step): $(-1)\cdot(-1) = 1$
7. Seventh blank (right, final step): $\text{Additive Inverse Property}$