Simplify (5 square root of 2+4)(5 square root of 2-4)

Solution:

Step 1: Simplify the expression using the difference of squares formula.

Step 1.1: Recognize the expression as a difference of squares.

$(5\sqrt{2}+4)(5\sqrt{2}-4)$

Step 1.2: Apply the difference of squares formula: $(a+b)(a-b)=a^2-b^2$.

$(5\sqrt{2}{)}^2-(4{)}^2$

Step 2: Calculate the squares of the terms.

Step 2.1: Square $5\sqrt{2}$.

$(5\sqrt{2}{)}^2=25\cdot 2$

Step 2.2: Square $4$.

$(4{)}^2=16$

Step 3: Subtract the squares.

Step 3.1: Perform the subtraction $25\cdot 2-16$.

$50-16$

Step 4: Simplify the result.

Step 4.1: Complete the subtraction to find the final answer.

$50-16=34$

Knowledge Notes:

The problem involves simplifying an expression that is structured as a difference of squares. The difference of squares is a mathematical identity that states for any two terms $a$ and $b$, the product of their sum and difference equals the difference between the square of the first term and the square of the second term: $(a+b)(a-b)=a^2-b^2$.

To solve the problem, one must recognize the expression as a difference of squares and apply the formula accordingly. This involves squaring each term separately and then subtracting the square of the second term from the square of the first term.

Squaring a term like $5\sqrt{2}$ involves using the property that $(\sqrt{a}{)}^2=a$. Therefore, $(5\sqrt{2}{)}^2=5^2\cdot (\sqrt{2}{)}^2=25\cdot 2$.

Squaring a whole number is straightforward: $(4{)}^2=16$.

Finally, subtracting the squares gives the simplified result of the original expression. In this case, $50-16=34$.

This problem does not involve complex numbers, and the original solution process seems to have incorrectly introduced the square root of a negative number and the imaginary unit $i$. The correct process involves real numbers only and does not require the use of complex numbers.