use the binomial expansion for (a+b)^3 to write this equation in the form a+b root 2 ---Mathematical Expression:
(5 + $\sqrt{2}$)$^3$
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Let's expand the expression (5 + square root 2) cubed using the binomial expansion formula. We identify a equals 5 and b equals square root 2, then apply the formula (a plus b) cubed equals a cubed plus 3 a squared b plus 3 a b squared plus b cubed.
Now we substitute our values into the binomial formula. We get 5 cubed plus 3 times 5 squared times square root 2, plus 3 times 5 times square root 2 squared, plus square root 2 cubed. Let's identify each of these four terms that we need to calculate.
Let's calculate each term carefully. First, 5 cubed equals 125. Second, 3 times 5 squared times square root 2 equals 3 times 25 times square root 2, which is 75 square root 2. Third, 3 times 5 times square root 2 squared equals 3 times 5 times 2, which is 30. Finally, square root 2 cubed equals square root 2 squared times square root 2, which is 2 square root 2.
Now we combine all the terms we calculated. We have 125 plus 75 square root 2 plus 30 plus 2 square root 2. We group the rational numbers together: 125 plus 30 equals 155. And we group the irrational terms: 75 square root 2 plus 2 square root 2 equals 77 square root 2. Therefore, our final answer is 155 plus 77 square root 2, where a equals 155 and b equals 77.
We have successfully expanded (5 + square root 2) cubed using the binomial expansion. Our final answer is 155 plus 77 square root 2. This is in the required form a plus b square root 2, where a equals 155 and b equals 77. The binomial expansion method allows us to systematically expand expressions with irrational terms and express them in standard form.