Final answer:
In order to use Carol's equations to show that (ab)⁻¹ = (cb)(db) and (ac)⁻¹ = (bcidc)⁻¹, we need to understand the concept of inverting an equation. We can rewrite (ab)⁻¹ as (cb)(db) and (ac)⁻¹ as (bcidc)⁻¹ using Carol's equations. Additionally, we can prove that (ab)⁻¹ (ac)⁻¹ = (cb)(db) (bc)(dc) by applying the properties of inverting an equation and multiplying two inverse equations together.
Step-by-step explanation:
In order to use Carol's equations to show that (ab)⁻¹ = (cb)(db) and (ac)⁻¹ = (bcidc)⁻¹, we need to understand the concept of inverting an equation. In algebra, when we have an equation of the form AB = C, we can invert both sides of the equation to get (AB)⁻¹ = C⁻¹. This means that (ab)⁻¹ can be written as (cb)(db) and (ac)⁻¹ can be written as (bcidc)⁻¹.
Now let's prove that (ab)⁻¹ (ac)⁻¹ = (cb)(db) (bc)(dc) using Carol's equations. We start with (ab)⁻¹ (ac)⁻¹, and using the property of inverting an equation, we can rewrite it as ((cb)(db))⁻¹ ((bc)(dc))⁻¹.
Next, we follow the rule that if we have two equations of the form (AB)⁻¹, we can multiply them together to get (AB)⁻¹ (CD)⁻¹ = (ABCD)⁻¹. Applying this rule to ((cb)(db))⁻¹ ((bc)(dc))⁻¹, we get (cb)(db)(bc)(dc)⁻¹⁻¹.
Finally, using the property that a number raised to the power of -1 is the reciprocal of that number, we can simplify (cb)(db)(bc)(dc)⁻¹⁻¹ to (cb)(db)(bc)(dc), which is equal to (cb)(db) (bc)(dc). Therefore, we have shown that (ab)⁻¹ (ac)⁻¹ = (cb)(db) (bc)(dc).