Fill in the Blanks of the Following Proof by Contradiction $7 + 4 \sqrt { 2 }$

Essay

Fill in the blanks of the following proof by contradiction that $7 + 4 \sqrt { 2 }$ is an irrational number. (You may use the fact that $\sqrt { 2 }$ is irrational.)
Proof: Suppose not. That is, suppose that $7 + 4 \sqrt { 2 }$ is (i). By definition of rational, $7 + 4 \sqrt { 2 } =$ $\frac { a } { b }$ , where (ii). Multiplying both sides by $b$ gives
$$7 b + 4 b \sqrt { 2 } = a ,$$
so if we subtract $7 b$ from both sides we have
$$4 b \sqrt { 2 } = \underline { \text { (iii) } } .$$
Dividing both sides by $4 b$ gives
$$\sqrt { 2 } = \text { (iv). }$$
But then $\sqrt { 2 }$ would be a rational number because (v). This contradicts our knowledge that $\sqrt { 2 }$ is irrational. Hence (vi).