O. Prove that the following expressions are frue, by using the distribultre propeny a. \( m \times(n+p)=(m \times n)+(m \times p) \) b. \( d=(z+h)=(d, z)+(d+h) \) c. \( r \times(s+t)=(r \times s)+(r \times t) \) Use zero as the idenify element of addifion and

To prove the given expressions using the distributive property, we will apply the property that states: \[ a \times (b + c) = (a \times b) + (a \times c) \] Let's go through each expression step by step. ### a. Prove that \( m \times (n + p) = (m \times n) + (m \times p) \) 1. Start with the left side: \( m \times (n + p) \). 2. By the distributive property, we can expand this as: \[ m \times (n + p) = (m \times n) + (m \times p) \] 3. Therefore, the expression is true. ### b. Prove that \( d = (z + h) = (d, z) + (d + h) \) This expression seems to have a typographical error. It should likely be \( d \times (z + h) = (d \times z) + (d \times h) \). Assuming this correction, we proceed: 1. Start with the left side: \( d \times (z + h) \). 2. By the distributive property, we can expand this as: \[ d \times (z + h) = (d \times z) + (d \times h) \] 3. Therefore, the expression is true. ### c. Prove that \( r \times (s + t) = (r \times s) + (r \times t) \) 1. Start with the left side: \( r \times (s + t) \). 2. By the distributive property, we can expand this as: \[ r \times (s + t) = (r \times s) + (r \times t) \] 3. Therefore, the expression is true. ### Conclusion All three expressions have been proven true using the distributive property. The key steps involved applying the property correctly to expand the left-hand side of each equation into the right-hand side.