Question

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.