Of course! Let's tackle the given expression [tex]\( \sqrt{i}(7 \sqrt{11} - 3)(7 \sqrt{11} + 3) \)[/tex] step by step.
1. Expression Simplification Using the Difference of Squares:
The expression [tex]\( (7 \sqrt{11} - 3)(7 \sqrt{11} + 3) \)[/tex] can be simplified using the difference of squares formula, which states:
[tex]\[
(a - b)(a + b) = a^2 - b^2
\][/tex]
In this case, let:
[tex]\[
a = 7 \sqrt{11} \quad \text{and} \quad b = 3
\][/tex]
Applying the formula, we get:
[tex]\[
(7 \sqrt{11} - 3)(7 \sqrt{11} + 3) = (7 \sqrt{11})^2 - 3^2
\][/tex]
2. Simplifying Each Term:
Now, let's compute each part of the expression:
[tex]\[
(7 \sqrt{11})^2 = 7^2 \times (\sqrt{11})^2 = 49 \times 11
\][/tex]
[tex]\[
3^2 = 9
\][/tex]
3. Calculating the Products:
Next, calculate [tex]\( 49 \times 11 \)[/tex]:
[tex]\[
49 \times 11 = 539
\][/tex]
4. Subtracting Terms:
Now, subtract the squared term of [tex]\( b \)[/tex]:
[tex]\[
539 - 9 = 530
\][/tex]
Thus, the simplified product of the expression [tex]\( (7 \sqrt{11} - 3)(7 \sqrt{11} + 3) \)[/tex] is 530.
5. Including the Factor [tex]\( \sqrt{i} \)[/tex]:
Now, consider the original term [tex]\( \sqrt{i} \)[/tex]:
[tex]\[
\sqrt{i} \cdot 530 = 530 \sqrt{i}
\][/tex]
Combining all the steps, the simplified expression is:
[tex]\[
530 \sqrt{i}
\][/tex]
