x(x+7i)-7i(x+7i) use the distributive property to simplify the expression explain why none of the terms in the simplified expression contain i

Answer 1

To simplify the expression \( x(x + 7i) - 7i(x + 7i) \) using the distributive property, we will first distribute each part of the expression.

1. Distributing the first term: \[ x(x + 7i) = x^2 + 7xi \]

2. Distributing the second term: \[ -7i(x + 7i) = -7ix - 7i(7i) = -7ix - 49i^2 \] Since \( i^2 = -1 \), we have: \[ -49i^2 = -49(-1) = 49 \] Therefore, the second distribution gives us: \[ -7ix + 49 \]

Now we can combine the results from both distributions: \[ x^2 + 7xi - 7ix + 49 \]

3. Combining like terms: Notice that \( 7xi - 7ix = 0 \), so we can simplify the expression to: \[ x^2 + 49 \]

Now, the final simplified expression is: \[ x^2 + 49 \]

Explanation of why none of the terms in the simplified expression contain \( i \):

In the original expression, \( -7i(7i) \) contributed a term involving \( i^2\), which we simplified to a real number \( 49 \) because \( i^2 = -1 \). All other terms involving \( i \) canceled each other out (i.e., \( 7xi \) and \( -7ix \)). Thus, after simplification, we are left with purely real terms:

• \( x^2 \) is a real number (the square of a real number).
• \( 49 \) is a real constant.

Hence, the final expression \( x^2 + 49 \) contains no imaginary part, as all terms simplify to real numbers.